/
Text
DEGRUYTER
EXPOSITIONS
IN
MATHEMATICS
4
Klaus Doerk
Trevor Hawkes
Finite
Soluble
Groups
de Gruyter Expositions in Mathematics 4
Editors
O.H. Kegel, Albert-Ludwigs-Universitat, Freiburg
V.P. Maslov, Academy of Sciences, Moscow
W.D. Neumann, Ohio State University, Columbus
R.O. Wells, Jr., Rice University, Houston
Finite Soluble Groups
by
Klaus Doerk
Trevor Hawkes
W
DE
G
Walter de Gruyter • Berlin • New York 1992
Authors
Klaus Doerk
Fachbereich Mathematik
Universitat Mainz
Saarstr. 21
D-6500 Mainz
Germany
Trevor Hawkes
Mathematics Institute
University of Warwick
Coventry CV4 7AL
England
1991 Mathematics Subject Classification: Primary: 20-02; 20C05,20D10, 20F16, 20F17
@ Printed on acid-Ггее paper which Tails within the guidelines of the ANSI to ensure permanence and durability.
Library of Congress Cataloging-in-Publication Data
Doerk, Klaus, 1939-
Finite soluble groups/by Klaus Doerk. Trevor Hawkes.
p. cm.—(De Gruyter expositions in mathematics, ISSN
0938-6572; 4)
Includes bibliographical references and indexes.
ISBN 3-11-012892-6 (alk. paper)
1. Finite groups. 2. Solvable groups. I. Hawkes, Trevor
1936- . II. Title. III. Series.
QA177.D64 1992 92-1261
512'.2 dc20 CIP
Die Deutsche Bibliothek Cataloging-in-Publication Data
Doerk, Klaus:
Finite soluble groups/by Klaus Doerk; Trevor Hawkes.—
Berlin; New York : de Gruyter, 1992
(De Gruyter expositions in mathematics , 4)
ISBN 3-11-012892-6
NE: Hawkes, Trevor:; GT
Ci 1992 by Walter de Gruyter & Co., D-1000 Berlin 30.
All rights reserved, including those of translation into foreign languages. No part of this book
may be reproduced or transmitted in any form or by any means, electronic or mechanical,
including photocopy, recording, or any information storage and retrieval system, without
permission in writing from the publisher. Printed in Germany.
Typesetting: Asco Trade Typesetting Ltd., Hong Kong. Printing: Ratzlow Druck, Berlin.
Binding: Dieter Mikolai, Berlin. Cover design: Thomas Bonnie, Hamburg.
For our children
Thomas Naomi Steffen Duncan
Table of Contents
Preface
Notes for the reader
хш
Chapter A
Prerequisites—general group theory
1. Groups and subgroups—the rudiments 1
2. Groups and homomorphisms 5
3. Series 7
4. Direct and semidirect products 9
5. G-sets and permutation representations 17
6. Sylow subgroups 21
7. Commutators 22
8. Finite nilpotent groups 25
9. The Frattini subgroup 30
10. Soluble groups 34
11. Theorems of Gaschiitz, Schur-Zassenhaus, and Maschke 38
12. Coprime operator groups 41
13. Automorphism groups induced on chief factors 44
14. Subnormal subgroups 47
15. Primitive finite groups 52
16. Maximal subgroups of soluble groups 57
17. The transfer
18. The wreath product ®
19. Subdirect and central products 73
20. Extraspecial p-groups and their automorphism groups 77
21. Automorphisms of abelian groups 83
Chapter В
Prerequisites—representation theory
90
1. Tensor products
2. Projective and injective modules '
3. Modules and representations of K-algebras
4. The structure of a group algebra (
5. Changing the field of a representation
6. Induced modules
vjjj Table of Contents
7. Clifford's theorems 139
8. Homogeneous modules 153
9. Representations of abelian and extraspecial groups 157
10. Faithful and simple modules 172
11. Modules with special properties 182
12. Group constructions using modules 190
Chapter I
Introduction to soluble groups
1. Preparations for the p^-theorem of Bumside 204
2. The proof of Bumside’s p^11-theorem 210
3. Hall subgroups 216
4. Hall systems of a finite soluble group 220
5. System normalizers 235
6. Pronormal subgroups 241
7. Normally embedded subgroups 250
Chapter II
Classes of groups and closure operations
1. Classes of groups and closure operations 262
2. Some special classes defined by closure properties 271
Chapter III
Projectors and Schunck classes
1. A historical introduction 279
2. Schunck classes and boundaries 282
3. Projectors and covering subgroups 288
4. Examples 302
5. Locally-defined Schunck classes and other constructions 321
6. Projectors in subgroups 328
Chapter IV
The theory of formations
1. Examples and basic results 333
2. Connections between Schunck classes and formations 344
3. Local formations 356
4. The theorem of Lubeseder and the theorem of Baer 366
5. Projectors and local formations 375
6. Theorems about /-hypercentral action 386
Chapter V
Normalizers
1. Normalizers in general 394
2. Normalizers associated with a formation function 396
Table of Contents jx
3. g-normalizers
4. Connections between normalizers and projectors 40g
5. Precursive subgroups
Chapter VI
Further theory of Schunck classes
1. Strong containment and the lattice of Schunck classes 426
2. Complementation in the lattice 440
3. D-classes 453
4. Schunck classes with normally embedded projectors 461
5. Schunck classes with permutable and CAP projectors 471
Chapter VII
Further theory of formations
1. The formation generated by a single group 479
2. Supersoluble groups and chief factor rank 483
3. Primitive saturated formations 497
4. The saturation of a formation 502
5. Strong containment for saturated formations 509
6. Extreme classes 516
7. Saturated formations with the cover-avoidance property 528
Chapter VIII
Injectors and Fitting sets
1. Historical introduction 535
2. Injectors and Fitting sets 537
3. Normally embedded subgroups are injectors 548
4. Fischer sets and Fischer subgroups 554
Chapter IX
Fitting classes—examples and properties related to injectors
1. Fundamental facts 563
2. Constructions and examples 574
3. Fischer classes, normally embedded, and permutable Fating classes 600
4. Dominance and some characterizations of injectors 617
5. Dark’s construction—the theme 630
647
6. Dark’s construction—variations
Chapter X
Fitting classes—the Lockett section
1. The definition and basic properties of the Lockett section 677
2. Fitting classes and wreath products
3. Normal Fitting classes
Table of Contents
4. The Lausch group 720
5. Examples of Fitting pairs and Berger’s theorem 737
6. The Lockett conjecture 761
Chapter XI
Fitting classes—their behaviour as classes of groups
1. Fitting formations 775
2. Metanilpotent Fitting classes with additional closure properties 783
3. Further theory of metanilpotent Fitting classes 799
4. Fitting class boundaries I 806
5. Fitting class boundaries II 816
6. Frattini duals and Fitting classes 824
Appendix a. A theorem of Oates and Powell 833
Appendix p. Frattini extensions 846
Bibliography 855
List of Symbols 871
Index of Subjects 873
Index of Names 889
Preface
This is our account of the theory of finite soluble groups as it has developed during
the past 30 years. We have concentrated on those parts of the subject where a coherent
and unified body of knowledge has emerged: the theory of Schunck classes and
formations with their associated subgroups, the projectors and normalizers; the dual
theory of Fitting classes with their injectors and radicals. All this material can be
viewed as one vast and splendid generalization of the subgroups of Sylow and Hall;
indeed, to have engendered an expansion of knowledge of such cosmic proportions,
Sylow’s theorem might well be compared to the Big Bang. Historical introductions
to these themes can be found at the beginning of Chapters III and VIII. We have
made no attempt to treat what is generally known as Hall-Higman theory and its
manifold applications; only a separate monograph could hope to do justice to that
In order to make the account as accessible as possible, we have collected together
all the basic and prerequisite results from group theory (in Chapter A) and from
representation theory (in Chapter B). For many of these standard theorems, we simply
cite the relevant proofs from the following two volumes:
[H] “Endliche Gruppen I” by B. Huppert and
[HB] “Finite Groups II” by B. Huppert and N. Blackburn.
Where a result, or its proof, is not available in these reference works, we usually state
and prove it in full.
We hope that this book will serve as a basic reference in the subject area, as a text
for postgraduate teaching, and also as a source of research ideas and techniques. With
the latter uses in mind, we have given special emphasis to the construction and
analysis of examples. Many interesting results which, for reasons of space or exposi-
tion, could not be incorporated in the main narrative are cited in the exercises at the
end of each section.
We warmly thank the many students and colleagues who have read parts of the
manuscript and made helpful suggestions; while acknowledging the improvements
they have brought about, we exonerate them from any responsibility for the short-
comings that remain. We are especially grateful to Peter Forster, who read drafts of
Chapters III, IV. and VI with great care and insight; and also to Owen Brison and
Peter Hauck for their painstaking work and thoughtful comments on large parts of
the Fitting class chapters. Our thanks are also due to Frau Rita Gerlach for typing
parts of the manuscript. Finally we are pleased to express our indebtedness to our
publishers for making a prompt and unequivocal commitment to bring our long-
gestated manuscript into print.
Klaus Doerk and Trevor Hawkes
Notes for the reader
This book is about finite groups, but of course infinite groups will turn up from time
to time, either surreptitiously, as in the guise of additive or multiplicative subgroups
of infinite fields, or quite openly, as in the case of the Lausch group of a Fitting class.
The unspoken rule is that all groups are finite, except where they are obviously not!
As our title suggests, our groups are also soluble. But here we are less prescriptive
and wander quite frequently into insoluble territory to see where the soluble theory
leads. Nevertheless, for certain stretches of the narrative, we confine ourselves exclu-
sively to the universe of finite soluble groups. Much of our work is concerned with
classes of groups. When these are viewed as sets of isomorphism classes, they become
subsets of countable sets, and so, with this interpretation, we are able to talk about
sets of classes of groups without challenging the conventions of set theory.
Chapters A and В contain the basic material we need about finite groups and their
representations and are intended for reference only. In Chapters I and II we set the
scene, first giving Bender's group-theoretical proof of Burnside’s p^-theorem and
then presenting Hall’s celebrated characterization of soluble groups by the existence
of Sylow complements. The embedding properties for subgroups and the closure
operations for group classes that are introduced here play a central part in all that
follows. Chapters III VII are about the canonical conjugacy classes of subgroups
called projectors and normalizers; and also about the various classes of groups which
give rise to them, in particular. Schunck classes and saturated formations. The last
four chapters of the book may be read more or less independently. They deal with
the dual theory of injectors and with the Fitting sets and classes that engender them;
since the duality is really only a loose analogy, it is not surprising that the Fitting
classes chapters reveal a quite different pattern from the earlier ones. Appendices a
and P deal in some detail with two themes which, while an important part of the
story, do not fit comfortably into the main flow of the book and which are not easily
accessible elsewhere.
The numbering of results. Formally-stated results (theorems, propositions, lemmas,
and the like) are numbered by section and by position within the section. Thus
Proposition 6.5 on page 330 is the fifth result in Section 6 of Chapter III; within
Chapter III it is referred to simply as (6.5), but elsewhere it is cited as 111,6.5.
Equations, inequalities, and other displayed statements are sometimes also labelled
for reference purposes. Their labels have the form of the section number followed by
a lower-case Greek letter. Thus (6./?) on page 330 is the second labelled display in
Section 6 of Chapter III. Since these items are rarely cited outside the confines of
their chapter, no chapter annotation is attached to them.
Chapter A
Prerequisites—general group theory
1. Groups and subgroups—the rudiments
A group is a non-empty set G together with an associative binary operation (which
we will usually denote multiplicatively by juxtaposition) with the property that G has
an identity element 1 = 1G satisfying
1g = gl = g for all g e G,
and such that each element g of G has a two-sided inverse g~l satisfying
= SS-1 = 1-
If hg = gh for all g, h e G, we call G abelian. The order of G is the cardinality of the
underlying set and is written |G|. If |G| is finite, we say that G is finite.
(1.1) Examples, (a) If fl is a non-empty set, the symmetric group on fl. denoted by
Sym(fl), is the set of all permutations of fl with “composition of maps” as the
associative binary operation. If fl = {1, 2,..., n}, then we write Sym(n) instead of
Sym(fl). It is well known that |Sym(n)| = nl.
(b) Let К be a (commutative) field, and let V = V(n, K) be a vector space of
dimension n over K. The set of all automorphisms of V (viz. non-singular linear
transformations from V to T), again with composition as the binary operation, forms
a group; it is called the general linear group of degree n over К and is denoted by
GL(n, K). If p is a prime and К - the finite field with pf elements, we will write
GL(n, pf) instead of GL(n, K). Clearly |GL(n, pz)| is the number of ordered bases of
K(n, Fp/), and therefore
|GL(n, pz)| = (p"f - l)(pn/ - p')...(p"' - p,n’nz).
A subset V of G is called a subgroup if it is a group in its own right with respect to
the binary operation defined on G; for this we write U < G and V < G when U # G.
If U < G, we call V a proper subgroup of G. We will use the symbol 1 to denote the
identity subgroup {1} of a group.
If U is a proper subgroup with the property that U = V whenever U < T < G, we
call V a maximal subgroup of G and write U < G; thus maximal subgroups are
precisely the maximal elements of the set of proper subgroups partially ordered by
inclusion.
2
A. Prerequisites—general group theory
If X £ G, we denote by <X> the subgroup generated by X; thus
<X> = n{G:X£ U<G}.
We call X a set of generators for <X>, and call a group cyclic if it is generated by a
single element.
If U and V are subgroups of G, we call the subgroup <U и F> their join and usually
write it <G, F). More generally, <GA: 2 e A) will denote the join of an arbitrary set
{17л}ЛбЛ of subgroups. We will use //(G) to denote the set of all subgroups of G. With
respect to the operations of set-theoretical intersection n and join < , > the set .T(G)
forms a lattice, the so-called subgroup lattice of G.
If X and Y are subsets of a group G, we define their Frobenius product XY by
XY = {xy: x 6 X, у 6 У}.
From the associativity of the group binary operation it is obvious that (X Y)Z =
X(YZ) for all subsets X, Y and Z of G. Let U, V 6 .'/’(G). If UV = YU, we say that
U and V are permutable and write U XV. Evidently U ± V if and only if < U, Vy = U V,
and so, in particular, the condition U J_ V is equivalent to U V < G.
The following lemma, attributed to J. Tits, will prove to be very useful.
(1.2)
Lemma.
Let
U, V, and
W be subgroups
of a group
G.
Then the following
statements are equivalent:
(a) U n VW = (U n F)(G n W);
(b) UVnUW= U(Vn W).
Proof. (a)=>(b): Clearly U(Vn W) s UVn UW. To prove the reverse inclusion
let uv = utwe UV n UW with u, ut eV, ve V, and we W. Then u~iui = aw-1 e
U n VW = (U r> V)(U n W), and so there exist elements vt e U n V and w, e U n W
such that aw-1 = tqtv,. It follows that tf‘v = e V nW and uv = (unjb^'a) e
U(V n W). Hence UV n UW £ l/(fn W), and equality holds.
(b) => (a): The inclusion (U n V)(U nW) £ U n VW is obvious. Let и = vw e
U n VW with и e U, v e V, and w e W. It will suffice to show that и e (U n V)(U n W).
Now v = uw~* e UV nUW = U(V n W) by hypothesis. Therefore there exist
elements iq e U and vt e V nW such that v = iq vt, and hence ut = vvj* e U n V. It
follows that и = iw = tq(iq w) e (U n V)(U n W) since a, w = uj'u e U nW. □
Condition (b) of Lemma 1.2 is obviously satisfied when V < U, and we can deduce
the well-known Dedekind identity (the modular law).
(1.3) Let U, V and W he subgroups of a group G with V<U. Then Un VW =
V(U n W).
If U < G and g e G, we write Ug instead of U {g} and call Ug a right coset of U in
G. Left cosets are analogously defined. Calling two elements g and h of G equivalent
if and only if hg 1 e U, we obtain an equivalence relation on G whose equivalence
1. Groups and subgroups—the rudiments
3
classes are exactly the right cosets of U in G. (The set of right cosets of U in G will
be denoted by G/G.) It follows that these right cosets form a partition of G and, in
particular, that there exists a subset T of G such that
G = \J Vt, and Usr^Vt = 0
teT
whenever s, t c T and s # t. This partition is called the right coset decomposition of
G by U. A set T of the kind just described is called a right transversal of U in G; it is
a set containing exactly one element from each right coset and so the number of right
transversals is PJ, e T | Ut |.
There is a corresponding partition of G into the left cosets of G, and a complete
set of left coset representatives is called a left transversal of G in G. The map
Ug -»g l G is a bijection from the set G/G of right cosets to the set of left cosets of
G in G; the common cardinality of these two sets is called the index of G in G and is
written |G: G|.
If g 6 G, the map и -» ug is a bijection from G onto the coset Ug. Thus all right
cosets have the cardinal |G|, and we obtain the following celebrated theorem of
Lagrange.
(1.4) Lagrange’s Theorem. If V is a subgroup of a group G, then |G| = |G: G| |G|, In
particular, if G is finite, | G| is a divisor of | G|.
If G, V < G, the set UV is obviously a union of right cosets of G; moreover, for
F,
Uv, = Uv^t^vf1 eUn Vo(V n Hi’i = (G n F)t>2,
and so GF contains | F: (G n F)| right cosets of G. Since I F| = |F:(Un F)||Un F|,
we obtain the following product formula.
(1.5) Lemma. Let G and V be subgroups of a group G. Then | G V11G n V| - | G11V|;
in particular, |GF: F| = |G: Gr> F|.
The following elementary result will be used often in the sequel.
(1.6) Lemma. Let V, V and W he subgroups of a finite group G.
(a) If W±G and И' 1 F, then W _L<G, F>.
(b) If(\G: Gl |G: Fl) = 1. then G = GF and |G: (Gn F)| = |G: G||G: F|.
(c) If (| G : G|’, |G: F|) = 1, and if W _L G and W IF, then W = (Wn U)(Wn F)
and W(U r> V) = IFG n IFF; in particular H’UGn F).
Proof, (a) Since G is finite, there exists an n 6 N such that <G, F> - (GF)". Since
IVG = GIF and IFF 1И', clearly H'<G, F> = <G, F> IF
(b) Let D = G n F. By (1.4) the coprime numbers |G: G| and |G: F| both divi e
IG-il and therefore |G.D| = m|G: G| |G: F| for some m e IU By (1.5) we then have
4
A. Prerequisites—general group theory
IVIIH
m|G| = |D| =ltrl-161'
Hence m = 1, G = UV and |G: D| = |G: U| |G : F|.
(c) Since WU is a subgroup of G, and since \W: И7 r> U| = ]WU : U|, the index
|H/:H,r>U| divides |G:U|. Similarly |И': H'rUz| divides |G: K|, and hence
W = (И'п U)(K'n К) by Part(b). But G = UK, and so
HznUK = (Hzr1U)(IVr1 K).
An application of (1.2) then yields W(U r> K) = WU r> WV.
□
If U and V are subgroups of G and if UK = G, we call К a supplement to U in G;
if further U r> К = 1, then V is said to be a complement to U in G.
If W < К the symbol V/W will denote the set of right cosets of W in К (When
W < К this set inherits the structure of a group and then К/ W will denote the quotient
group.) Clearly W £ W(U n К) £ V. If W(U r> К) = К we shall say that V/W is
covered by U, and if W(U n V) = W, we shall say that V/W is avoided by U.
(1.7) Lemma. Let V be a subgroup of a finite group G.
(a) Let W < V < G. Then V/W is covered by U if and only if |U r> К : U r> Wj =
| К: W\, and V/W is avoided by U if and only if |U о V: U r> Wj = 1.
(b) Let 1 = Vo < Kj < ••• < К = G be a chain of subgroups of G with the property
that Vj/Vj^i is covered by U for jeJ and avoided by U for j e {1,..., r}\J. Then
1^1 = ГЪе4^:»5-11-
Proof (a) This follows at once from the fact that
|H'(Un K): W\ = |Ur> V: Ury W\ by (1.5).
Part (b) follows directly from (a).
□
If g is a element of a finite group G, there exists a smallest natural number n such
that g" = 1; this is called the order of g and written o(g). It is easy to see that
o(g) = I <<7> I. the order of the cyclic subgroup generated by g, and therefore o(g)
divides |G|. The least common multiple of the integers {o(p): g e G] is called the
exponent of G and written Exp(G).
If it is a set of primes and if every prime divisor of o(g) belongs to it. we call g a
к-element. If every element of a group G is a л-element, we call G a n-group. The set
of distinct primes dividing |G| is denoted by <r(G), and it follows easily from Sylow’s
theorem that G is a л-group if and only if <r(G) £ it. The complementary set of primes,
P\ n, will be denoted by rf. If it = {p}, a singleton, we talk of p-elements and p-groups
and write p' instead of {p}'.
In the sequel we shall need the following simplified version of the Schreier subgroup
theorem.
2. Groups and homomorphisms 5
(1,8) Theorem ([H] I, 19.10). Let G = s„>, and let v he a subgroup of G
finite index. Then U has a generating set with 2n|G: l/| elements.
In Chapter IX, Section 5 we will also need the following elementary result.
(1.9) Lemma. Let At and Bt he subgroups of a group G such that is a subgroup
fori= 1,... n. Assume that A, < Bi for all I < i * j < n. Then
П(Л,ВЛ = ^Л2-. лД QB,).
Proof. The conclusion holds trivially when n = 1. Let n > 1 and set X = A2...a„
and Y = B2 r> • • n B„. By induction on n we can suppose that A2B2 n • • • n Л„В„ =
X Y. Since by hypothesis A, < Y and X < Bt, twice applying the Dedekind Law we
obtain Л1В1п-пЛ„В„ = Л1В1пА'У= Л1(В1пХУ) = Л1А'(В,пУ), and the
desired conclusion follows. rn
2. Groups and homomorphisms
Let G and H be groups. A map a: G -»H is called a homomorphism if
«(xy) = a(x)a(y)
for all x, у e G. (We will not observe any hard and fast rules about writing maps on
the left or on the right, but simply choose whichever side seems more appropriate in
a given context.) As usual, an injective (respectively surjective, bijective) homo-
morphism is called a monomorphism (respectively epimorphism, isomorphism).
Occasionally, the notation a: G i-» H will signify a monomorphism and a: G -* H an
epimorphism. If there exists an isomorphism a: G -> H, we say that G is isomorphic
with H (or that G and H have the same isomorphism type) and write G s H. A
homomorphism a from G to itself is called an endomorphism and when a is bijective,
an automorphism of G. The set of all automorphisms of G forms a group under the
binary operation “composition of maps”; this is called the automorphism group of G,
is denoted by Aut(G), and is obviously a subgroup of Sym(G).
If g, h 6 G, we set h9 = g lhg, and if X is a non-empty subset of G, we define Xя to
be the set {x’:xe X}. Furthermore we use XG to denote the set
Xc = {X9:geG}
of conjugates of X in G. The map p9: G —» G defined by
pg(h) = h9
for all h e G is easily seen to be an automorphism of G; it is called the inner
6
A. Prerequisites—general group theory
automorphism induced by g. An automorphism a of G is called inner if a = pg for some
g e G, and the set of all inner automorphisms is denoted by Inn(G). Evidently a group
is abelian if and only if Inn(G) = 1.
A subgroup N of G which is invariant under all inner automorphisms (for which
therefore № = N for all g e G) is called a normal subgroup of G. If N is a normal
subgroup of G, we denote this symbolically by hl < G (and by N <G when N G).
Clearly 1 and G are always normal subgroups of G, and a group G ± 1 with no other
normal subgroups is called simple. Thus G is simple if and only if it has precisely two
normal subgroups.
If a: G -» H is a homomorphism, its kernel is defined to be
Ker(a)= {geG:alg)= !„}.
If keKer(a), then a(g~lkg) = a(g~1)a(k)a(g) = a(g 1)a(g) = a(g'g)^a(l)=l, and
therefore Ker(a) < G. If g e G and a e Aut(G), then
(a-1p9a)(/i) = a-1(g-1a(h)g) = (a-1 (s))-1 ha-1(g)
for all /1 e G, and so alpga is the inner automorphism of G induced by the element
a-1(g). It follows that a-1 Inn(G)a = Inn(G) for all a 6 Aut(G) and therefore that
Inn(G) < Aut(G).
A subgroup hl of G is a normal subgroup if and only if Ng = ghl for all g e G
(which happens if and only if the left and right coset partitions of N in G coincide).
Thus, with the help of the associative law, we see that
' (Ng)(Nh) = N(gN)h = N(Ng)h = N(gh);
in other words, the set of right cosets is closed under the binary operation “subset
multiplication”. This binary operation inherits associativity from G, the coset hl
behaves like an identity, and g~'N is the inverse of ghl. Therefore, when N < G, the
set G/N of cosets of N in G forms a group. This group is called the quotient group (or
factor group) of G by N, and is also denoted by G/N; clearly | G/N | = | G: TV |. The
map v = vN: G -> G/N defined by v(g) = Ng is clearly an epimorphism, called the
natural epimorphism from G to G/N. Since Ker(vw) = N, the normal subgroups of G
are precisely the kernels of homomorphisms of G.
Let fl be a set. A group G is called an Q-group if there is associated with each
element to e fl an endomorphism of G denoted by
g->go>
for all g 6 G. A subgroup V of G is called Q-admissible if ua> e U for all и e U and
<» e fl. Evidently the intersection and the join of fl-admissible subgroups are again
fl-admissible. If N is an fl-admissible normal subgroup of G, the quotient group G/N
may be regarded naturally as an fl-group via the action defined thus
(Ng)w = N(ga>)
3. Series
7
for all g e G and w e fl. Finally if G and H are П-groups, a homomorphism a: G -»H
is called an Sl-homomorphism if
a(9<u) = a(p)oj
for all g eG and or e fl. Such concepts as ti-nionomorphism, Sl-automorphism are
defined in the same way. If fl = 0, every group is an figroup, every subgroup is
fl-admissible, and every homomorphism is an fl-homomorphism.
(2.1) The Isomorphism Theorems. Let Si be a set, and let G and H be Sl-groups
(a)([H] 1,3.8) I/cc.G-H is an Sl-homomorphism, then К = Ker(a) and Irn(a) =
{Л 6 H: 11 = a(p) for some g eG] are Sl-admissible subgroups of G and H respectively,
and the map у defined by
y(Kg) = a(p)
is an Sl-isomorphism from G/K onto Im(a).
(b) ([H] I, 3.12) If U and N are Sl-admissible subgroups of G and U normalizes N,
then VN/N = U/(U n N) as Sl-groups.
(с) ([H] I, 3.10) If M and N are Sl-admissible normal subgroups of G and N < M,
then the Sl-groups (G/N)/(M/N) and G/M are Sl-isomorphic.
For any group G the subset
Z(G) = {g e G: gx = xg for all x e G}
is called the centre of G. Obviously Z(G) is a normal subgroup of G, and the
map g -»pe is an epimorphism from G onto Inn(G) with kernel Z(G). Thus
G/Z(G) s Inn(G) by (2.1)(a).
3. Series
A subgroup U of a group G is said to be subnormal in G if there exists a chain of
subgroups 17O, If,..., Vr of G such that
U = Uo< Ut < < 4r i < Vr = G.
This is called a subnormal chain from V to G. If U is subnormal in G, we shall write
U An П-group is called Sl-simple if 1 and G 1) are the only fl-admissible nor-
mal subgroups of G. By the term Si-series we shall understand a subnormal chain Vo,
V ... U from V to G all of whose terms are fl-admissible. We shall call it an
Sl-composition series if each of its factors (i = 1,..., r) is fl simple, in whic
case the factors are called Sl-composition factors of G.
8
A. Prerequisites—general group theory
Since the results of this section will be applied to modules (including finite
dimensional vector spaces over fields), we need finiteness conditions which will ensure
the existence of fl-composition series. A group G is said to satisfy the maximal
(respectively minimal) condition for Q-subgroups if every non-empty set of fl-subgroups
has a maximal (respectively minimal) element.
(3.1) Theorem ([H] 1,11.2). If G satisfies both the maximal and minimal conditions for
Sl-subgroups, then G has a composition series.
The following celebrated theorem of Jordan-Holder, which we will sharpen further
in (9.13), is of central importance in the sequel.
(3.2) The Jordan-Holder Theorem ([H] I, 11.5). Let G be an Q-group, and let
1 = Vo <3 G, <3 • • • <3 Ur = G and
1 = |/0 <3 <3 • • <з Ц = G
be two Ll-composition series of G. Then r = s, and there exists a permutation n e Sym(r)
such that for i = 1,..., r the factor VJV^ is Sl-isomorphic with .
We now consider the most important special cases of this theorem.
(3.3) Take fl = 0. Then each group is an fl-group, and “fl-simple” means the same
as simple. The fl-composition series and factors are just called composition series and
composition factors. Thus a composition series is a chain of the form
1 = Uo <3 G, <3 <3 Vr = G
where each factor Vi/Vi^l is simple (i = 1,..., r).
(3.4) Take fl = Inn(G). Then the fl-admissible subgroups are just the normal sub-
groups of G, and the fl-simple, fl-admissible subgroups are called minimal normal
subgroups. (We shall use the notation “N < G” to mean that N is a minimal normal
subgroup of G.) The fl-composition series and factors are called chief series and chief
factors, and so a chief series of G is a chain
1 = Go< G, < • • •< Gr = G
of subgroups G, < G such that VJU^i is a minimal normal subgroup of G/G,_, for
i — 1,2,..., r. A chief factor H/K of G is called central if H/K < Z(G/K) and eccentric
otherwise. If the chief factor H/K of G has finite order and if the prime p divides this
order, the chief factor H/K is called a p-chief factor of G. An fl-endomorphism will
be called a G-endomorphism when fl = Inn(G).
(3.5) Let fl = Aut(G). The fl-admissible subgroups are called characteristic subgroups
of G. (We shall write “G char G” to denote the fact that G is a characteristic subgroup
4. Direct and semidirect products
9
of G.) If G < H, then H induces automorphisms on G by conjugation, and from the
assertion: V char G we can conclude that U < H. An fl-composition series is called
a characteristic series of G in this case, and if G is fl-simple, we say that G is
characteristic simple.
(3.6) Let M be an additively-written (not necessarily finite) abelian group, and let fl
be a ring with a 1 (multiplicative identity). (Our rings are assumed to be associative
and distributive.) We call M a right Ll-module if the following axioms are satisfied:
RM1: M is an fl-group, i.e. (m, + m2)w = + т2<л for all mb m2 e M and
well;
RM2: m(to, + w2) = mai1 + ma>2 for all m e M and , ш2 e fl;
RM3: m(oi|O)2) = (m«>,)«>2 for all m e M and и,, a>2 e fl;
RM4: ml = m for all me M.
Clearly an fl-admissible subgroup of a right fl-module is an fl-module (called a
submodule), and if N is a submodule of an fl-module M, the quotient fl-group M/N
is also an fl-module (called a quotient module). Thus the isomorphism theorems (2.1)
and the Jordan-Holder theorem (3.2) are valid for right fl-modules.
We end this section by stating formally the useful elementary fact that the order
of a subgroup is the product of the orders of its projections onto the factors of a
subnormal chain; it can be easily proved by induction on the length of the chain.
(3.7) Lemma. Let 1 = Uo < L’t <! • <! Ur = G be a subnormal chain, and let H be a
subgroup of G. Then
i=1
4. Direct and semidirect products
All the results about fl-groups in this section are also valid for fl-modules when
expressed in the appropriate additive notation (see (3.6)).
(4.1) Definition (The internal [restricted] direct product). Let {G,}ie/ be a set of
subgroups of a group G. Then G is said to be the direct product of the subgroups G,
(i e I) if the following three conditions are fulfilled:
DPI: G; < G for all i e I;
DP2: G = <G,:i 6/>;
DP3' G n j e /, j i) ~ 1 f°r all i £
It is straightforward to’ verify that these three conditions together are equtvalent to
DPI together with the following condition:
DP4: Each g e G can be written uniquely (up to the order of the factors) as a
product
9 = П » ’
id
10
A, Prerequisites—general group theory
where g, e G and gt = 1 for all but finitely many values of i. In this case we write
G = X, E, G,, unless the group is written additively (as with Q-modules, for example),
whereupon we write G = G, instead and call G the direct sum of its subgroups
G, (i e /).
The internal direct sum is a description of the structure of a given group in terms
of certain subgroups. The external direct sum shows how to construct a group with
such a structure out of a set of given abstract groups.
(4.2) Definition (The external restricted direct product). Let {GJ, e, be a set of groups,
and let G be the set of all maps f : I -» Gf satisfying
(a) f(i) e G, for all i e I, and
(b) f(i) # 1 for only finitely many i e I.
The set G becomes a group under the operation of pointwise multiplication defined
by
(fg)(i) = f(i)g(i)
for all i e I. We call G the external restricted direct product of the groups G, (i e I).
If each Gj (i e I) is an П-group, the group G can be given the structure of an П-group
by setting
(/<a)(i) = (f(i))co
for/e G,сое П, andi e I. Iff is finite, say I = {1,2,..., n}, then each f e G is uniquely
determined by the n-tuple (/(1),..., /(n)). In this case the group G may be considered
as the group whose underlying set is
{(0i, 02,--,0л): g, e GJ
with componentwise multiplication as its binary operation.
The i th coordinate subgroup of G is
Gj* = {/ e G: f(j) = 1 for j # i}.
It is clear that G* is a subgroup of G isomorphic with G, and that G is the internal
restricted direct product of its subgroups Gj* (i e I). Since it is usually clear from the
context whether a direct product is “internal” or “external”, and since to within
isomorphism they are the same, we will not make the distinction in the sequel; in
particular, we will use the same notation: X> e i Gf for both (and G, G, for the direct
sum when the notation is additive).
(4.3) Definition. If we omit the requirement ‘/(i) # 1 for only finitely many i e Г
from Definition 4.2, the set
{/: / is a map from I to (J Gf with f(i) e GJ
iel
is also a group under pointwise multiplication. It is called the unrestricted direct
product, or the Cartesian product of the groups Gf (i e I).
4. Direct and semidirect products
tl
(4.4) Lemma. Let G be an Q-group with the maximal condition on Q-subgroups and
assume that lnn(G) < Q. If G is generated by a set {GJ,., of Q-simple Q-subgroups
Gh then there exists a finite subset J of I such that G = ){ G
Proof. The set ,t! of all fl-subgroups of G which are the direct product of a finite
subset of the subgroups G, is certainly non-empty; therefore by hypothesis ..tt has a
maximal element, M say. If M # G, there exists an i e I such that G, f, M. Since
Inn(G) £ fl and G, is fl-simple, we have M n G, = 1 and M, G, < MG,. Therefore
MG, = M x G, e .//, contrary to the choice of M. Hence M = G. n
(4.5) Definition. An fl-group G is called Q-semisimple if G is a direct product of
finitely many fl-simple fl-subgroups. (For fl-modules the terms irreducible and
completely reducible are sometimes used in place of simple and semisimple.)
(4.6) Lemma. Let G be an Q-group with the maximal condition on Q-subgroups, and
assume that Inn(G) £ fl. Then any two of the following statements are equivalent :
(a) G is generated by Q-simple Q-subgroups;
(b) G is Q-semisimple;
(c) If U is an Q-subgroup of G, then G has an Q-subgroup V such that G = U x V.
Proof. The equivalence of (a) and (b) follows from (4.4).
(b) => (c); Assume that (b) holds, let
J{ = {X : X is an fl-subgroup of G such that U c, X = 1},
and let V be a maximal element of it. If UV < G, then there exists an fl-simple
fl-subgroup Y of G with Y f, UV, and it follows that Fn UV = 1. We claim that
U ci YV = 1. Let и = yv with yeY and v e V. Then у = utT1 e Yc UV = 1, and so
v = ue U eV = 1. Thus U n YV = 1, and YV e Л, contrary to the choice of V.
Hence G = UV = U x V, and Statement (c) follows.
(c) => (a): Assume (c) holds, and let U be a proper fl-subgroup of G. If M is a maximal
fl-subgroup of G containing U, then G/M is fl-simple. By assumption there exists an
fl-subgroup TV such that G = M x N, and therefore N is fl-simple. Take for U a
maximal element in the set of subgroups which are generated by fl-simple fl-
subgroups. If U # G, we obtain an fl-simple fl-subgroup N not contained in U. This
contradiction shows that U = G and hence that Statement (a) follows from (c). □
(4.7) Definitions, (a) An fl-group is called directly Q-indecomposable if G # 1 and if
G has no direct decomposition G = Gj x G2 with G, and G2 both non-trivial fl-
subgroups. When fl = Inn(G), we call G simply directly indecomposable.
(b) An fl-endomorphism a of an fl-group G is called normal if a commutes with
all inner automorphisms of G.
(4.8) Schur’s Lemma ([H] I, 10.5). Let G be an Q-simple Q-group anda:G -rGa
normal endomorphism. Ifa*0(<>. IIm(a)| > D, then a is an automorphism of G, and
a 1 is also an Q.-endomorphism of G.
12
A. Prerequisites—general group theory
(4.9) The Krull-Remak-Schmidt Theorem ([H] I, 12.2 and 1, 12.3). Let G be an
non-identity Q-group satisfying the maximal and minimal conditions for Q-subgroups.
Then G admits a direct decomposition
G = G, x • x G„
into directly Q-indecomposable Q-subgroups Gj. If G = Ht x • • • x Hmis another such
decomposition, then m = n, and the subgroups Я, can be numbered so that
G = Gt x • • x Gj.j x H] x • •• x H„
for allje{l,...,n}. Furthermore, there exists a normal Q-automorphism a of G such
that a(G,) = for i = 1....n.
The following theorem (also due to Krull, Remak and Schmidt) gives a criterion
for the indecomposable factors to be unique, up to the order of the factors. The
criterion states that there should be no non-trivial homomorphism from G,- to Z(Gj)
for each distinct pair of indecomposable factors Gf and Gy; it can be reformulated as
the condition (|G;: GJ, |Z(Gj)|) = 1, where G- denotes the derived subgroup of G;
defined in (7.1 )(b).
(4.10) Theorem ([H] I, 12.6). Let G be a finite group. The direct decomposition
G = G, x • • • x G, into directly indecomposable factors G, is unique (up to the order of
the factors) if and only if (| G;: GJ, |Z(Gj)|) = 1 for all 1 < i j < r.
(4.11) Lemma. Let G,,..., G„ he groups.
(a) Z(Gj x x G„) = Z(Gj) x x Z(G„).
(b) If N < G, x G2 and Nr-G, = 1, then N < Z(Gt) x G2.
Proof, (a) This follows from the definitions by easy calculation.
(b) Since fN, G,] < N n G, = 1, we have N < CCiX<li(Gt) = Z(GJ x G2. □
(4.12) Definition. Let я be a set of primes. The n-socle of a group G, denoted by
Soc„(G), is the subgroup generated by the identity and its minimal normal я-subgroups.
When я = P, we call it simply the socle of G and write Soc(G). Thus
Soc(G) = <N:N-<G>.
By (4.4) the socle of a group is a direct product of a subset of tts minimal normal
subgroups.
(4.13) Proposition, (a) A characteristic-simple finite group is a direct product of sub-
groups, each isomorphic with a fixed simple group G. (We call such a group a direct
power of G.)
(b) Let G = Gj x • x G, with each G, a non-abelian simple group. Then a subgroup
S is subnormal in G if and only if it is a (direct) product of a subset of the factors Gj.
4. Direct and semidirect products
13
(c) Let N be a minimal normal subgroup of a finite group G, and let N < M < G.
Then N is a direct product of minimal normal subgroups of M; in particular, Soc(G) <
Soc(M).
Proof, (a) Seel, 9.12(a) of [Н].
(b) Let S < S, < • < S,_] <S, = G. Arguing by induction on t. we may suppose
that S < G, and without loss of generality we may suppose that the direct components
are so numbered that Gj c, S # 1 for i = 1,.и and G.r, S = 1 for i = и + 1,r.
Since Gj n S < G, and G, is simple, we have Gj x • • x G„ < S. Now set T =
S n (Gu+1 x x G„), and note that S = S r\G — (G, x x GU)T by the modular
law. Since T < Gu+l x * x Gr and Tr^Gj — 1 for j = и 4- 1, ..., r, it follows that
T < Z(G„+1 x x Gr) = Z(Go+1) x ••• x Z(G,)by(4.11).SinceZ(Gj) = 1 foralliby
hypothesis, we conclude that T = 1, and hence that S = Gt x • • x G„.
(c) Let К be a minimal normal subgroup of M contained in N. By the minimality
of /V we have N = <№: g e G), and therefore, applying (4.4) to M with О = Inn(M),
we see that N is a direct product of suitable G-conjugates of K, each of which is a
minimal normal subgroup of M. □
The following application of Theorems 4.10 and 4.13(b) provides useful informa-
tion about the socle of a finite group.
(4.14) Lemma. Let M be a normal subgroup of a finite group G. If M = Gj x • • • x Gr
is a direct product of non-abelian simple subgroups G,,..., Gr, then M is a product of
minimal normal subgroups of G; in particular, M < Soc(G).
Proof. For g e G we have M = Me = Gf x x Gf, and therefore by (4.10) there
exists n e Sym(r) that Gf = G„(i, for i = 1, ..., r. It follows that for 1 < i < r, the
subgroup <G,?> is the direct product of the subgroups in the G-orbit G®, and from
4.13(b) we conclude that <G,G> is a minimal normal subgroup of G. If {G/ j e J} is a
set of representatives of these G-orbits, we have M = Xje j (Gf). □
(4.15) Definitions. Let Gt,..., Gr be groups, and let D = Gt x • • • x Gr be their direct
product.
(a) The map nt: D -» G, defined by
79(01,02.....9r) = 0i
is called the projection of D onto the ith component.
(b) A subgroup U of D is called subdirect if njl!} = G, for each i = 1,.... r.
The following observation is a direct consequence of the definition of л,.
(4.16) The ith projection Tr.-iG, x - x Gr-► G, is an epimorphism with kernel
G, x x G,., x 1 x Gi+1 x x Gr.
The next elementary result will be frequently used in the sequel.
14 A. Prerequisites—general group theory
(4.17) Lemma. Let Nt, Nr be normal subgroups of a group G. Then the map
p: G -> (G/N,) x x (G/Nr) defined by
iAg) = (gN,.....gbir)
is a homomorphism; its kernel is 1\\ and its image /1(G) is subdireet. In particular,
G/(fj'=1 Nf) is isomorphic with a subdirect subgroup of (G/N,) x • • • x (G/Nr).
Proof. From the definitions of a quotient group and of a direct product we have
i‘(gg') = (gg'b\, ...,gg'Nr)
= (gL\.....gNr)(g'Ni.......g'N,)
= i‘(g)p(g').
and so /1 is a homomorphism. Moreover, p(g) = (N„..., Nr), the identity of
G/N] x x G/N,, if and only if g e N, for all i = 1, ..., r, and therefore Ker(p) =
Ai=i Nf. Lastly, the projection л, of p(G) is obviously (i/N,: g e G} = G/Nh whence
/i(G) is subdirect, and then the final assertion is clear by the isomorphism theorem.
□
Next we state the well-known structure theorem for finite abelian groups.
(4.18) Theorem ([H] I, 13.12). Let A be a finite abelian group. Then
A = <X]> x x <_x„>
with 1 # о(.к;) = p“‘ for suitable (not necessarily distinct) primes Pi,...,p„. The number
n and the set of prime powers pf are uniquely determined by A. By suitable numbering
it can be arranged that pt < p2 < " < p„ and that the exponents a, of each fixed prime
are in non-decreasing order. The n-tuple (p“',..., p“") so obtained is called the type of
A. There is exactly one isomorphism class of abelian groups of each type.
(4.19) Definition. A finite abelian p-group A of type (pm, pm) is called homo-
cyclic of exponent pm, order p™ and rank n. If m = 1, then A is called an elementary
abelian p-group of rank n.
Thus an elementary abelian group A of order p" has a direct decomposition
Л = <x,> x • • x <_x„>,
with o(.Xj) = p. The map
(.x?,...,x:")->(ai + pZ....a„ + pZ)
from A onto the additive group of the vector space F(n, p) of dimension n over
4 Direct and semidirect products
15
- Z/pZ is an isomorphism. Since, over a finite prime field, linearity and additivity
have the same meaning, it follows that Aut(4) S GL(n, p). We shall make frequent
use of this dichotomy of viewpoints.
Our next observation follows at once from (4.13)(a).
(4.20) Lemma. A chief factor (in particular, a minimal normal subgroup) of a group G
is either an elementary abelian p-group for some prime p or the direct power of a
non-abelian simple group.
We end this section with a short description of the semidrect product (internal and
external). In many situations where П-groups arise, three additional properties hold:
(1) 12 it itself a group;
(2) For each co e Q, the map <rw: g -> geo is an automorphism of the Q-group G;
(3) The map co -> is a homomorphism from П into Aut(G).
In this situation we conventionally use a Roman letter, such as H, instead of Q
Given two groups G and H, we often want to describe an “action” of H on G which
fulfils these three conditions. The obvious approach is simply to specify a homo-
morphism a : H -> Aut(G) and to define gh (written, more usually, gh in this case) to
be the image of g under the automorphism ah. More often though, it is easier in
practice to work from the other direction, defining first the element gh in G and then
checking the following facts:
(i) The map ah: g -> gh is an automorphism of G;
(ii) g№ = (g>fk for all g e G, h, к e H.
It follows from (ii) that o: h -»oh is a homomorphism from H into Aut(G). We
formalize these observations in the following definition.
(4.21) Definition. Let G and H be groups, and suppose that for each g e G and he H
an element gh e G is defined such that
(4.a) the map g -> gh is an automorphism of G, and
(4/1)
= (0v
for all g e G and h,keH. Then we say that H is a group of operators for G (acting
by automorphisms) or, more succinctly, simply that G is an H-group. If a homo-
morphism a: H -► Aut(G) is specified and the Я-acticn on G then defined by g =
(g)<rh, we will say that H is a group of operators for G via a .
Whenever G is an Я-group in this sense, it is possible to construct a group X which
contains subgroups G and H (actually isomorphic copies of G and H) with G Л -
GH and G n H = 1 such that the conjugate /1 lgh is the image g of g under the
prescribed Я-action. This construction is called the semidirect product; it has an
“internal” and an “external” version, which we formalize m the following definitions.
(4 22) Definition (The semidirect product), (a) Let X be a group with subgroups G
Lnd Я such that G < GH = X and G n Я = 1. Then we say that X >s the (tntemal)
16
A. Prerequisites—general group theory
semidirect product of G with H. (With the Я-action: g* = h 'gh, it is clear that H is
a group of operators for G.)
(b) Let H be a group of operators for G, and define a binary operation on the
Cartesian product X = G x H by
(4.y) (g, h)(g', h'} = (g(g')h hh'}.
Then it is straightforward to verify that with respect to this binary operation the set
X becomes a group in which (lc, 1H) is the identity and the element ((g-1)‘, h1} is
the inverse of (g, h). We call X the (external} semidirect product of G with H via a
(where о: H -> Aut(G) determines the Я-action on G), and write
X = [G]H (via <t).
If the Я-action is clear from the context, we will suppress <r.
It is clear from the definition that X = [G]H is the internal semidirect product of
the subgroup G x 1 (s G) with 1 x Я (S Я). Moreover, conjugation in X by the
element (1, h} corresponds to the Я-action on G as the following calculation shows:
(g,l)'1’'" = (l,/1-1)(.<?, DU, h)
= (l,h 1)(g, h) = (gh, 1).
It will be usually be convenient to make no formal distinction between the internal
and external semidirect products; in particular, we identify G x 1 with G, 1 x Я with
Я, and regard the external version [С]Я as an internal semidirect product of G with
Я.
Finally, we remark that if the Я-action on G is trivial (in other words, if <т(Я) =1),
then [С]Я = G x Я. Thus the direct product is a special case of the semidirect
product.
(4.23) Definition. Let A be a group of operators for a group G. We say that A acts
fixed-point-freely on G if Сс(Л) = 1, that is to say, if 1 g e G, there exists a e A such
that да # g. We say that A acts regularly on G if Cc(a) = 1 for all 1 a e A, (in other
words, if the orbits of A permuting the elements of G are all regular.)
Remarks, (a) If A Л 1 and A acts regularly on G, then A acts fixed-point-freely on G.
(b) If A acts regularly on G, then A can be viewed as a subgroup of Aut(G).
(c) Some authors use the term “fixed-point-free” to mean “regular” in our sense;
of course, when A has prime order, the two concepts coincide.
5. G-sets and permutation representations
17
5. G-sets and permutation representations
(5.1) Definition. Let £2 be a set and G a group.
(a) 12 is a right G-set if there exists a map
(co, g} —• a>g
from 12 x G to 12 satisfying
GS1: (cog)h = co(gh), and
GS2: colc = co
for all to eQ and allg,heG.
[Of course, a given set 12 may be endowed with the structure of a G-set in many
different ways, depending on the map (5.a)]
(b) A permutation representation of a group G is a homomorphism
(5/0 a: G -» Sym(£2)
for some set 12. If Ker(a) = 1, the representation is said to be faithful; then a(G) is a
subgroup of the symmetric group isomorphic with G.
Let да denote the image of an element g e G under the homomorphism a of (5./?),
and if я e Sym(£2), let am denote the image of co e £2. Then the G-action on 12 defined
by
(co, g) -> cog = co(ga)
obviously satisfies GS1 and GS2, and therefore converts £2 into a G-set. Conversely,
if 12 is some G-set, then the map да; 12 -> £2 defined by
(5.y) co(ga) = cog
for all co e Q is a permutation of £2, and moreover the map a; g -> да is a homo-
morphism from G into Sym(£2). Thus Equation 5.y defines a bijection between G-sets
and permutation representations, and so the two concepts are equivalent.
Let 12 be a G-set. The relation — on £2 defined by:
co ~ r if and only if т = cog for some geG
is easily seen to be an equivalence relation. The equivalence classes are called the
orbits of G on £2. Evidently for co e £2 the subset
coG = {cog ;geG}
is the orbit containing co. If £2 is itself an orbit of G on £2, we call £2 a transitive G-set
(and the associated a: G - Sym(£2) a transitive permutation representation). If co e 12,
we call
18 A. Prerequisites—general group theory
0» = {06 G:co0 = b)}
the stabilizer of co in G. Clearly G„ is a subgroup of G. The following elementary fact
is very useful in counting arguments.
(5.2) The Orbit-Stabilizer Theorem ([H] I, 5.10(a)). Let G be a finite group, and let
co be an element of a G-set 12. Then
|ruG| = |G:Gj.
In particular, if 12 is transitive, then |I2| = |G: GJ for all co eQ.
(5.3) Example. A group G may itself be viewed as a right G-set with the G-action
defined by group multiplication thus:
(x, g)->xg
for all x, g e G. Axiom GS1 is just the associative law for G, and Axiom GS2 is a
property of the identity of G. We call G, so regarded, (or any isomorphic G-set) the
right regular G-set and the associated permutation representation a: G -> Sym(G), the
regular permutation representation of G. It is clear that it is transitive and that each
point stabilizer Gg = 1, the identity subgroup; in particular, the regular representation
is faithful.
(5.4) Lemma. Let G be a p-group, and let Q be a G-set with (|fi|, p) = 1. Then G has
a fixed point on £2 (namely, an orbit of length 1).
Proof Let £2 = {Jjfij be a partition of £2 into G-orbits. By (5.2) we have |£2J = p"1
for some integer nt > 0. If л, > 0 for all i, then |£2| = £|£2,| = O(p), against the
hypothesis that (|£2|, p) = 1. Therefore there exists some j with nj = 0 and hence
I£2J = 1. The element co e £2; is the desired fixed point. □
We now give an application of (5.4).
(5.5) Proposition. Let P and Q be p-groups with Q # 1, and let P be a group of
operators for Q. Then the subgroup CQ(P) = {y e Q: y“ = у for all x e P} is non-
trivial. In particular, if 1 N <1 P, then N n Z(P) # 1.
Proof. Since P fixes l0, the set 12 = <2\{l0} is a G-set with pf |£2|, and the first
statement follows at once from (5.4). For the final assertion, we take Q = N and note
that P acts as a group of operators on Q by conjugation. □
(5.6) Definitions. Let M be a subset of a group G. We set
WC(M) = {g e G: g~'Mg = M}
5. G-sets and permutalion representations ig
and call NC(M) the normalizer of M in G. Similarly we define the centralizer C,(M\
of M in G by
Q(W) = {9 e G : g~'mg = m for all m e M}.
Clearly CG(M) and NG(M] are subgroups of G with Cc(M) < NC(M). If U < G, then
fVG(C) is a group of operators for U (via conjugation) and we obtain a homomorphism
a: NG(U) -> Aut(L')
given by (u)ax = x-1toc for и e U and xe NC(U). Since Ker(a) = Cc(U), we have
CG(C) < NG(U)and see that NG(U}/CG(U] is isomorphic with a subgroup of Aut(G).
If X is a group of operators for Y (via a: X -► Aut(y)), we can form the semidirect
product [У]Х and thus give meaning to NX(Y), Nr(X), CX(Y} and Cr(X).
(5.7) Lemma. If M is a non-empty subset of a group G, we have |G:Nc(Af)| =
|{M»: g e G}|.
Proof. Since the set of G-conjugates of M is a transitive G-set under conjugation, the
conclusion follows directly from the Orbit-Stabilizer theorem. □
Another important illustration of the G-set viewpoint is the following:
(5.8) Remarks. A group G is a G-set via the action
{g,h)->h'gh
for h,g e G. The G-orbits in this case are called conjugacy classes of G, and so we can
write
G = U
i=l
where K, c, K} = 0 for i *j, K, = {1}, and K, = {gf :geG} for any gt e K, <i =
2, The number к of conjugacy classes is called the class number of G, and
|G| = f |K,-|
«=1
where |K,| = |G : Ccts,)!/or i = 1, к fey (5.2).
(5.9) Definition. Let U be a subgroup of a group G. We define the core of L in G by
CoreG(C) = П С».
geG
Clearly CoreG(<7) is the largest normal subgroup of G contained in U.
20 A. Prerequisites—general group theory
(5.10) Lemma. Let U < G, and let Cl = {Ug,,..., Ug„} he the complete set of right
cosets of U in G. Then the G-action defined by
(5.6) (Ug„g)^U(g,g)
makes Ci a transitive G-set. The associated representation a: G-> Sym(I2) has kernel
Corec(l7). and so G/Corec(l/) is isomorphic with a subgroup of Sym(n).
Proof. The map a: G Sym(I2) (s Sym(n)) is defined by (5.y), viz. (Ug,)(ga) = Ug,g
for all g e G and i= 1,..., n. Thus Ker(a) = {g e G: Ug,g = Ug, for i = 1...n} =
{g e G: g e g, 1 Ug, for i = Since each g in G can be expressed in the form
g = ug, for some ue U, we have Ue = I/9, for some i e {1,..., n}, and therefore
Ker(a) = P|96C Ue = CoreG(l/). □
(5.11) Definition. Let G be a group, and let Г2, and I22 be G-sets. A map 0:12, -♦ I22
is called a homomorphism (of G-sets) if
(n>9)0 = (to6)g
for all to e 12] and g e G. If furthermore 0 is a bijection, we say Г2j is isomorphic (as
G-set) to I22. Two permutation representations a, and a2: G Sym(I2) are said to be
equivalent if their associated G-sets are isomorphic.
(5.12) Theorem. Let Г be a transitive G-set, let у e Г, and set U = Gy. If Cl =
{Ug,,..., Ug„} is a G-set according to the G-action of (5.0), then Г and Cl are
isomorphic G-sets.
Proof. Let J e Г, and let 0: Г -♦ Cl be defined by
(У9)0 = Ug
for all g e G. Then в is well-defined and is a bijection by (5.2). Since
<fyg)h)6 = (y(gh))6 = U(gh) = (Ug)h = ( (yg)0)h
for all g, h e G, the desired conclusion holds. □
(5.13) The Frattini Argument. Let G be a group, let Cl be a G-set, and let N <3 G.
Ij Cl is transitive when viewed as an N-set by restriction, then G = GaN for any
we Cl.
Prooj. Let to e Ci. If g e G, by hypothesis tog — wn for some ne N. Thus gn' e Gm,
and hence g e GaN. □
In many applications of (5.13), we have Cl = UG = UN for some subgroup U of G,
and then obtain the conclusion G = NG(U)N.
6. Sylow subgroups
6. Sylow subgroups
21
(6.1) Definitions, (a) Let G be a finite group, let p be a prime, and let |G| = p"m with
pin. A subgroup U of G with |U| = p“ is called a Sylow p-subgroup of G. We shall
denote the set of Sylow p-subgroups of G by Sylp(G).
(b) If U < G and = {Ge: g e G}, we call X a characteristic conjugacy class of
subgroups if a(Ue) e for all a e Aut(G) and g e G.
(6.2) Sylow s Theorem ([H] I, 7.3 and I, 7.5). Let p be a prime and G a finite group.
Then Sylp(G) is a (non-empty) characteristic conjugacy class of G. If U is a p-subgroup
of G and if P e Sylp(G), then U < Pe for some g e G. Furthermore, we have | SyL(G) | =
|G : AC(P)| = 1 (mod p).
(6.3) Remarks, (a) Sylp(G) coincides with the set of maximal p-subgroups of G. For
by Lagrange’s theorem Sylow p-subgroups are maximal p-subgroups of G and by
(6.2) any p-subgroup is contained in some Sylow p-subgroup of G.
(b) (The Frattini argument for Sylow subgroups). Let A < G and let P e Sylp(A).
Since Sylp(A) is a G-set under conjugation and a transitive А-set by Sylow’s theorem,
by (5.13) we have G = NC(P)N.
(c) If P e Sylp(G) and NC(P) < U < G, then we assert that NC(U) = U. For U <
Ng(U) and P e Sylp( 17); therefore by Remark (b) above we have NC(U) — UNHe№j(P)
< UNg(P) = U, and the assertion is justified.
(d) Define
Op(G) = n{P:PeSylp(G)}.
Then 0p(G) char G, and 0p(G) is the largest normal p-subgroup of G.
(6.4) Theorem. Let G be a finite group and p a prime.
(a) Let P e Sylp(G) and N < G. Then PnNe Sy lp( A)andPN/N e Sylp(G/A);more-
over, Ngin(PN/N) = NC(P)N/N.
(b) If Nt, N2<G and PeSylp(G). then N,N2oP = (A,nP)(A2nP) and
N1PnN1r = (NlfyN1)F.
(c) Let {pt,..., pr} be the complete set of prime divisors of | G|, and let Pj e SylP1(G)
for i = 1,..., r. Then G=<Pt,..., P,>, and if r = 2, then G = PjP2.
Proof, (a) [H] I, 7.7.
(b) Clearly (Nt n P)(A2 n P) < A, A2 n P. If M < G, let |M|P denote the highest
power of the prime p dividing |M|. By (a) we have | А,- n P| — | А;|р or i , an
\NiN2nP\ = |A> A2|p. Therefore
|(JV, n P)(N2 n P)| = ITVi |pI Ajlp/I^i n w2lp = IMA2Ip = lNtN2 n P|,
and consequently (A. n P)(A2 n P) = A, A2 n P. The second assertion follows from
this by (1.2).
22 A. Prerequisites—general group theory
(c) Let U = <Pn ..., Pr>. Then |Pf| divides |C/| for i = 1,.... r, and so
|G| = |Pi|...|Pr|
divides |L7|. Therefore U = G. If r = 2, by (1.5) we have
IP1P2I = 1ЛП^1/1А = |G|
since P^Pj = 1.
7. Commutators
(7.1) Definitions. Let G be a group.
(a) For g, h e G we set [g, /1] = g-'lT'gh and call [3, /1] the commutator of g
with h.
(b) If A and В are subsets of G, we set
[Л, B] = <[a, b] : a e A, b e B).
The special case [G, G], the subgroup generated by all the elements of G which can
be expressed as commutators, is called the commutator subgroup and is denoted by
G'. The higher commutator subgroups GM are defined recursively by G1"1 = G and
G(l+I1 = (G10)' for i > 1. We will write G" instead of G(21.
(c) If U is a subgroup of a group G, the normal closure of U in G is the
subgroup
<G°> = <l/9: g e G>.
Evidently <L°> is the smallest normal subgroup of G that contains U.
The next observations follow by direct calculation.
(7.2) Let g, h, and к be elements of a group G.
(a) [0, h] = [h, 0Г1.
(b) [9. Ю = [9ДК9, 4*-
(с) =
(d) (The Witt identity; [H], Satz III, 1.4):
EES, /Г1], *]*[[&, К’1], 9]‘[[*. «С1], Л]9 = 1.
(7.3) Lemma. Let g and h be elements of a group G.
la) If [[9, h], 3] = 1, then [3", /1] = [3, /1]" for all n e LJ.
(b) If [9, h] commutes with both g and h, then (gh)" = g"h"[h, g]"ln-1)/2.
(c) [9, h”] = [g, h] [gh, h] ... [g1’"', h] for all n e N.
7. Commutators
23
Proof, (a) and (b) are in [H] III, 1.3.
(c) The equation is certainly true when n = 1. Suppose that n > 1 and that
r3’ * 7 With the helP of <7-2)(b) we then see that
[0, " J — Ей, h h ] = [3, h" *] [g, h]*" and the desired conclusion now follows
from the fact that Eff, h]x = Eff*. /1х] for any x e G. □
The next portmanteau’ lemma contains many standard facts about commutator
subgroups.
(7.4) Lemma. Let A, B, and C be subgroups of a group G.
(а) ЕЛ B] = Ев, Л] < <Л, ву.
(b) EA B] < A if and only if В < NC(A).
(c) If a: G-* a(G) is a group homomorphism, then а(ЕЛ, В]) = [а(Л), а(В)].
(d) Ij A and В are normal (characteristic) subgroups of G, then (A, B] is a normal
(characteristic) subgroup of G.
(e) A subgroup U is normal in G with G/G abelian if and only if G‘ < U.
(f) If В normalizes A and C, then (AB, С] = [А, С] ЕВ, С].
(g) Let A, В < G with В < A. Then ЕЛ, G] < В if and only if A/B < Z(G/B).
(h) <ЛВ> = А ЕЛ.В].
(i) If N <G and heG, then [TV, {/1}] = E1V, </i>].
Proof. Statements (a), (b) and (c) are proved in EH] III, 1.6, and (d) follows from (c).
Statement (e) is EH] I, 8.2, and (f) is EH] III, 1.10 (a). To prove Statement (g) we ap-
peal to (c) to obtain the following chain of equivalent statements: [A, G] < Bo
[A, G]B < Bo[A, G]B/B < B/Bo[A/B, G/B] = lo A/B < Z(G/B). To prove
Statement (h) we first note that A[A, B] is a group by (a). Since ab = a[a, b] for all
aeA and b e B, we have (АВУ < A[A, В]. Since A and ЕЛ, B] are obviously
subgroups of <ЛВ>, Statement (h) is now clear. Finally, we observe that Statement (i)
follows directly from (7.3) (с). О
(7.5) Definition. Let G be a group.
(a) For n > 2 and g,, ..., g„ e G we define the n-fold commutator Effi, •, 0„]
recursively as follows:
Esi>--Snl = EEsi,-’Sn-ils2-
(b) If G],..., G„ are subgroups of G, we set
[Gi,...,GJ = <Effi,
(7.6) The Three Subgroups Lemma (EH] Ш, 1.10). Let N < G, and let А, В, C < G.
If [В, С, Л] < N and EC, A, B] < N, then ЕЛ, В, C] < N.
24
A Prerequisites—general group theory
(7.7) Definition. For any group G we set Kt(G) = G, and for i > 1 we define a
subgroup K,+1(G) recursively by
Kj+1(G) = [K,(G), GJ
The chain
g = k1(G)>k2(G)> ••
of (evidently characteristic) subgroups A-',(G) of G is called the lower {or descending)
central series of G. (Note that by (7.4) (g) we have Ki(G)/Ki+l(G) < Z(G/Ki+1(G)), so
that each factor of the series is central.)
(7.8) Theorem. Let G and H be groups.
(a) K,(G) = [G,.! ., G] for all i e M.
(b) [КДС), K/G)] < Ki+j(G) for all i,j e M.
(c) KfG x H) = K,(G) x KfH) for all i e 1U
Proof, (a) [H] III, 1.8.
(b) [H] 111,2.11 (b).
(c) The statement is certainly true for i = 1; let n > 1, and suppose inductively that
it is true for all i < n. Then with the help of (7.4) (f) we obtain
K„(G x H) = [^„-.(G) x K^IH), G*H}
= [K„_,(G). G] x [K„ ,(Я), H]
= K„(G) x K„(H). □
(7.9) Lemma. Let А, В, C < G with C < NG(B) and BnC = 1. If b e В and A <
C n Cb, then b e СВ(Л).
Proof. For each a e A, our hypothesis implies the existence of an element с e C
such that a = cb. Then C contains c~'a = с-1сь = [c, b], which also belongs to В by
(7.4) (b). Therefore, because С n В = 1, we conclude that c = a = cb and hence that b
centralizes a. □
(7.10) Lemma. Let G be a group and let glt..., gn+1 e G. Then
[[0i, • • • • ft,]"1. ft,+i] = ([0i, • • , ft,+iI191.e"1 T1-
(Here[0i,...,0„] = g, if n = 1.)
Proof. Apply (7.2)(c) with g = [<?,, ...,0„], h = д' and к = gntA. Since gh = 1, we
obtain
в. Finite nilpotent groups 25
1 = [(A 9n+i ] = [[919„+1 , gj- >, 9n+i]i
and the result follows.
□
The following theorem generalizes (7.8)(a).
(7 1!) Theorem. Let G„ .... G„ be normal subgroups of the group G, and lei n > 2
Then ' ~ '
[G1,...,G„] = [...[[g1,G2], GJ....GJ.
Proof. Let 4„ be the left side and B„ be the right side of the equation in question.
Obviously A„ < B„ and /f2 = [G,. G2] = B2. Assume we have shown 4n_, = B„_t.
Then B„ — [ Д.-1. G„] = [A„_,, G„], Therefore B„ is generated by elements of the form
У?. 91 with y, = [0( ....s, ], where c, = + 1, g e G„ and 3, , e G,. Repeated
application of (7.2)(c) yields
[y? • • • У?, = [У?, 9]”’ Ay? • • у*". 9]
= [У?, 9]/Л Ay?. (7 A * • • • [У?. »].
If г., = 1, then [уЛзА' ’ = [G...... GJ. If е,= -1.
by (7.10) we have [у, *, 9] = ([у,-, зУ' ) 1 6 [Gj,.... G„], and therefore also
’< ► i c*
[y. .9]’"' ' e [G,,.... GJ. This shows that B„ < 4„ and hence that B„ = An.
□
8. Finite nilpotent groups
Because many of the results in this section are either not easily accessible in the
literature or else appear in a formulation unsuited to ou' needs, we will give their
proofs in full.
(8.1) Definitions, (a) A group G is said to be nilpotent of class c(= c(G)) if G = 1
when c = 0 and if the lower central series of G satisfies KC(G) 1 = Kc+1(G) when
c > 0. We will use to denote the class of finite nilpotent groups of class at most c,
and 91 to denote the class (J;=o 91c of all nilpoteni groups.
(b) For any group G a subgroup Z,(G) is defined recursively by setting Z0(G) = 1.
ZJG) = Z(G), and Z1+1(G)/Z,(G) = Z(G/Z,(G)) for i > 1. Evidently each subgroup
Z,(G) is characteristic in G, and by (7.4) (g) we have
[Z,+1(GJ, G] < Z,(G).
We call the chain 1 = Z0(G) < Z, (G) < • - the upper (or ascending) central series of
G. The characteristic subgroup ZJG) = U'=o 4(G) is called the hypercentre of G.
26
A. Prerequisites—general group theory
It follows easily from these definitions that a subgroup U of G is contained in the
hypercentre Za,(G) if and only if [17, G,, G] = 1 for some tie N.
(8.2) Theorem. Let G and H be finite groups.
(a) Let U < G and N < G. If G is nilpotent, then U and G/N are nilpotent', moreover
c(U) < c(G) and c(G/N) < c(G).
(b) If G and H are nilpotent, then G x H is nilpotent and c(G x H) =
Max {c(G), с(Я)}.
(c) Let M,N < G and assume that G/M and G/N are nilpotent. Then G/(M n N) is
nilpotent, and c(G/(M n N)) = Max {c(G/M), c(G/W)}. In particular, every finite group
G possesses a smallest normal subgroup with nilpotent quotient group (called the
nilpotent residual of G and denoted by G91—see II, 2.4).
Proof, (a) Suppose that Kc+1(G) = 1. Since obviously K,(17) < KfG) for all i > 1, we
have KC+1(17) = 1 and therefore U is nilpotent of class at most c.
By(7.4)(c)wehaveKc+1(G/N) = Kc+1(G)N/N = 1, and so the quotient G/N is also
nilpotent of class at most c.
(b) By (7.8) (c) we have
K,(G x H) = KfG) x K,(H).
If c = Max {c(G), с(Я)}, then Kf+1(G x H) = 1 and either Kt(G) 1 or KC(H) # 1;
in any case KC(G x H) # 1.
(c) By 9.4 the quotient group G/(M n N) is isomorphic with a subgroup of (G/M) x
(G/N) and is therefore nilpotent of class at most Max{c(G/M), e(G/N)} by Parts (a)
and (b). On the other hand, G/M and G/N are both epimorphic images of G/(M n N),
and so c(G/(M n N)) is at least Max [c(G/M), c(G/N)} by Part (a). Therefore equality
holds. □
(8.3) Theorem. Any two of the following statements about a finite group G are
equivalent:
(a) G is nilpotent;
(b) Zcl(G) < ZC(G) = G for some integer c(= c(G));
(c) IfU<G, then U < NG(U) (the normalizer condition);
(d) Every maximal subgroup of G is normal;
(e) G is the direct product of its Sylow subgroups;
(f) If H/K is a chief factor of G, then H/K < Z(G/K) (such a chief factor is called
central);
(g) If U < G, there exists a chain
U =U0<V1S " <Ur=G
such that | Gj/Gj.i | is prime for i = 1,..., r;
(h) All subgroups of G are subnormal.
[In particular, from (e) we see that groups of prime power order are nilpotent, and
from (f) we know that if N is a non-trivial normal subgroup of a nilpotent group G, then
ZIG) nN^iand [N, G] < /V].
8. Finite nilpotent groups
27
Proof (a) => (b). Suppose that G is nilpotent of class c, set K, = K(G) and Z - Z (G>
We show first, by induction on i, that ’ 1 ~
(8.a)
kc+1_( < z,
for i - 0, 1,. , c. For i = o we have Kc+I = Zo = 1, and (8.a) certainly holds then
(П we°Xin У Kc+'-‘ - Zi fOr SOme' - °- Appealing tO (7-4) <b>and
[Kc_1Z1.,G] = [Kc_i,G][Zi,G]
< Kc-.+iZj < Z,Z, = Zf,
and therefore Kc_iZJZi < Z(G/Z,). Thus Kc_t < Zj+1, the induction is justified, and
(8.o) holds. In particular, for i = c, we obtain G = K, = Zc.
To complete the proof, suppose, for a contradiction, that G = Zc_]. If this is so,
we prove by induction on i that
(8.Д)
Ki+1 < Zc_,_f,
which is certainly true for i = 0. If (8.Д) holds for i = к - 1, from (7.4) (g) we obtain
= [*'», G] < [Z(_k, G] < and the induction step is proved. Setting i =
c — 1, we obtain Kc < Zo = 1, contradicting the supposition that G has class c.
Therefore Zc_1 < G, and Statement (b) is proved.
(b) => (c): Since Z0(G) = 1 and ZC(G) = G, there exists an i e {0,..., c — 1} such that
Z;(G) < U and Zi+1(G) f U. By (7.4) (g) we then have
[Zi+1(G), G] < [Zi+1(G), G] < Z,(G) < U,
and from (7.4) (a) and (b) we deduce that Zi+1(G) < NC(G). Therefore U NG(U).
(b) => (d): If U < G, then U < NG(U), and therefore NG(U) = G.
(d)=>(e): Let P e Sylp(G). If NC(P) < G, let NG(P) <U < G. By (6.3) (c) we have
NC(U) = U and by our assumption that Statement (d) holds, we also have NG(U) = G.
This contradiction shows that all the Sylow subgroups of G are normal. If |G| =
p°'. . p°r with p, ? pj for 1 < i /7 < r, then by Sylow’s theorem G has a unique Sylow
p,-subgroup P;, and since P;n = 1» ’t follows that G = Pj x x Pr.
(e) => (f): Let G be a direct product of its Sylow subgroups. In order to prove that
chief factors of G are central, by the Jordan-Holder theorem we may suppose without
loss of generality that G is a p-group for some prime p. Let H/K be a chief factor of
G. Then by (5.5) we have 1 * (H/K) n Z(G/K) < G/K, and therefore H/K < Z(G/K)
since H/K is a minimal normal subgroup of G/K.
(f) => (a): Let 1 = Ho < Ht < < = G be a chief series of G in which each chief
factor is central. We prove that
(8.}’)
Ki+l(G) < H„.
for i = 0, 1,..., m, proceeding by induction on i. For i = 0 we have Kt(G) — G — Hm.
28 A. Prerequisites—general group theory
Suppose that (8.y) holds for i = r — 1. Then Kr+1(G) = [A-,(G), G] < [Hm_r+1, G] <
H„ r by (7.4) (g), and (8.y) holds for i = r. Therefore (8,y) holds for all values of i and
in particular for i = m, which yields KmJ.t(G) = Zo = 1. Therefore G is nilpotent.
(c) => (g): If U = G, there is nothing to prove. If U < G, then U < NG(U), and there
exists a subgroup G, of NG(U) containing U with Ut/U of prime order. The desired
conclusion follows by induction on |G: G|.
The implication: (g) => (h) is trivial, and so to complete the proof of the theorem it
will suffice to show that (h) => (d): If U < G, then U sn G, and therefore U < G. □
The next lemma is useful, especially in the study of soluble groups.
(8.4) Lemma. Let N be a nilpotent normal subgroup of a finite group G, and let M be
a maximal subgroup of G which does not contain N. Then N/(M n N) is a chief factor
of G complemented by M.
Proof. Since N f, M, we have M < MN. Therefore MN = G and M r\N < N. By
(8.3) we have Nn(M n N) > M n N. Therefore NG(M n N) is a subgroup of G contain-
ing, as well as M, elements of N\M, and it therefore coincides with G by the
maximality of M. Hence M n N <! G. If there existed a normal subgroup К of G
satisfying M n N < К < N, there would be a chain of subgroups M < KM < G,
against M < G. Therefore N/(M n N) is a chief factor of G. □
(8.5) Definition. Let n, a be sets of primes and G a finite group.
(a) The characteristic subgroup OJG) of G is defined thus:
0„(G) = : N < G, N a n-group).
[If M and N are normal subgroups of G, then MN is a normal subgroup and
iAIJV| = |Af||N|/|Mn JV|. It follows that 0„(G) is the largest normal subgroup of G
which is a n-group.]
The characteristic subgroup 0n a(G) is then defined as follows:
O^Gl/O^G) = Oa(G/OJG)).
(b) The characteristic subgroup On(G) is defined dually thus:
O"(G) = Q {N : N < G and G/N is a n-group}.
[Evidently O”(G) is the smallest normal subgroup of G with the property that its
quotient group is a л-group.] Correspondingly, we define О’1,"(G) = O"(O"(G)).
The subgroups O„(G) and OP’(G) can be characterized as follows.
(8.6) Lemma. Let G be a finite group and it a set of primes.
(a) If К sn G and К is a л-group, then К < On(G); thus O„(G) is the join of the
subnormal n-subgroups of G.
(b) If pis a prime, then OP'(G) = (P: P e Sylp(G)> = P[P, G] for any P e Sylp(G).
м
8. Finite nilpotent groups
fhatl' JoVobvti'1 7 ' К\Г G' With K 3 W- We the assertion
hat R S! 0„(G) by induction on r. If r = 1, then К < G, and К < OJG) by definition
Г nd;by,nf ““ - -i)char K,_t < G.whenceOJK^)™
G, and therefore 0„(R,_,) < 0„(G) by definition. Thus К < О (G) as asserted
bv ?6 5h‘ f 7h°P P a₽d 3 =t> <F7 6 Sylp<C)>-K P 6 Sy‘₽(C)’ th“ PRIR 6 SW«
by (6.5) (a). Therefore P < R, and consequently J < R. On the other hand, J is a
normal subgroup of G because Sylp(G)is a conjugacy class, and |G : J| divides |G PI
a p -number. Hence G/J is a p'-group, and R < J. Therefore R = J Finally by (7 4Wh)
we have J = P[P, G] for any P e Syl (G) ’ q
(8.7) Definition. The Fitting subgroup F(G) of a finite group G is defined as follows:
F(G) = <Op(G): p e <r(G)>.
It is clear from this definition that F(G) is the direct product X Op(G) over the prime
divisors p of |G| and that 0p(G) e Sylp(F(G)). Therefore by (8.3) the Fitting subgroup
is a characteristic nilpotent subgroup of G.
(8.8) Theorem. Let G be a finite group.
(a) F(G) = (S : S sn G and S is nilpotent}; in particular, F(G) is the largest nilpotent
normal subgroup of G.
(b) If S,,Sr are nilpotent subnormal subgroups of G, then (St.......Sr) is also a
nilpotent subnormal subgroup of G.
(c) If Nt and N2 are nilpotent normal subgroups of G such that G = NtN2, then G
is nilpotent.
Proof, (a) If S is a nilpotent subnormal subgroup of G, then Op(S) sn G and so
Op(S) < O„(G) by (8.6). By (8.3) we have OP(S) e Sylp(S), and therefore by (6.5)(c) it
follows that S < <Op(G): p e P> = F(G).
(b) From Part (a) we know that <S,,..., S,> < F(G), and therefore by (8.2)(a) the
join <S],..., Sr> is nilpotent. From the implication: (a) => (h) of (8.3) we know that
the join <S!,..., Sry is subnormal in F(G); it is therefore subnormal in G.
(c) This is a special case of Part (b). C
(8.9) Remark. Let H/K be a chief factor of a finite group G. If the composition
factors of H/K are abelian, then H/K is an elementary abelian p-group for some prime
p; in particular, nilpotent chief factors are elementary abelian p-groups.
Proof Since a chief factor is obviously characteristic-simple, by (4.13)(a) it is either
an elementary abelian p-group, or else a direct power of a non-abehan simple group,
and the latter possibility is ruled out by the assumption that its composition factors
are all abelian. That nilpotent groups have abelian composition factors clearly follows
from Theorem 8.3, (a) =>(f).
30 A. Prerequisites-—general group theory
9. The Frattini subgroup
(9.1) Definition. Let G be a finite group. The Frattini subgroup Ф(С) is defined to be
1 when G = 1 and otherwise by setting
®(G) = Q {M: M <• G}.
Clearly ®(G) is a characteristic subgroup of G.
(9.2) Theorem. Let G be a finite group.
(a) Let S s G. Then G = <S> if and only if G = (S, ®(G)>.
(b) Let N < G. Then N has a supplement distinct from G in G if and only if
N Ф(6).
(c) Let N < G and let U be minimal in the set of supplements to N in G. Then
Nr^U< Ф(1/).
(d) If N < G, U < G, and N < Ф(С), then N < O(G).
(e) If N < G, then G>(N) < Ф(О and <b(G}N/N < ®(G/N). Moreover, if N < ®(G),
then <t>(G/N) = Ф(С)//У.
(f) If A is an abelian normal subgroup of G and A n ®(G) = 1, then A is comple-
mented in G.
Proof. If a non-generator of a group means an element which is redundant in every
generating set, then ®(G) may be characterised as the set of non-generators of G when
G # 1. This fact, proved in [H] III, 3.2, implies Statement (a); Statement (b) is also
proved there.
To prove Statement (c), suppose that N c\U Ф(С). By (b) there exists a proper
supplement V to N n U in V, which implies that G = NV = N(N n t/)F = NV with
V < U, in contradiction to the choice of V. Therefore N c-V < Ф(С).
Statement (d) is proved in [H] III, 3.3, Statement (e) in [H] HI, 3.3 and 3.4 and
Statement (f) in [H] III, 4.4. □
(9.3) Theorem. Let G be a finite group.
(a) ([H] III, 3.6) Ф(С) is nilpotent; in particular ®(G) < F(G).
(b) ([H] III, 3.7) G is nilpotent if and only if G/G>(G) is nilpotent.
(с) ([H] III, 4.2) If N <> G and N < Ф(С), then F(G/N) = F[G)/N.
(d) ([H] III, 3.12) G' n Z(G) < ®(G).
(9.4) Lemma. Let Gj, ..., Gr be finite groups. Then Ф(С, x x Gr) =
©(GJ x x ®(Gr).
Proof. Let D = Gj x • • x Gr, and identify G( with the subgroup
lx”-xlxG|Xlx"Xl
of D. Since G, < D, by (9.2) (e) we have ®(G;) < Ф(Л) and therefore
(9.a) ®(Gt) x • x ®(Gr) < Ф(Л).
9. The Frattini subgroup
31
Let K. — G, x • • • x G , x 1 * c. «
ОДВД < O(D/K,),and furthermore Ф(Р/К) = ф^ /к к - X haVe
D/K = G^/K s Gf;itfollowsthat|Ф(Р)ЛФ(Р)\ s.L7^
we know from (417) that Ф(О) is isomorphic with a subgroup of the direct product of
the r groups Ф(PO/WPOr' K,). Непсе |Ф(О)| < p[' , = ^(GJ x . Px ф(С)|
and this fact, together with (9.a), now yields the desired conclusion. □
(9 5) D!?z"!‘iOn’ Let P a Prime and G a p-group. We define characteristic sub-
groups S2(G) and U(G) (pronounced “agerno G”) as follows:
ЭДО) — (.9 6 G : gp = 1 >, and
U(G) = <.gp: g e G>.
We also use the notation Q,(G) = <0 e G: gp‘ = 1> for i e M. Hence Q^G) = 11(G).
(9.6) Theorem ([H] III, 3.14). Let G be a finite p-group.
(а) Ф(С) = G'U(G); Ф(С) is the smallest normal subgroup of G with elementary
abelian quotient group.
(b) If p = 2, then Ф(С) = U(G).
(c) If V < G, then Ф(П) < Ф(С).
(d) If N<G, then &(G/N) = d>(G)N/N.
(9.7) Burnside’s Basis Theorem([H] III, 3.15). Let G bea p-group with |С/Ф(О)| = pd.
Then G contains a set of generators with d elements, and no set with less than d elements
generates G. Every element of С\Ф(С) belongs to some minimal generating set.
(9.8) Lemma. Let N < G with G/N cyclic. Then a minimal supplement to N in G is
also cyclic.
Proof. By (9.2)(c) we have Л7,1'<Ф(1;). Since U/(N r.G) s UN/N = G/N, it
follows that и/Ф(и) is cyclic. From (9.3)(b) we deduce that V is nilpotent and from
(9.7) we conclude that each Sylow subgroup of U, and hence U itself, is cyclic. □
(9.9) Definition. Let H/K be a chief factor of a finite group G. We call H/K Frattini
if H/K < Ф(С/К) and complemented if the minimal normal subgroup H/K is comple-
mented in G/K.
We note that by (9.2)(c) a chief factor can not be simultaneously Frattini and
complemented. Since a chief factor H/K is characteristic-simple, by (4.13)(a) it is
either an elementary abelian p-group or the direct power of some non-abehan simple
group. Our next result shows that a soluble group (whose chief factors are necessarily
all abelian) has the important property that each of its chief factors is either
complemented or Frattini.
(9.10) Lemma. Let H/K be an abelian chief factor of a finite group G. Then
(a) H/K is either complemented or Frattini, and
(b) if H/K is complemented, then each complement is a maxtmal subgroup of G.
32
A. Prerequisites—general group theory
Proof, (a) Suppose that H/K f Ф(С/К). We must show that the minimal normal
subgroup H/K of G/K is complemented in G/K, and, in so doing, may clearly suppose
without loss of generality that К = 1. Since H f. Ф(6), we have H M for some
maximal subgroup M of G. Since H is abelian, Lemma 8.4 implies that H/(M H)
is a chief factor of G complemented by M, and therefore M n H = 1 because H is a
minimal normal subgroup of G.
(b) Let L be a complement to H/K in G, and let L < M < G. Since H f M, the
argument of Part (a) shows that M complements H/K in G. Hence | G: M\ = \H/K\ =
|G: L|, and therefore L = M. □
(9.11) Lemma. Let К and bi be normal subgroups of a finite group G with N < К and
К nilpotent. If K/N < <b(G/N), then К < ®(G)N.
Proof. We argue by induction on |G|. If N = 1, there is nothing to prove. Therefore
suppose that N # 1, and let L be a minimal normal subgroup of G contained in bi.
Since the hypotheses evidently carry over to G/L, we conclude by induction that
K/L < &(G/L)(N/L). If L < Ф(С), then ®(G/L) = Ф(С)/£ by (9.2)(e), and then К <
Ф(С)7У, as desired.
Therefore suppose that L Ф(С). Because К is nilpotent, by (8.9) the subgroup L
is abelian and by (9.10) is therefore complemented in G, by M say. Since M < G and
К is nilpotent, by (8.4) we have К n M < G. The isomorphism x -»xL (x e M) from
M onto G/L yields <b(M)L/L = <t>(G/L), and therefore К < <f>(M)LN =
However, Ф(М) n К is normalized by M and is also centralized by L because
(К с, M) c, L < M r L = 1. Therefore Ф(М) n К < G, and then from (9.2)(d) we can
conclude that Ф(М) n К < Ф(С). Consequently, we obtain
К = Ф(М)7У n К = (Ф(М) n K)N < ®(G)N. □
(9.12) Lemma. Let bf and N2 be distinct minimal normal subgroups of a finite group
G. Then there exists a bijection
r. {Nt, NtN2/N2} - {N2, N2N2/N2}
such that corresponding chief factors are G-isomorphic and Frattini chief factors
correspond to one another.
Proof. Put N = N2, and first suppose that Nr <, ®(G). Then N/N2 = N2N2/N2's
Frattini by (9.2)(e). If N/N, is also Frattini, then N < Ф(С) by (9.2)(e), and all four
chief factors in the statement are Frattini. In this case the map t with tbf = N/N2
and t(NlN2/Nl) = N2 satisfies the stated requirements. If, on the other hand, N/N3
is not Frattini, then N2 is not Frattini by (9.2)(e), and the same choice of r will suffice;
likewise if all four chief factors are not Frattini.
It remains to consider the case where bf n Ф(С) = N2 n Ф(С) = 1 and (say)
bl/bl2 < ®(G/N2). Since <b(G/N2) is nilpotent, the chief factor N/N2 is abelian, and
hence so is bf f = N/N2}. By (9.10) there exists a complement M to N, in G; let
N3 = M c\N. Then N37Vi = (M n N)Ni = MNj г-b/ = bl, and so N3 s N/Ni = N2.
9. The Frattini subgroup
33
LXtNN/nZ S’ iS LCOm^ment in ‘° N/N2, a contradiction to the
fact that N/N2 is Frattini. Hence N3 N2, N = N3N2 and N3 s N/N, a N, Con-
tW N n ф7с\аП л/ к06 N' и abeHan’ and Ьу (91 ° NAN r*(G» = N- R follows
that N n Ф(О = N3, hence that N/Nt = N3N2/N2 is Frattini, and that all four chief
factors in question are G-isomorphic. If we therefore take tN. = N2 and tIN/N ) =
N/N2, the desired conclusions hold. □
We can now state and prove, as promised, the following strengthened form of the
Jordan-Holder theorem for chief series in finite groups.
(9.13) Theorem. Let and JF2 be chief series of a finite group G. Then there exists
a one-to-one correspondence between the chief factors of and those of ,#2 such that
corresponding factors are G-isomorphic and such that the Frattini chief factors of
correspond to the Frattini chief factors of X2.
Proof. From (3.2) (with Q = Inn(G)) we know that and 3f2 have the same length
and so we may denote them thus:
T1:G = Vo> U2 >> Ur= 1, and
3P2:G= Fo> F, > ••> Fr= 1.
We prove the assertion by induction on r, noting that it is trivially true when r = 1.
Let r > 1, and suppose that the theorem is true for all groups with chief series of
length <r — 1. If Ur_, = Frl, the induction hypothesis applied to G/U^ yields a
suitable correspondence between the factors of the two chief series lying above GF_,,
and by making Vr 2 correspond to l^_, we have the desired conclusion.
Therefore suppose that the minimal normal subgroups Ц and are distinct,
and set N = Gr_, Kr_,. In this case N/U,-i and N/Vr_t are chief factors of G, and there
exist chief series Jf3 and of the following form
^3:G= W0>--->IFr_3>N>Gr_1>l, and
.?F4: G = Wo > ••• > Wr_3 > N > K-i > L
Let us call two chief series “equivalent” if there exists a one-to-one correspondence
between their factors satisfying the requirements of the theorem. This obviously
defines an equivalence relation on the chief senes of G. Since and .#3 have
the minimal normal subgroup U-, in common, by induction they are equivalent.
Similarly and Ж4 are equivalent. Furthermore, as the senes and jF4coincide
above N, it clearly follows from (9.12) that X3 and Jf4 are also equivalent. Therefore
JF, and :K2 are equivalent.
Finally we state Philip Hall’s estimate for the order of Aut(G) in terms of
|Aut(G/4>(G))| and |O(G)|.
34
A. Prerequisites—general group theory
(9.14) Theorem ([H] 111, 3.17 and III, 3.18). Let G be a finite group generated by d
elements.
(a) | Aut(G)| divides | Aut(G/O(G))| |O(G)|d.
(b) If a e Aut(G) and [G, a] < Ф(С), then the order of a divides | ®(G) |d; in particular,
if (o(a), |®(G)|) = 1, then a = 1.
10. Soluble groups
(10.1) Definition. Let л be a set of primes. A finite group G is said to be n-soluble if
SI: every chief factor of G is either a л-group or a л'-group, and
S2: the л-chief factors of G are abelian.
[A group satisfying Condition SI is called n-separable.]
A group is soluble if it is л-soluble for л = P, in other words, if its chief factors are
all abelian.
Because Frattini chief factors of a finite group are elementary abelian p-groups by
(8.9), and in view of (9.13) (the strengthened Jordan-Holder theorem), in order to
establish л-solubility it is sufficient to check that Conditions SI and S2 are satisfied
by just the non-Frattini chief factors of a single chief series. Since abelian chief factors
have prime power order, it is clear that a finite soluble group is л-soluble for every
set л of primes.
The class of л-soluble groups enjoys many of the closure properties proved in
Section 8 for nilpotent groups, for example, those described in Parts (a), (c) and (d) of
the following theorem.
(10.2) Theorem. Let G be a finite group.
(a) Let V < G and N < G. If G is n-soluble, then U and G/N are also n-soluble.
(b) If N < G, and if N and G/N are n-soluble, then G is n-soluble.
(c) If Nj< G and G/N! is n-soluble for i = 1, 2, then G/(Nt n N2) is also n-soluble.
(d) If Nt and N2 are n-soluble normal subgroups of G, then Nt N2 is n-soluble.
(e) G is n-soluble if and only if G/Q>(G) is n-soluble.
Proof, (a) Suppose that G is л-soluble. Since each chief factor of G/N is isomorphic
with a chief factor of G above N, it follows at once that G/N is л-soluble. Further-
more, if
G = Gq t> Gt t> * • t> Gr = 1
is a chief series of G, then a refinement of the series
V = V nG0> V nGj > V c.Gr = 1
to a chief series of U obviously inherits both properties SI and S2 of Definition (10.1),
and so U is л-soluble.
10. Soluble groups
35
N with V< СЫе[ fe i‘°L?f ° WUh H~N' and let M’K be a chief factor
N with M < H. Then clearly H/K = «М/Ку-. g e G>, and so by (4.4) there exist
elements gY....g„ in G such that
H/K = X (м/ку<.
i=l
If N is я-soluble, then M/K is either a л'-group or an abelian я-group. If M/K is a
я'-group, so also is H/K, and when M/K is an abelian я-group, clearly H/K is too.
If G/N is я-soluble, the chief factors of G, both above and below N, are therefore either
я'-groups or abelian я-groups, and so G is я-soluble by (3.2).
(c) Suppose that G/N, and G/N2 are я-soluble. Since N,N2/N2 < G/N2, we con-
clude from Part (a) that N,N2/N2, and hence the isomorphic group N,/(N, n N2), is
ir-soluble. But then G/(N, n N2) is я-soluble by Part (b).
(d) Since N, is я-soluble, by Part (a) so is N, N2/N2 s N,/(N, n N2), and then by
Part (b) we see that N, N2 is itself n-soluble.
(e) This follows from the fact that by (8.9) the chief factors of G below Ф(С) are
elementary abelian groups of prime power order. □
The criteria for solubility given in the next theorem are often used as definitions.
Indeed, the word “soluble”, as applied to groups, derives from the fact that the Galois
group of the splitting field of a polynomial over Q which is “soluble by radicals” has
composition factors of prime order.
(10.3) Theorem. Any two of the following statements about a finite group G are
equivalent-.
(a) G is soluble;
(b) G<n) = 1 for some n e N;
(c) The composition factors of G have prime order.
Proof, (a) => (b): Let
G = Go t> G, t> t> G„ = 1
be a chief series of G. Since each factor G,/G;+1 is abelian, it follows from (7.4) (e) by
induction on i that G10 < G, for i = 0, 1,.... Therefore G1 1 = 1.
(b)=>(c): Since a refinement of an (abelian) chief series into a composition senes
clearly has prime order factors, the desired conclusion follows from the Jordan-
Holder theorem.
(c) => (a): This is clear by (8.9). u
(10.4) Definition. If G is a finite soluble group, by (10.3) there exists a smallest
non-negative integer n such that C"> = L and this is called the dertved length of G.
complemented or Frattini.
36
A. Prerequisites—general group theory
(b) Let M he a maximal subgroup of G and УС a chief series of G. Then M avoids
just one chief factor in and covers the rest.
(c) Each maximal subgroup of G has prime power index in G.
Proof, (a) This follows from (8.9) and (9.10)(a).
(b) Denote the terms of .# thus: G = Go t> t> • t> Gn = 1. Since MG0 = G and
MG„ = M, there exists an integer i such that G = MG0 = = and M =
MG, = = A1GO-Clearly the chief factors of .#above G,-! and below G, are covered
by M. Since Gj G^, nM < , we can apply (8.4) to G/Gj and the abelian normal
subgroup Gi-i/Gi to conclude that 0;1/(С,-, n M) is a chief factor of G. Therefore
G( = Gj., n M, and so M avoids G^/Gj.
(c) By (b) the index |G: Ml equals |G,_,/G;|, which is a prime power by (a). □
Some of the most important facts about the fundamental “normal” structure of a
finite soluble group are included in the following theorem.
(10.6) Theorem. Let Gbe a finite group:
(a) Cc(F(G))F(G)/F(G) contains no non-trivial soluble normal subgroup of G/F(G);
in particular, Cc(F(G)) < F(G) when G is soluble.
(b) If N is a minimal normal subgroup of G, then F(G) < CG(N); furthermore, if N
is abelian, then N < Z(F(G)).
(c) (i) The Fitting subgroup FfG/OfG)) equals F(G)/®(G) and is the product of the
abelian minimal normal subgroups of G/Q>(G), all of which are complemented. Further-
more, F(G)/O(G) is complemented in G/®(G) and G/d>(G) is isomorphic with the semi-
direct product [F(G)/O(G)] (G/F(G)).
(ii) Assume that
EITHER
G is soluble
OR
G is p-soluble and Op.(G) = 1.
Then
(10.a)
F(G)/4>(G) = GCWC)(F(G)/®(G)) = Soc(G/®(G)).
In particular, G/F(G) is represented faithfully as a group of automorphisms of
F(G)/O(G) and if G 1, then Ф(С) is a proper subgroup of F(G).
(d) If G is soluble and F(G)/®(G) < Z„,(G^(G)), then G is nilpotent.
Proof, (а) [H] III, 4.2 (b).
(b) [H] III, 4.2 (e).
(c) (i) By (9.3)(c) we have
(10.0) F(G/O(G)) = F(G)/®(G),
and so, in proving the statements of Part (c)(i), we can suppose without loss of
10. Soluble groups
37
STS(?Г T = ' Slnte Ф(^(С)) ~ Ф(С) by (9-2) Ы we conclude from (9.6)
! < r / i. °P ° e ementary abelian for all primes p and hence that F(G) is abelian
Let U be a normal subgroup of G contained in F(G). By (9.2) (f) there is a complement
S ° ‘J?®1subgrouP v = c n F(G) is normal in C and is centralized by U
because F(G) is abelian. Therefore V is normal in UC = G, and by the Dedekind
modular law V is a complement to V in F(G). It then follows from (4.6) (with
D — Inn(G)) that F(G) is a product of abelian minimal normal subgroups of G all of
which are complemented by (9.2) (f). Since the Fitting subgroup of a group obviously
contains every abelian minimal normal subgroup, F(G) is therefore the abelian
component of the socle of G. Theorem 9.2 (f) also implies that F(G) is complemented
in G, and consequently G s [F(G)](G/F(G)), where the action of G/F(G) on F(G) is
conjugation by a coset representative.
(c) (ii) Suppose first that G is soluble. Since the Fitting subgroup F(G/<I>(G)) is
abelian, it is self-centralizing by the final assertion of Part (a) of this theorem. From
Equation 10. [f we then conclude that
Cc/eiC)(F(G)/a>(G)) = F(G)/®(G).
Since the socle is abelian in this case, Equations 10.a are now clear.
Now suppose that G is p-soluble and that Op.(G) = 1. In this case F(G) = Op(G) and
again the socle of G is abelian. To prove that F(G/<I>(G)) is self-centralizing in G/<I>(G),
we may again suppose that Ф(С) = 1 and hence that F(G) is an abelian p-group.
Suppose, by way of contradiction, that F(G) is a proper subgroup of the normal
subgroup Cg(F(G)) of G, and let R/F(G) be a chief factor of G with R < Cc(F(G)).
Since F(G) = Op(G), the quotient R/F(G) is a p'-group, and by the Schur-Zassenhaus
theorem (see (11.3) below) R has a complementary subgroup, Q say, to F(G). Since
F(G) < Z(R), we have Q < R, and therefore Q = Op (R) char R < G. Hence Q < OP (G),
which is trivial by hypothesis. This contradiction proves that F(G) = CG(F(G)), as
desired. The rest of Part (c) (ii) now follows in either case. □
Next we state a well-known theorem of O. Schmidt.
(10.7) Theorem ([H] III, 5.4). If all the proper subgroups of a finite group G are
nilpotent, then G is soluble; in particular, if all the proper subgroups of G are abelian,
then G" = 1.
(10 8) Definition. Let V be a subgroup of a finite group G. If U either covers or avoids
each chief factor of G, we say that U has the cover-avoidance property and call U a
САР-subgroup of G. (Take note! If U either covers or avoids the chief factors of some
given chief series, it does not necessarily follow that U is a САР-subgroup. For
example let G = Л x C2, a direct product of an alternating group on four letters
example, « 2 = be a s low 2.Subgroup of
and a cyclic group C2 = <c> oi order z. ш \
Л and let U = (a bcf Then V covers or avoids the chief factors in
1 < C2 < KG < G,
but the chief factor is neither covered nor avoided by G.)
38
A. Prerequisites—general group theory
(10.9) Proposition. Let G be a finite soluble group. Then V is a САР-subgroup of G
if and only if eoch Sylow subgroup of V is a САР-subgroup of G.
Proof Let H/K be a p-chief factor of G and let P e Sylp(l/). Then PK/K e Sylp(l/K/K)
by (6.4)(a), and since (V n H)K/K < Op(UK/K) < PK/K, it follows that (U n H)K g
PK oH = (Pri H)K. Hence (V n H)K = (P n H)K, and we see that H/K is covered
(avoided) by U if and only if H/K is covered (avoided) by P. □
11. Theorems of Gaschiitz, Schur-Zassenhaus, and Maschke
The following theorem of Gaschiitz [1], published in 1952, shows i.a. that the
existence of a complement to an abelian normal subgroup is governed by the structure
of the Sylow subgroups.
(11.1) Theorem ([H] I, 17.4). Let A be an abelian normal subgroup of a group G, let
A < В < G, and assume that the index |G : B| = к is finite. Assume further that the
map а-гак (ae A) has an inverse.
(a) If A has a complement in B, then A has a complement in G.
(b) If A has a complement in В and all such complements are conjugate in B, then
all complements to A in G are conjugate in G.
If Л is finite and (|Л|, к) = 1, then the congruence kl = I (mod |Л|) has a solution
and the map a -»a1 is the inverse of the map a -» ak. In view of this, the following
theorem follows easily from (11.1).
(11.2) Theorem. Let A be an abelian normal subgroup of a finite group G. Then A is
complemented in G if and only if for each prime p dividing |Л| there exists a Sylow
p-subgroup P of G such that A is complemented in AP.
Another important splitting theorem is the following.
(11.3) The Schur-Zassenhaus Theorem ([H] I, 18.1 and 18.2). Let N be a normal
subgroup of a finite group G such that (| JV|, | G/N]) = 1. Then N has a complement in G,
and if either N or G/N is soluble, then all complements to N in G are conjugate in G.
Remark. If (|N|, |G/N|) = 1, clearly at least one of the groups N and G/N has odd
order. Therefore, if one is prepared to cite the celebrated theorem of Feit and
Thompson that groups of odd order are soluble, the solubility assumptions can be
omitted from the hypotheses of (11.3).
As another application of Gaschiitz’s Theorem 11.1, we give a short proof of
Maschke’s theorem.
(11.4) Maschke’s Theorem. Let G be a group of operators acting as automorphisms
on a (not necessarily finite) abelian group A. Let H be a subgroup of G of finite index,
11 Theorems of Gaschiitz, Schur-Zassenhaus, and Maschke 39
that T:A^A d*'ined by I(fl) = ,s bijective- f“"ber assume
‘f‘ ~ ‘,X A1’ W,th A' a G-^iant subgroup and A2 an H-invariant subgroup
of G. Then there extsts a G-invariant subgroup A? of A such that A = At x Af
Proof Let G* = [A]G and H* = AH(< AG}. Then A, has a complement in H*
namely A2H, and since |G*: H*| = |G: H\, we deduce from (11.1) that A. has a
complement in G*, call it C. Let A*2 = C n A. Since A is abelian, C n A is normal
in <C, A> = G*, and so, in particular, A* is G-invariant. Furthermore, Л,Л? =
A,(Cri Л) = AjCn Л = G*r> A = AandA, ci AJ < Лх nC = l.andsoour A? has
the desired properties. 1-1
If we now take G to be a finite group, H = 1, and A to be a KG-module of finite
dimension n over a field К (Л written additively), then the map r: a -> |G|a has an
inverse r 1: a —> (l/|G|)a provided that Char(K) is either zero or does not divide |G|.
Thus from (4.6) and (11.4) we obtain the following more familiar version of Maschke’s
theorem.
(11.5) Theorem. Let G be a finite group, and let К be a field which contains |G| *.
Then each finite dimensional KG-module is semisimple.
As an application of Theorem 11.4, we describe the invariant-subgroup structure
of a finite abelian group admitting a group of operators of relatively prime order.
(11.6) Theorem. Let A be a finite abelian p-group, and let G be a group of operators
for A with (IG/Cg(A)|, | Л|) = 1. Then A has a direct decomposition
A = At x x As
into G-admissible subgroups А,- with the following properties for each i = 1,..., s.
(i) A, is indecomposable as a G-module;
(ii) Л(/Ф(Л;) is an irreducible G-module;
(iii) Л,- is homocyclic.
Proof. We proceed by induction on |Л|.
Case 1: Suppose that П(Л) f Ф(Л). Since П(Л) is elementary abelian the subgroup
Я(Ф(Л)) is complemented in Q(A), and therefore by (11.4) (with H - CG(A)) we have
П(А) = Г2(Ф(Л)) x В,
and since |Л| = ofajofaj) К i>
40
A. Prerequisites—general group theory
\<i j, tin) x В. Since В is a G-module, it follows again from (11.4) that there exists
a G-submodule C of A such that A = С x B. Clearly C satisfies the hypotheses and
hence by induction the conclusions of this theorem. Since (11.5) applies to В with
К = f we conclude that В is semisimple, and the theorem is true in this case.
Case 2: We now suppose that O(.4) < Ф(.4) and set A = .4/fl(.4). By induction we
can write
A = A, x • • x As,
with At homocyclic and G-indecomposable, and with Л,-/Ф(Л() G-irreducible. Let
(p*'-1, , p‘‘~') denote the type of .4^, where > 2 and fcf > 1 for i = 1,.... s. Since
fl(.4) < Ф(4), we evidently have d(A) = d(A) = = d (say). (Here d(X) denotes
the minimal number of generators of a group X.) Then
|.41 = |П(Л)||.4“| = pd f] p^-1*' = f] p‘*
i=l i=l
Let Aj denote the inverse image of Л,- in A under the natural homomorphism from
A to A. Then the type of Л, is clearly
, (d-*i) e. k, e.
(p, ... ,p,p‘-,... ,p‘f.
First consider the case s = 1. Then A = At, d = kt, and -4j has type (p‘],..., pe‘).
Since A! /Ф(.4,) = .4 1/Ф(.4|) is G-irreducible, certainly A, is G-indecomposable, and
in this case the theorem is true.
Now let s > 1. Then for each i = 1,..., s we have Ai < A, and by induction
.4,- = Bj x Cf
with Bj of type (pe\..., pef, B; indecomposable and В1/Ф(В1) irreducible as a G-
module. Furthermore, C, has exponent p, and so С, < П(.4) < Ф(4). Hence
Л = <4j,.4S> = <B,,Bs,
= (B,» * • -» fO ~ Bj &2 - B,.
However, П?-] |Bf| = P[?=1 p'1*'= |Л|, and therefore finally we have A =
Bt x • • x Bs. □
(11.7) Theorem. Let A and G be as in Theorem 11.6. Then any two of the following
statements are equivalent:
(a) A is G-indecomposable;
(b) А/Ф(А) is G-irreducible;
(c) If A has exponent p", then the only non-trivial G-admissible subgroups of A are
0,(4) for i = 1,.e;
(d) OJA) is irreducible.
12. Coprime operator groups 41
Proof. The equivalence of Statements (a) and (b) is clear from (11.6).
(a) => (c). Suppose that A is G-indecomposable, and let 1 < i < e Since A is homo
cyclic by Statement (lii) of (11.6), the map ~ ~ ’ nce'”shom°-
a,: a -» aF‘ ‘
is a G-homomorphism from A onto £2,(4), and consequently a, induces a G-
isomorphism from С/Ф(.4) onto £2,(/l)/£21_1(/l). Hence by Statement (ii) of (11.6) the
section £21(.4)/£2,_1(.4) is irreducible (by convention, £2O(,4) = 1). Let В be a non-trivial
G-subgroup of A. There clearly exists a natural number j such that В < £2+1(/l) and
В f, 12Д.4), and then B12y(4)/12j(4) is a non-trivial G-subgroup of £2+,(,4)/12 (,4).
Consequently В12Д.4) = £2Jtl (.4), and since £2,(.4) = Ф(£2,+](4)), it follows that В =
£1J+1(.4).
Since the implication: (c) => (d) is obvious, it remains to prove that (d)=>(a).
If A = ,4, x ,42 were a non-trivial G-decomposition of A, then £2,(4) =
121(4]) x £2](.42) would be a non-trivial decomposition of£2](,4). □
(118) Theorem. Let G be a finite group and p a prime.
(a) If p| |Ф(С)|, then p| | G/®(G)|; in particular, if л ср, then Q js a n-group if and
only if G/4>(G} is a n-group.
(b) If the non-Frattini chief factors of G are all p'-groups, then G is a p'-group; in
particular, if G is soluble and not a p'-group. then G has a complemented p-chief factor.
(c) If G is soluble and has a non-trivial cyclic Sylow subgroup, then G has a
complemented chief factor of order p.
Proof, (a) Let P e 8у1р(Ф(С)). If р)|б/Ф(С)|, then PeSylp(G), and since Ф(б) is
nilpotent, we have P char Ф(С) by (8.3) (e). Therefore P < G by (3.5), and consequently
P has a complement, U say, in G by (11.3). But then G = PU < Ф(С)С = U by
(9.2)(a), which contradicts the hypothesis that p||G|. Therefore p||С/Ф(С)|.
(b) This follows easily from (a) by induction on the length of a chief series.
(c) By Part (b) the group G has a complemented p-chief factor, and since this is
cyclic and elementary abelian, it has order p. □
12. Coprime operator groups
We recall our convention that the elements of a group of operators induce auto-
morphisms on the group they act upon. The following fundamental theorem shows
that, under a “coprime” action, a fixed coset of an admissible subgroup always
contains a fixed element. Since the known proofs of this theorem appeal in some form
to the conjugacy statement oftheSchur-Zassenhaus theorem, one must either assume
that at leas'one of the groups A and H is soluble or else cite Fed-Thompson
theorem, that a group of odd order is soluble; the same caveat applies to the
subsequent results of this section which depend on this theorem.
42
A. Prerequisites—general group theory
(12.1) Theorem ([H] I, 18.6). Let A be a group of operators for a group G. and let H
be an A-admissible subgroup of G such that (|.4|, |H|) = I. If Hg“ = Hg for all a e A,
then there exists an element x in Hg such that x“ = x for all ae A.
(12.2) Definition. Let Q be a group of operators for a group P. We say that Q
stabilizes a chain of subgroups
P = P0> P, > ' >P„= 1
if [Pj_i, Q] < P, for ii = 1,..., n. (This is equivalent to saying that Q fixes every coset
of Pj in Pj-j for i = 1,..., n.)
(12.3) Proposition. Let Qbe a group of operators for a group P with (|6|, |P|) = 1. If
Q stabilizes a chain of subgroups of P, then [P, Q] = 1.
Proof. If n = 1, there is nothing to prove. Proceeding by induction on n, we may
suppose that n = 2 because [P,, Q] = 1 by the induction hypothesis. Since Q leaves
invariant each coset of P, in P, by (12.1) it fixes a transversal S' to P, in P. Each x e P
can be written x = yt for some у e Pt and t e S'. Since [Plt Q] = 1, it follows that
v“ = (yt)“ = y“t“ = X for all a e Q. Thus [P, Q] = 1. □
It is perhaps worth remarking that the group of operators Q in the preceding
theorem can be replaced by Q/Ca(P) without changing the conclusion. Therefore the
coprimeness hypothesis can be replaced by the weaker condition (|Q/Cc(P)|, |P|) = 1.
(12.4) Corollary. Let n be a set of primes, let P be a n-group, and let Q be a group of
operators for P.
(a) If Q stabilizes a chain of subgroups of P, then Q/Cq(P) is a n-group.
(b) If Q/Cq(P) is a n'-group, then [P, Q, Q] = [P, Q].
Proof, (a) If x is a л'-element of Q, then x e Cc(P) by (12.3). Since 0"((2) is generated
by the n'-elements of Q, it is therefore contained in Cc(P), and Assertion (a) is clear.
(b) Let N = [P, Q, Q], and note that N < [P]Q by (7.4)(a). Then Q stabilizes the
chain
P/W > [P, Q]/A > 1
of P/N. and so by Part (a) we have Q/Cq(P/N) e (£„ n <£r. = (1). Thus [P, Q] < A,
and the Statement (b) now follows. □
(12.5) Proposition. If Q is a n'-group of operators for a n-group P, then P =
[P, Q] CP(Q). Moreover, P = [P, Q] x CP(Q) when P is abelian.
Proof. Since Q leaves invariant each coset of the Q-invariant subgroup [P, Q] in P,
by (12.1) each coset contains an element of CP(Q).
A proof of the final statement can be found in [H] III, 13.4. □
12. Coprime operator groups
43
(12.6) Corollary. If Q is a
[P, Q] = P, then [R, Q] =
л -group of operators for an abelian n-group P, and if
R for every Q-admissible subgroup R of P.
Proof. Let Q be a л'-subgroupof R. By (12.5) we have P = [P, e]CP(O) = Ф(Р)С (О)
and so from (9.2)(a) we conclude that P = CP(Q), that is to say [P, Q] = 1. Sm'ce R
is л-perfect, it is generated by its л'-subgroups, and therefore [P. R] = 1 □
We note in passing that the Feit-Thompson theorem is not required in the proof of
(12.7), this is because R is generated by its Sylow q-subgroups for q e л', and q-groups
are certainly soluble.
(12.8) Theorem. Let Q be a p-perfect group of operators for a p-soluble group G.
Assume that
(a) Op.(G) = 1, and
(b) [F(G), Q,Q] < Ф(С) for some ne N.
Then Q centralizes G.
Proof. Since Q is generated by its Sylow subgroups whose orders are prime to p, it
will suffice to assume that Q is an q-group for some prime g / p. We will also suppose
without loss of generality that Q acts faithfully on G and will aim to show that (2=1.
We first observe that a p-soluble group G with Op.(G) = 1 satisfies F(G) = 0p(G).
In view of Hypothesis (b) it follows from 12.4 (b) that the «/-group Q/Cq(F(G)/Q>(G))
is also a p-group; hence Q centralizes F(G)/®(G). Form the semidirect product
H = [G]C,
and note that H, like G, is p-soluble. Let N — OP(H). Since [N, G] < N n G <
0p.(G) = 1, it follows that QN n G is a p'-group contained in Cc(F(G)/®(G)). There-
fore by (10 6)(c)(ii) we have QN n G < F(G) and consequently QN n G = 1. Since Q
is a complement to G in H, we conclude that QN = Q and hence that N < C^G).
But О is supposed to act faithfully on G and therefore 0p(H) — 1.
ft now follows easily that F(H) = O„(H) = OP(G) = F(G), and because Ф(С) < Ф(Я)
by (9.2)(e), we see that Q centralizes Р(Н)/Ф(Н). Applying(10.6)(c)(n) again, this time
to H, we obtain
Q < СИ(Р(Н)/Ф(Н)) = F(H).
Since Q and F(H) have relatively prime orders, we conclude that Q = 1, as desired
44
A. Prerequisites—general group theory
13. Automorphism groups induced on chief factors
(13.1) Definition. Let p be a prime. A finite group G is called p-nilpotent if G =
Opp(G), or, equivalently, if each Sylow p-subgroup has a normal complement in G.
It follows easily from (8.3) that a group is nilpotent if and only if it is p-nilpotent
for all primes p.
(13.2) Lemma ([H] VI, 6.3). Let M and N (< M) be normal subgroups of a finite
group G with N < Ф(С). If М/N is p-nilpotent, then M is p-nilpotent.
(13.3) Proposition ([H] IV, 4.4). A finite group G is p-nilpotent if and only if G is
p-soluble and all p-chief factors of G are central.
Many of the closure properties of nilpotent groups carry over to p-nilpotent
groups, as our next portmanteau theorem shows.
(13.4) Theorem. Let p be a prime and G a finite group.
(a) If G is p-nilpotent, then so also is every subgroup and quotient group.
(b) If Nj < G and G/Nt is p-nilpotent for i = 1,2, then G/(N, n N2) is also p-nilpotent.
(c) Op.p(G) = <S: S sn G, S is p-nilpotent).
(d) If St, ..., Sr are p-nilpotent subnormal subgroups of G, then (,Sl,...,Sr) is a
p-nilpotent subnormal subgroup of G.
(e) If N < G, then Op.JN) = N n 0p.,p(G).
(f) Opp(G/<I>(G)) = Op..p(G)/®(G).
(g) F(G) = Qp6 p Op. p(G); in particular, G is nilpotent if and only if G is p-nilpotent
for all primes p.
Proof, (a) Let U < G = Op. p(G). Clearly Op.(G) n U < Op.(U} and U/(Qp.(G) c.U)s
l/Op (G)/Op.(G), which is a p-group. Therefore U = Op. p(l7). If N < G, then Op.(G)N/N
is a normal p'-subgroup of G/N of p-power index. Thus G/N is p-nilpotent.
(b) From the G-isomorphism between an<J ^/(TV, it follows that
the chief factors of G/(Nr n N2), regarded as G-groups via conjugation, are a subset
of the chief factors of G/N, and G/N2. The assertion therefore follows from (13.3).
(c) Let S = Op. P(S) sn G. Since Op.(S) < G by (3.5), we have Op.(S) < Op.(G) by (8.6),
and then a similar argument applied to the factor group G/Op.(G) yields Op. p(S) <
Op. p(G). Statement (c) now follows from the fact that Op. p(G) is itself p-nilpotent.
Statements (d) and (e) both follow immediately from (a) and (c), and Statement (f)
is a direct consequence of (13.2). Finally, to see that Statement (g) is justified we note
that F(G) is obviously contained in QpE p Op. p(G) by (c) and that the intersection is
itself nilpotent by (13.3) and the implication: (f) => (a) in (8.3). □
(13.5) Definition. Let A be a group of operators (acting by automorphisms) on a
group G, and let H and К be -4-admissible subgroups of G such that К < H. Then
there exists a homomorphism p: A -> Aut(H/K) defined by a -> pa, where p„: Kh -»
Kh° for all h e H. The image of p is called the group of automorphisms induced by A
on H/K and is denoted by Nut A( H/K). Since Ker(p) = CA(H/K), we have hutA(H/K)
= А/СЛ(Н/К) by the isomorphism theorem.
1
13. Automorphism groups induced on chief factors
45
(13.6) Lemma Let .4 be a group of operators for a finite group G, and let H/K be an
A-composit,on factor ofG. If H/K is insoluble,assume further that Inn(G) < Aut,(G)
(a) If H/K .s .„soluble, then AutA(H/K)possesses a unique minimal normal subgroup
and this is A-.somorphic with H/K.
'Ь> H/K 'S soluble',hen H!K is an elementary abelian p-group for some prime p
and Op(AutA(H/K)) = 1. r
In particular, if p is a prime divisor of \H/K\, in both cases O„(Aut„(H/K)) = 1.
Proof. Since H/K is characteristic-simple, by (4.13) (a) it is either a direct power of
some non-abelian simple group or else an elementary abelian p-group for some prime
p. Set C = CA(H/K).
(a) Let H/K be a direct power of some non-abelian simple group. Then Inn(G)
(S G/Z(G)) contains some .4-invariant chief factor H*/K* which is .4-isomorphic
with H/K, and by the hypothesis that Inn(G) < Aut^fG), we can view H*/K* as a
chief factor of A. Moreover, Cc.H* = K*, and therefore the group H*C/C, which is
.4-isomorphic with H*/(H* r.C) = H*/K*, is a minimal normal subgroup of A/C and
is .4-isomorphic with H/K. It remains to show that it is unique. Let N/C be a mini-
mal normal subgroup of .4/C distinct from H*C/C. Then [A', H*] < N n H* =
C n H* = K*, and therefore N < CA(H*/K*) = CA(H/K) = C, a contradiction of the
definition of N. Therefore H*C/C is the unique minimal normal subgroup of
.4/C S Aut^(H/K).
(b) Let L/C be a normal p-subgroup of .4/C (where p|\H/K|), and let P denote
the semidirect product [H/K](L/C). Since P is a p-group, by (8.3)(h) we have
(H/K) n Z(P) # 1, in other words, CHIK(L) 1. Since L < .4, the subgroup CH/K(L)
is .4-invariant, and consequently CH/K(L) = H/K because H/K is .4-simple. But then
L < CA(H/K) = C, and we have proved that OP(A/C) =1. □
(13.7) Lemma. Let A be a group of operators on a group G of the form G =
Gj x x G„ where Gt, ..., G, are non-abelian simple groups. If A stabilizes an
А-composition series of G, then A centralizes G.
Proof. Let 1 = K„ < Kj < • • • < Kr = G be an ,4-composition series of G stabilized
by ,4. By (4.13)(b) we can choose the notation so that Kr_, = G, x ••• x G„ and
proceeding by induction on r, we may suppose inductively that [Kr_,, A] = 1. Since
(4.13)(b) also implies that .4 permutes the set {G,}'=]. the fact that Kr_t is A-
admissible means that the subgroup N = G,+i x - x G, is A-admissible^ Therefore
[G. .4] = [Kr_, N, A] = [K,^, .4] [N, .4] < N. But [G, .4] < Kr_, by hypothesis.
Thus [G, .4] = 1. D
The following theorem will prove to be a valuable tool in the study of finite soluble
groups and of saturated formations of finite groups.
JX p. J c; be -be -f °" ""
are not Frattini. Then Cp — Cp — 0p-p(G).
46
A. Prerequisites—general group theory
(b) The Fitting subgroup F(G) of G is precisely the centralizer of all chief factors
of G. Moreover, if G is soluble and i«' write
F(G)/<I>(G) = К i x x K„
with each Kt a minimal normal subgroup of G/<b(G) (according to (10.6)(c)), then
f(G) = П;=1 cG(K,).
Proof (a) Obviously Cp < Cp, so it will suffice to prove that Op. p(G) < Cp and then
that C* < O„.JG).
Step 1: The proof of that Op p(G) < Cp. Let H/K be a chief factor of G with p|| H/K |,
and let C= CG(H/K). Since 0p(G) avoids H/K, we have 0p.(G) < C, and therefore
0p- p(G)C/C is a normal p-subgroup of AG(H/K}. It then follows from (13.6) that
0p. p(G} < C and hence that 0p-.p(G) < Cp.
Step 2: The proof that C* < 0p- p(G). By (13.4)(c) it will be enough to show that
C* is p-nilpotent. First suppose that 0p- p(G) = 1, and let N be minimal normal
subgroup of G; then p| |N| and N is not a p-group. It follows that N is a direct power
of a non-abelian simple group and is therefore certainly not Frattini because Frattini
chief factors are abelian by (9.3)(b) and (8.9). Moreover CG(N) c. N < Z(N) = 1, and
we have shown that С* гл N = 1 for every minimal normal subgroup N of G. Since
C* < G, we conclude that C* = 1 in this case.
Now suppose that 0p- p(G) # 1, and let N be a minimal normal subgroup of G
contained in 0p. p(G). By Step 1 we have N < Cp, and by induction on |G| applied to
G/N we may suppose that Cp/N is p-nilpotent. If N Ф(С), then N lies in the centre
of C* by definition of Cp, and therefore by (13.3) (together with the Jordan-Holder
theorem) C* is p-nilpotent. On the other hand, if N < Ф(С), then C* is p-nilpotent
by (13.2), and Step 2 is complete.
(b) This follows directly from Part (a), (13.4)(g) and (10.5)(c). □
Our final result in this section shows how a chief factor of a group can be passed
on intact to a subgroup.
(13.9) Lemma. Let H/K be a chief factor of a finite group G, and let X be a sub-
group of G which covers H/K and G/Cg(H/K). Then (H n X)/(K n X) is a chief
factor of X', moreover, X r\CG(H/K) = CX((H с, X)/(K г, X}), and Autc(H/K) S
AutA((H nX)/(KnX)).
Proof Since G = XCG(H/K), the group G and X induce identical groups of auto-
morphisms on H/K', in particular, H/K is simple as an X-group. Since X covers H/K,
we have H/K = (H c\ X}K/K = (H c\ X)/(K с. X), an X-isomorphism. Therefore
(H n X)/(K n X) is simple as an X-group, CX(H/K) = CX((H n X)/(K n X)) since iso-
morphic modules have the same kernels, and Autx((H n X)/(K n X)) s Autx(H/K)
Autc(H/K). □
14. Subnormal subgroups
14. Subnormal subgroups
47
We recall that a subgroup U of a group G is said to be subnormal
U sn G) if there exist subgroups Go, G,__Gr of G such that
in G (written
G = Gr < Gr_j < • • < Gj < Go = G.
If G is a finite group, then the subnormal subgroups of G are precisely the terms of
the composition series of G.
Here we limit our account of subnormal subgroups to just a few elementary
properties that we shall need later. For a comprehensive and up-to-date survey of
the deep and fascinating theory that has grown out of Wielandt’s seminal paper [1]
of 1939, we refer the reader to Lennox and Stonehewer [1].
Our first objective will be to show that the subnormal subgroups of a finite group
form a sublattice of the subgroup lattice.
(14.1) Lemma. Let U be a subnormal subgroup of a finite group G.
(a) If V < G, then U nV sn V.
(b) If A <G, then UN/N sn G/N.
Proof. Let G = Gr < - < Gt < Go = G. Then evidently
(a) G n V < • • <1 Gj r> V < Go r> V = V, whence G n V sn V, and
(b) UN/N < ^UtN/N^UgN/N = G/N, and so UN/N snG/N. □
As an obvious consequence of (14.1) (a) we have the following.
(14.2) Corollary. If U, V sn G, then U nVsn G.
(14.3) Lemma (Wielandt [1]). If U is a subnormal subgroup of a finite group G, then
Soc(G) < Ng(U).
Proof. Let G = Ur < < Cj < Go = G, and let A be a minimal normal subgroup
of G. We prove the lemma by induction on r, noting the obvious validity of the
conclusion when r = 1. Since N is minimal normal, either A n Gt = 1, in which case
[A, Gj] = 1 and A centralizes G, or else A < U, and by (4.13)(c) we have A <
Soc(Gj). In the latter case, the induction hypothesis yields A < AC(G). □
(14.4) Theorem (Wielandt [1]). Let {Gf: i e 1} be a set of subnormal subgroups of a
finite group G. Then their join J = (,Ui'.ieIj is also subnormal in G.
Proof. We argue by induction on |G|. Let NoG. Then UiN/N sn G/N by (14.1)(b),
and so by induction we have JN/N = <G,A/A: i e /> sn G/N. Since A normalizes
each Ц by (14.3), it follows that J < JN sn G.
Thus by (14.2) and (14.4) the subnormal subgroups of a finite group form a lattice
with respect to intersection and join.
48
A. Prerequisites—general group theory
Our next goal is to define a canonical shortest subnormal chain from U to G when
U is subnormal in G. As the example of the subnormal subgroup <(121(34)) of Sym(4)
shows, the chain of repeated normalizers: U < NC(U) < Wg(^g<^)) < ’ ’ ’ is deficient
because it may become stationary before it reaches G. The most satisfactory choice is
to work down from the top, taking iterated normal closures.
(14.5) Terminology. Let U be a subgroup of a group G.
(a) For a non-negative integer i we define a subgroup < U'jG) recursively by setting
<17 oc> = G and by defining <G lG) to be the normal closure of U in <G' (‘-,,G) for
i> 1. Thus <17' OG) !> <17 1C>> .
(b) For a non-negative integer i we define a subgroup [G, f L7] by setting [G, 017] =
G and [G, iL'] = [[G, (i - 1)U], U] for i > 1. Thus
[G,iG] = [...[[G,G],G]...,G].
i
(14.6) Lemma. Let U be a subgroup of G. In the terminology of (14.5) we have
(a) <17 ic> = G[G, il7] < <G-"“”C> for all i > 0, and
(b) if U < Ur < Gr_j <3 • • - < Uo = G, then <U-rG) < Ur.
Proof (a) We write = <17’,G> for i = 0, 1, .... By (7.4)(h) we have
L'[G, 17] = < L7®: g e G) = < G = Wo. Suppose inductively that we have already
shown that 17[G, ilf] = WJ <1 lVi_I. Then it follows, with the help of (7.4)(h) and (f),
that
U[G, (i + 1)17] = V[U, 17] [G, (i + 1)17] = 17[17[G, ilf], 17]
= 17[H<, 17] = <U’:xe И/>
= Wi+1 < Wit
and the induction step is complete.
(b) Since Wo = G = Uo, we argue by induction on r and suppose that it has already
been shown that Wit < If i for some i e {1,..., r}. Since V < If < 17;_,, we have
= <fJx: x e H<_i> < <L'X: x e If-f) < If,
and therefore by induction Wr < If. □
(14.7) Definition. Let U be a subnormal subgroup of a group G. Then there exists a
subnormal chain
U = Ur < < U, < Uo = G
of length r joining U to G. The smallest length of such a chain joining 17 to G is called
the subnormal defect of U in G.
14. Subnormal subgroups
49
It is dear from (14.6)(b) that G & <GG> t> <G •2C'> t> t> rr,
and so by (14.6)(a) we have the following characterization of subnormal subgroups’.
!?g u be a sub9roup °f a 0roup a Then u is subnormal of defect d
(a) [G, (d - 1)G] jt U and (b) [G, dG] < U.
r
Remark. In (7 5) the notation [G,G,...,G] is used to denote the subgroup
<[0, ur]: 0 e G, e G> of G. It turns out that if G is finite, the condition
[G, G, G] < G for some r c N
is also necessary and sufficient for a subgroup G to be subnormal in G, although
it can fail to be sufficient when G is infinite. (See Lennox and Stonehewer Г 11 Theorem
7.3.6 (ii).)
The following lemma, due to Wielandt [7], is sometimes helpful in establishing
subnormality by induction.
(14.9) Lemma. Let H be a proper subgroup of a finite group G, and suppose that H
is subnormal in К whenever H <K < G but is not subnormal in G. Then H is contained
in a unique maximal subgroup of G.
Proof. We argue by induction on |G: H|, noting that the conclusion certainly holds
if H < G. Since NC(H) < G by hypothesis, there exists a maximal subgroup M of G
containing WG(H). Let H < L <• G. Since H sn L, by (14.6) and (14.8) for the defect d
ofHinL we have H = <3 = <H': .x e Hj. 2> < L, where Wj = Write
J = and let J < К < G. Since each conjugate Hx satisfies the same hypotheses
as H, we have Hx sn К whenever Hx < K. Therefore J sn К by (14.4). Furthermore,
from the fact that H <3 J, we deduce firstly that J is not subnormal in G, secondly
that J < NC(H) < M, and thirdly that |G: ,/| < |G:H|. Thus J satisfies the same
hypotheses as H, and so by induction L = M. □
An interesting application of this lemma is the following criterion for subnormality.
(14.10) Theorem (Wielandt [7]). A subgroup Hof a group G is subnormal in G if and
only if H sn <H, H9> for all g e G.
Proof. If H sn G and H < L < G, it is clear from (14.8) that H sn L, and it follows
that the condition is necessary.
To prove the sufficiency we argue by contradiction, assuming that there exists a
group G with a non-subnormal subgroup H satisfying H sn <H, H9> for all\ g e G and
further that among such counter-examples G has minimal order. This choice of G
ensures that H sn К whenever H < К < G, and therefore H is contained in a unique
maximal subgroup M of G by (14.9). Let g e G. Since H is not subnormal in G, by
hypothesis <H, H9 > is a proper subgroup of G and is therefore contained in M.
50
A. Prerequisites—general group theory
Consequently H = < M", and it follows that M = Me for all ge G. But then
we have H sn M < G, and so H sn G. This contradiction proves that the stated
condition is also sufficient. □
(14.11) Corollary (Baer [2], Alperin and Lyons [1]). The following statements about
a subgroup H of a finite group G are equivalent:
(a) H<F(G)nO,(G);
(b) H9) is a nilpotent n-group for all g e G.
Proof. The implication: (a) => (b) is obvious. To prove the reverse implication suppose
that (b) holds. By (8.3) the subgroup H is subnormal in <H, He) for all ge G.
Therefore H is a subnormal nilpotent л-subgroup of G by (14.10), and then Statement
(a) follows at once from (8.6)(a) and (8.8)(a). □
In passing, we should mention that the concept of the “normalizer" of a subgroup
has a counterpart in the realm of subnormality: If H is a subgroup of a group G, a
subnormalizer of H in G is a subgroup S containing H such that
(a) H sn S, and
(b) if H sn T < G, then T < S.
A subnormalizer, if it exists, is obviously unique. Since we will meet a generalization
of the concept of a subnormalizer in Chapter IV, Section 5, it might be helpful to see
that it need not always exist.
(14.12) Example. Let N = Z3 x Z3. Since Aut(A) GL(2,3) has order
(32 — 1)(32 — 3) by (l.l)(b), by Sylow’s theorem Aut(A) contains a group T of order
16. Let G denote the semidirect product [A] T, and let T and T* be distinct Sylow
2-subgroups of G. Since | TT*\ = | T| | T*|/| T n T*| < |G| = 9 • 16, we have | T n f*| >
16/9 > 1. Therefore T n T* is a non-trivial subgroup of G which is subnormal in T
and T*. An easy calculation shows that N is a minimal normal subgroup of G and
hence that G = <T, T*>. If Tn T* were subnormal in G, it would follow from (14.3)
that [A, T n T*] < N n T n T* = 1. But since T consists of automorphisms of N,
no non-identity element of T centralizes A. Therefore T n T* is not subnormal in G
and hence has no subnormalizer in G.
We conclude this section with some results of a special nature about subnormal
closure and single-headed groups. They will be applied to the study of Fitting classes,
mainly in Chapter XL
(14.13) Definitions. Let G be a finite group.
(a) If U < G, the subnormal closure (U c> of U in G is defined as follows
= P){S: U < S sn G}.
(b) G is called single-headed if G 1 and G has exactly one maximal normal
subgroup.
(c) G is said to be perfect if G = G'.
14. Subnormal subgroups
51
It is clear from (14.2) that <U «> is the smallest subnormal subgroup of G
containing U, and from (14.6) that <1/ c> = Q/=o ,c> g p °
(14.14) Lemma. Let U be a subgroup of a finite group G, and let S = <<J c\ the
subnormal closure of U in G. Let R = S’1, the smallest normal subgroup of S with
nilpotent quotient group Then RU = S. Moreover, if U is single-headed, then either
S-Ror S/R is single-headed, and if further S is soluble, then S itself is single-headed.
Proof. By (8.3) every subgroup of the nilpotent group S/R is subnormal, and so, in
particular RU sn S. But by definition of S, no proper subnormal subgroup of S
contains U. Hence RU = S.
Since a non-trivial quotient of a single-headed group is obviously single-headed
it follows from the isomorphism S/R s U/(R n U) that S/R either has order 1 or is
single-headed. If S ф 1 and S is soluble, then a maximal normal subgroup T of S has
prime index in S; in particular, R < S' < T <S, and so T/R must concide with the
unique maximal normal subgroup of S/R. □
(14.15) Lemma. Let N be a proper normal subgroup of a finite group G, and let S be
a minimal subnormal supplement to N in G. If G/N is single-headed, then S is single-
headed, and if, in addition, G is perfect, so also is S.
Proof. Let M/N be the unique maximal normal subgroup of G/N. Since
G/M = SM/M = S/(S n M), then S n M о S. Let L <• S. Then L sn G, and so
LN A Gby the minimality of S. Since LN sn G by (14.4), we have LN < M, and hence
L < S n M. Consequently Sn M is the unique maximal normal subgroup of S. If G
is perfect, then G/M (and hence S/(S M)) is perfect, and therefore S is perfect. □
(14.16) Theorem. Let G ± 1 be a finite group, and let У denote a set of subnormal
subgroups of G containing at least one minimal subnormal supplement to each maximal
normal subgroup of G. Let -V* denote the perfect groups in У
(a) Then G = <S: S e У >, and, in particular. G is generated by its single-headed
subnormal subgroups.
(b) If G is perfect, then G = (S: S e У*).
Assume further that ff* = 0 (which clearly holds when G is soluble) and that У
comprises all single-headed subnormal subgroups of G. If К = \M'. there exists Se У
with M <• S>, then К < G, and the group G/K is nilpotent and has Sylow subgroups of
prime exponent.
Proof, (a) Let R = <S: S e У>, a subnormal subgroup of G by (14.4). If R < G, then
G has a maximal normal subgroup M containing R. But by definition У contains a
supplement So to M in G, and so G = MSQ < MR < M < G, which .s absurd.
Therefore R = G, and since themembersof У are single-headed by (14.15), the second
assertion is clear. By appealing to the final statement of (14.15). one can use a simila
argument with У* in place of У to prove Assertion (b). mniuoacv
To justify the last part of the theorem, observe that Уa un,°n £ X has
classes and hence that К < G. Assume that У* - 0. If M <• S e У, /
IS. Primitive finite group* S3
(15.2) Theorem (Baer fl]) Let G be a primitive finite group with stabilizer M. Then
exactly one of the following three statements holds:
(1) G has a unique minimal normal subgroup N. this subgroup N is self-centralizing
(in particular, abelian}, and N is complemented by M in G;
(2) G has a unique minimal normal subgroup N. this N is non-abelian, and N is
supplemented by M in G.
(3) G has exactly two minimal normal subgroups N and N*. and each of them is
complemented by M in G. Also Cc(N) " N*. CclN*} = N, and N Si N* « NN* m M.
Moreover, if V < Gand VN = VN* - G. then V r\ N = Vr\N* = I.
Notation. We will use ф to denote the class of all primitive groups in the universe
under consideration, and if G e Ф. we set G e 4), if G satisfies Statement i (i — 1.2 or
3) tn the above theorem. Thus 'B = D, О О Dj.
Proof. Let G be a primitive group with stabilizer M. Let I # К s G. and set
C - Cc(K) (s G). Since К 4 M. we have
(15. a) KAf-G.
Moreover, since C n M is normalized by M and centralized by K. we also have
C ri Af s KM = G and therefore
(15.0) Cn.tf-I.
If D 4 G with UDSC, then C — Cr^DM — D(CnM) D. and therefore
(IS.;-) either(i) C - I.
or (ii) C is a minimal normal subgroup of G.
We now consider in turn each of three possible cases that can arise.
11) G has an abelian normal subgroup N Take К = .V above Since
As C G.lAi. n follows from (15.;) that N = Cc(.V) and hence that N is the unique
minimal normal subgroup of G- From (15.s) and (15.0) we conclude that M comple-
ments N in G.
(2) G has a unique minimal normal subgroup N and N is non-abelian. In this case
NM -Gby(l5.z).
(3) Neither tl) or (2) obtains. In this case G has at least two minimal normal
subgroups. .V and N* say. and both are non-abelian. Take К - N above. If N** were
a third minimal normal subgroup, we should have N*N** S C - Cc(N}, against
(15.y). Therefore N and N* are the only minimal normal subgroups of G. Moreover.
(15.7) implies that Cc(JV) = N*. and so M complements .V* by (15.x) and (15.0)
By the same reasoning Cc(<V*) “ A' and M also complements N in G. Next, ap-
plying the modular law and an isomorphism theorem, we have N(NN* r\M}~
NN* n NM = NN*, and so
N* a NN*/N - N(NN* n M}/N s NN* m M.
A. Prerequisite*—general group theory
prime order, and therefore by Pan la) the group G/K is generated by subnormal
subgroups of prime order in particular, by (8.81(b) it is therefore nilpotent. We
suppose that GiK has a cyclic subgroup V/K with lE'KI - p' (ре P) and denve a
contradiction. Let S be a minimal subnormal supplement to И in K. Since P sn G by
(8.3) and P is single-headed, it follows that Se.Z and hence by the definition of К
that ISK/KI is I or a prime. But i5K/K| |V/K| « p*. and we have the desired
contradiction. Therefore the Sylow p-subgroups of G/K have exponent p, as claimed.
□
The following technical lemma will be needed in Chapter XI. Section 4.
114.17) Umnu. Let N, and N. be normal subgroups of a finite group G such that G/N,
is single-headed for i » 1.2. and G 4 N, N2. Let Shea minimal subnormal supplement
to N, N2 in G. Then
(a) S is a single-headed group satisfying S/lSnN,)3i G/N, for i 1.2
lb) If G/N, N2 is a p-group. then SAS rv N,)(SnN2) is a non-trivial cyclic p-group.
Proof, (a) By (14.15) the supplement 5 is single-beaded Let Af/N, be the unique
maximal normal subgroup of G/N,. Since N, N2 A G. we have N, N2 £ M. and so
if SN, were contained in .W. we could conclude that G - SN, N2 £ .VfN2 =• M,
which is absurd. Hence SN, = G and G/N, SN.N, a S.-fSnN,). Similarly
GiN. a S/tfnN.).
lb) We will prove this assertion by induction on iGi. Since G/N,N2 ;S a single-
headed p-group. it is cyclic, and if S •» G. there is nothing further to prove. Therefore
suppose that SsKsG and set К, = К nN, for t = I. 2. Since KN, « G. we
have K/K, 3 G/N, and SK, = K. Thus K/K, is single-headed (i = I. 2). and S is a
minimal subnormal supplement to K,K2 in K. Now K.(K nN, N2) — G/N,N. is a
non-trivial cyclic p-group. and [K n N, N-. K] S (N,. K][N«. K] S К, К2. whence
(K nN,N2b'K,K. s ZfK/K,K2). Thus K/K,K. is a centre-by-cydic group, is
therefore abelian, and since it is single-headed, it is a cyclic p-group. Since the
hypotheses of Part (b) arc satisfied with К. K,. K: and S in place of G. N,, N- and
X. we can conclude by induction that S/(S n К, )(Xn K2)( - X.fX n N, )(Sn N.i is a
non-lnvial p-group. □
15. Primitive finite groups
(15.1) Definition. A finite group G is called primitive if it has a maximal subgroup .M
such that Corec(M) * I. In this situation we call M a stabilizer of G.
The term "primitive'' refers to the fact that the transitive G-sel G/M affords a
faithful permutation representation of G which is primitive: the subgroup M and ns
conjugates in G are the point-stabilizers in this representation.
The first fundamental fact is that primitive groups fall into three different categories,
the first of which includes all the soluble primitive groups.
54
A. Prerequisites—general group theory
Similarly N = NN* c M. Finally, if V < G with VN = VN* = G, then Fn N <
<F, N*> = G, whence I'nA1 = I. Likewise knN* = 1. □
(15.3) Examples. Each of the classes (i = 1,2, 3,) is non-empty.
(1) Sym(3) belongs to 5J3t; its stabilizers are Syl2(Sym(3)).
(2) Any non-abelian simple group G belongs to the stabilizers are all the
maximal subgroups of G.
(3) If G is a non-abelian simple group, then B = G x G is primitive with the
diagonal subgroup {(g,g):ge G] as a stabilizer. Clearly D e ф3.
(15.4) Lemma. Let G be a non-trivial finite group.
(a) If M <• G, then G/Corec(Af) is primitive.
(b) If К <G and G/K is primitive, then G has a maximal subgroup M such that
К = Co rec(M).
Proof, (a) This is immediate from the definition because A//CoreG(Af) is a maximal
subgroup of G/Corcc(M) with trivial core.
(b) Let M/К be a stabilizer of G/K. Then M < G, and Corec(M) = K. □
(15.5) Proposition. Let H/K be a complemented abelian chief factor of a finite group
G, and let Mbea complement to H/K in G. Let R = CG(H/K) and S = CM(H/K). Then
S = Corcc(M), G/S is a primitive group of type 1, and R/S is the unique minimal normal
subgroup of G/S. Furthermore, R = HS, H nS = K, and R/S is G-isomorphic with
H/K.
Proof. By (9.10)(b) the complement M is a maximal subgroup of G. Since [Я, S] <
К < S, we have H < NG(S) by (7.4)(b); therefore S MH = G, and so S < Corec(M).
On the other hand, [H, Corc(;(M)] < H г.М = K, and therefore Corec(M) < S.
Consequently S = Corec(Af), and G/S is primitive by (15.4)(b). Since H < R, we
have R = RciHM = H(RcsM) = HS, and H csS = K. Hence R/S = HS/S g
H/(H n S) = H/K', in particular, R/S is an abelian minimal normal subgroup of G/S.
ThereforeG/S e 43], and R/S is the unique minimal normal subgroup of G/S by (15.2).
□
Primitive groups play a very important role in the study of finite soluble groups.
The next theorem shows that a primitive soluble group is the semidirect product of
an elementary abelian p-group N with a soluble subgroup H of Aut(/V). In the
language of representation theory, N is a faithful irreducible H-module over the finite
field Fp, and for every such module the semidirect product [/VJH is primitive. The
construction and analysis of primitive soluble groups is therefore closely tied up with
representation theory.
(15.6) Theorem. Let G be a primitive soluble group with stabilizer M.
(a) G has a unique minimal normal subgroup N, the stabilizer M complements N in
G, and N = CG(N) = F(G).
15- Primitive finite groups
(b) If p is the prime dividing | N|, then О (M) = 1
(c) All complements to N in G are conjugate to M.
Proof, (a) By (15.2) we know that G has a unique minimal normal subgroup W that
M complements N and that /V = Cc(N). By (10.6)(b) we have F(G) < CcJv) = N
and therefore N = F(G). ' ' ’
(b) Let P = Op(M). Then NP is a normal p-subgroup of G, and so P < F(G\ r-,M =
N riM =1.
(c) Let L be a complement to N in G. If L = 1, there is nothing to prove Therefore
suppose that N < G, and write R = O„„.(G). Since N = 0,(G) and G is soluble,
it follows that N<R and that |R:/V| is a p'-number ± 1. Now N(LoR) =
NL r\R = R, and therefore L n R is a complement to N in R. Since R < G, we have
L < Ng(L r . R}^ G because Op.(G) = 1. Since L < G, it follows that L = NG(L о R),
and by the same argument M = NG(M n R). By the Schur-Zassenhaus Theorem 11.3
the p-complements Lr.R and M n R of R are conjugate in G (in fact, in R), and
therefore so are their normalizers. n
The next elementary observation is obvious from (15.4) and (15.6)(b).
(15.7) Corollary. Let G be a finite soluble group. Then the map
MG -»G/Corec(M)
is a bijection from the set of conjugacy classes of maximal subgroups of G to the set of
primitive quotient groups of G.
(15.8) Proposition. Let G be a finite group.
(a) G is primitive of type 1 if and only if (i) G has a unique minimal normal subgroup
N, (ii) N is abelian, and (iii) N f Ф(С).
(b) Assume that G is soluble. Then G is primitive if and only if G has a self-
centralizing minimal normal subgroup.
Proof, (a) If G e ф,, then G has the stated properties by (15.2). Conversely, suppose
that G satisfies (i), (ii), and (iii). Since N is abelian and not Frattini, by (9.10) it is
complemented in G, by M say. Let К = Corec(M). If К 1, then К contains a
minimal normal subgroup of G distinct from N, contrary to hypothesis. Therefore
К = 1, and G is primitive.
(b) Theorem 15.2 implies the necessity of the condition. To prove its sufficiency,
let N = Cc(/V)<) G. By (10.6)(b) we have F(G) < Cc(W), and therefore N = F(G).
Since <D(G) < F(G) by (10.6)(c), we conclude that ®(G) = 1. Therefore G is primitive
by Part (a). D
The next result shows that the set of all primitive epimorphic images of a soluble
group G carries a lot of information about the structure of G/<P(G).
(15.9) Proposition. Let G be a finite soluble group. Then G/<P(G) >s isomorphic with a
subdirect subgroup of the direct product of the primitive eptmorphie tmages of G.
56
A. Prerequisites—general group theory
Proof. By (10.6)(c) the section F(G)/O(G) is the product of all the complemented chief
factors of G of the form Н,/Ф(С) for ' 61- By (15.5) there exist primitive epimorphic
images G/Sf of G such that S; n H, = Ф(С). Let R = Qie/S, (<JG). Suppose, for a
contradiction, that R f Ф(С). Then there exists an i e I such that Ht/<I>(G) < Л/Ф(С),
and then Hi < S„ against Sf nW, = Ф(С). Hence R = Ф(С), and by (4.17) the group
G/®(G) = G/R is isomorphic with a subdirect subgroup of the direct product
X,eI(g/s,). □
Concluding Remarks, (a) If G is a primitive soluble group, we can write G = NM
with N = Soc(G) and M a stabilizer; then all complements to N in G are conjugate
to M. It is not unreasonable to hope that such groups might posses a stronger
property, akin to the D-property of Hall subgroups, namely the property that if V is
a subgroup of G satisfying V oN = 1, then U is contained in a conjugate of M.
Unfortunately, this is not the case: a counterexample is described in the final para-
graph of Example VIII, 2.19.
(b) For the study of projectors in universes containing insoluble groups, it is
important to know whether primitive groups can have more than one conjugacy class
of stabilizers. For groups of type 1 this problem can be resolved with the help of the
first cohomology group. First we need the following theorem, a proof of which can
be found in [H] I, 17.3.
(15.10) Theorem. Let A be a finite abelian group which is also a G-module, and let S
denote the semidirect product S = [A]G. Then |7f‘(G, A )| is the number of conjugacy
classes of complements to A in S.
For simple modules the vanishing of the first cohomology group is controlled by
the second Loewy layer of the principal indecomposable module (see B, 4.20(a) for
its definition). This result appears in Exercise 34 on page 122 of Huppert and
Blackburn [1].
(15.11) Theorem. Let К be a field, G be a group, and A a simple KG-module. Then
Hl(G, A)^0 if and only if A is isomorphic with a composition factor of
where P, is the principal indecomposable KG-module and J = J(KG), the Jacobson
radical of KG.
If G is p-soluble, Gaschiitz’s Theorem B, 6.18 states that a simple FpG-module
appears as a composition factor of the second Loewy layer of Pl if and only if it is
isomorphic with a complemented p-chief factor of G, and so we can deduce the
following.
“Let G be a p-soluble group, and let V be a simple FpG-module. In the semidirect
product [F]G, the minimal normal subgroup V has more than one conjugacy class
of complements if and only if G has a complemented chief factor isomorphic with E”
Of course, in this case Cc(F) contains Op. P(G) by A, 13.8(a), and so [F]G cannot
be primitive unless G = 1.
Even when G is not p-soluble, the composition factors of P, belong to the first
block and have Op. p(G) in their kernels by Brauer’s Theorem B, 4.23(b). Thus a
SI
16. Maximal subgroups of soluble groups
necessary condition for a primitive group G of type t to have more than one
conjugacy class of stabilizers is that O,..,(G/Soc(G)) = 1. It is obvious from (15.10)
and (15.11) that another necessary condition is that P.J 0 and by В 46 and
B, 4.13 this holds if and only if p||G/Soc(G)|. 5 ’
Any non-abelian simple group S whose order is divisible by p appears as a stabilizer
of a primitive group of type 1 with more than one conjugacy class of stabilizers
For certainly P, J A 0, and we assert that S does not centralize For if
[Pj J, S'] < Pj J2, and if v g Pj \ P, J, then the map
g-*v(g- O + PjJ2
is a homomorphism from S to the p-group PJ/PJ2 and must therefore be the zero
map. But then S has trivial action on P, /Pt J2, which is consequently semisimple, a
contradiction. It follows that some composition factor, V say, of P, J/Pi J2 is non-
trivial, and hence faithful for S. It then follows from (15.10) and (15.11) that [F]S is
a primitive group of type 1 with more than one conjugacy class of stabilizers.
16. Maximal subgroups of soluble groups
The set of maximal subgroups of a finite soluble group G has some interesting
properties, not found in all finite groups; for example, if L, M are inconjugate maximal
subgroups of G, then L r> M is maximal in at least one of L and M. The elementary
facts presented in this section provide some answers to the following two questions
about a pair of maximal subgroups L and M of a finite soluble group G:
(i) When is L conjugate in G to M?
(ii) What can be said about L n M?
(16.1) Theorcm(Ore[lJ). Let Land M be maximal subgroups of a finite soluble group
G. Then L is conjugate in G to M if and only if Corec(L) = Corec(M).
Proof. For any subgroup V it is obvious that CoreG((J) ~ CorcG(C'fl) for all g € G.
Therefore L and M have the same core when they are conjugate. Conversely, if
К = Corcc(L) = CorcG(M), then L/K and M/K are complements to the socle of the
primitive group G/K; by (15.6) they are conjugate in G/K, and therefore L and M are
conjugate in G.
The next theorem shows that two inconjugate maximal subgroups of a finite soluble
group always permute.
(16.2) Theorem (Ore [1]). Let L and M be distinct maximal subgroups of a finite
soluble group G. Then any two of the following statements are equivalent;
(a) L is conjugate to M in G;
(b) LM A G;
(c) LM is not a subgroup of G.
58
A. Prerequisites—general group theory
Proof, (a) => (b): Suppose that M = U L and that LM = G. Then g = Im with
I e L and m e M. Consequently M = Llm = Lm and so M = Mm~‘ = L, contrary to
hypothesis. Therefore LM 4 G when (a) holds.
(b) => (c): If LM were a subgroup of G, the hypothesis that L^M <G would
imply that LM = G.
(c) => (a): We suppose that L and M are not conjugate in G, and conclude that
LM = G. Let К = Corec(L) and R = Corcc(M). By (16.1) we have К R and can
therefore suppose without loss of generality that R £ K. But then R f L, whence
LM > LR = G, and therefore LM = G. □
Our next goal is to describe the intersection of two conjugate maximal subgroups
of a soluble group G. If G is primitive with socle N = {1, n2,..., n,} and stabilizer
M 1, then clearly 7Vc(Af) = M and {M, MfM"'} is a complete set of distinct
conjugates of M in G. It follows at once from Part (a) of the next lemma that
M"‘ n M"1 = the centralizer in M"‘ of the element nflnr
(16.3) Lemma. Let G = NH be a semidirect product of a normal subgroup N with a
subgroup H.
(a) If ne N, then H суН” = C„(n);
(b) Corec(H) = CH(N).
Proof, (a) Let he H n H". Then h = k" for some ke H, and it follows that
k 'h = (k^n^kjn e N. Since k~'heH and HcyN= 1, we conclude that h = к
and h e CH(n). Hence HrH'< CH(n), and since the reverse inclusion is obvious,
Assertion (a) is proved.
(b) Since Corec(H) <J G, we have [N, Corec(H)] < N n H = 1, and so Corec(//) <
CH(N) (this also follows directly from (a)). On the other hand, H normalizes and N
centralizes CH(N), and therefore CH(N) < HN = G. Hence CH(N) < Соге0(Я), and
equality holds. □
(16.4) Proposition. Let H/K be a chief factor of a finite soluble group G complemented
by a (maximal) subgroup M of G. If L is a conjugate of M, then L = Mh for some
he H and
LriM = {m e M : [h, m] e М].
Proof. If M complements H/K, then M < G by (9.10)(b). Let L = M9 for some
g e G = MH. Writing g = mh with m e M and he H,we have L = Mh. Since G/K is
the semidirect product of H/K with M/K, by (16.3) we have (L n M)/K = CM/K(/iRj,
and therefore Lc-.M = {me M :hmK = hK} = {m e M: [h, m] e K} =
{me M: [h, m] e H n M) = {tn e M : [h, m] e M] since [h, m]e H for all me M.
□
Having described the intersection of two conjugate maximal subgroups, we now
turn to the intersection of two inconjugate ones. The following partial order on
16. Maximal subgroups of soluble groups 59
results^ ClaSSES °f maX'mal subBrouPs helPS in the efficient formulation of our
(16 5) Definition. Let L and M be maximal subgroups of G. We write Lc < MG if
and only if Corec(L) < CoreG((W). -
R^arks 1. It is obvious that the relation < just defined is transitive. Furthermore
if L M and M < L , then Corec(L) = Corec(M), and consequently L° = MG
by (16.1). Therefore < really is a partial order on the conjugacy classes of maximal
subgroups of G.
2. It is obvious that the maximal normal subgroups of G are the maximal elements
in this partial order. Also, it is not difficult to show that if M is a maximal supplement
to F(G) in G (a so-called critical maximal subgroup of G), then MG is a minimal element
in this partial order.
(16.6) Theorem. Let L and M be inconjugate maximal subgroups of a finite soluble
group G. If Mc jg L°, then LicM is a maximal subgroup of L.
Proof. Let R = CoreG(M) and S/R = Soc(G/R). The hypothesis implies that
R jt Corec(L) and therefore that
(16.a)
LR = G.
Since G/R is primitive, S/R is a chief factor of G, and since R centralizes S/R it follows
from (16.a) that S/R is L-irreducible. From (16.a) we also have S = S n LR = (Sr\ L}R,
whence
S/R = (S n L)R/R =(Sn L)/(R n L),
and therefore (S r> L)/(R n L) is a chief factor of L. Now M complements S/R
in G and LM = G by (16.2); hence |L:Lr-.MI > |(Sn L)(Ln M): Lc M| =
|SnL:RnL| = |S:R| = |G:M| = \LM: M\ = \L:Lr\M\. Consequently
(S c.L)(LnM)= L, and Lr- M complements the chief factor (S n L)/(R n L) in L.
Therefore L n M < L by (9.10) (b). □
Since < is a partial order, Theorem 16.6 has the following consequence.
(16.7) Corollary. Let L and M be inconjugate maximal subgroups of a finite soluble
group G. Then Lr>Misa maximal subgroup of at least one of L and M.
(16.8) Proposition (Gaschhtz [6]). Let M be a complement to an abelian питтЫ
normal subgroup N of a finite group G. Then Corec(M) is a complement to Nm C.(N}
Furthermore, if G is soluble, there is a bijection between the set
of complements to N in G and the set Л of complements to N m Cc(/V) that are normal
in G.
60
A. Prerequisites—general group theory
Proof. Let К = CoreG(M). Then by (16.3)(b) we have NK = N(M n CG(N)) =
NM n Cc(N) — Cc(N) since N is abelian. Because Nr. К <Nr.M= l,we conclude
that К is indeed a complement to N in CG(N).
For MB e Jt define r(Mc) = CoreG(M). Then r is an injection from .it to Ж by
(16.1) and the preceding paragraph. If К e .Г, then CG(NK/K) = CG(/V)/K = NK/K.
Therefore by (15.7)(b) the group G/K is primitive, and if L/K is a stabilizer, then
clearly Pe.H and CoreG(L) = K. Therefore т is surjective. □
(16.9) Corollary. Let L and M be inconjugate complements to a minimal normal
subgroup N of a finite soluble group G. Then Lr. M isa maximal subgroup of L and M.
Proof By (16.8) the classes LB and MB are incomparable in the partial order <, and
Theorem 16.6 then yields the desired conclusion. □
17. The transfer
The transfer map* v is a homomorphism from a group G into an abelian section
of G. It evolved out of a technique developed by Burnside, which exploited the
determinant of a monomial representation (namely, a representation induced from
a linear representation of a subgroup). Its main application is in establishing the
existence of a proper normal subgroup of G (as the kernel of v when t>(G) # 1). Since
finding normal subgroups is the least of one’s worries in the study of finite soluble
groups, until recently the transfer has not played a significant part in this area. But
it now transpires that ideas related to the transfer are important to the understanding
of normal Fitting classes (see particularly Section 5 of Chapter X). In this short section
we present only enough of the bare bones of the theory to satisfy our subsequent
needs.
(17.1) Definition. Let H be a subgroup of a finite group G and let {rt,.... r„} be a
right transversal to H in G. For each g e G and i 6 {1,..., n} we obviously have
П9 =
for some h^g) eH and some permutation i -»ig of the symbols {!,..., n}. If
H' < N < H, we define the map v = cG_I)/N: G -» H/N by
(17.a) gV = f] MsJN
i=i
and call v the transfer (map) from G to H/N. (Note that the order of the product on
the right-hand side of Equation 17.a is unimportant because H/N is abelian.)
The traditional symbol v is derived from “Verlagerung", the German word for transfer.
дггт -
17. The transfer £.
ol
(17.2) Theorem ([H] IV, 1 4 15 and 1 71 1 Pt H/м h„ к i-
* ’ , . L J ’ Let H'N be an Delian section of a finite
group G, and let v = vG^H/N. J J
(a) The map v is a homomorphism.
(b) The definition of v is independent of the choice of the transversal to H in G
t e ° Н^‘~'} {with HS^' = Hs‘)be the orbits of 9 on
G/n (i = 1,f(g)j. Then
He)
go = П W‘s,7‘)n.
i=l
Furthermore, = |G : H[ and s^-sf1 e H; in particular, av^cF-11' when
geZ(G).
(17.3) Lemma. Let H' < N < H < G.
(a) If 9> x e 6 and gvc_Hlfl = hN with he H, then gvG^HXIKX = hx№.
(b) If m e M < G, there exists an element he H c\M such that mvG~„/N = hN.
(c) Let К < H, and for g e G let gvG^ll/H. = hH' with heH. Then gvG .KIK- =
hVu-,KiK'-
(Since H‘ < Ker (гн_К/К.), by a slight abuse of notation the conclusion of (c) may be
formulated thus: vc_,IJtlr ° VH.K/K. = vg-.kik'-)
Proof, (a) If {rL,..., r„} is a transversal to H in G, then {rf,..., rx} is a transversal
to Hx in G, and if r,g = hfg)rifl, then rxgx = (hjtgffr?. Therefore by (17.2)(a) and (b)
we have = (йг’о -и/л'Г since gv = gxv for any homomor-
phism v from G into an abelian group.
(b) By (17.2)(c) we have mvG^H/K = ]”[(s1m/‘s1“I)N with Sjn/'s,"1 eHoM.
(c) This is proved in [H] IV, 1.6. □
(17.4) Definition. Let H be a subgroup of a finite group G. The focal subgroup of H
in G is denoted by {H: G} and defined as follows:
{H; G} = <[/i, gf. h e H, g e G, and [h, g] ~ H).
(Obviously {H; G] < H since H' < \H; G}.)
(17.5) Focal Subgroup Theorem. Let S e Syl„(G) for some prime p, and let n = |G: S|.
Let v = uG^s/S'- Then
(a) xv = x"(modulo {S; G}) for all xeS, and
(b) Ker(o) n S = G' nS = (S; G],
Proof, (a) Let л e S. By (17.2)(c) the following equations hold
x„ = П = n ^(>-4xV1)S'
1-1 '-1
= n 5-74s' = П xf‘ = x" <modul° G'*'
62
A. Prerequisites—general group theory
(b) Let К = Ker(t>). Since G' < K. we have S' < {S; Gj < S n G' < S n K. If
л 6 S n K, then x" = 1 (modulo {S; G}) by (a), and hence x" e {S; G}. Since (p, n) = 1
and x has p-power order, it follows that x e <x">; therefore S n К < {S; G}, and
Conclusion (b) is clear. □
18. The wreath product
The wreath product is a special kind of a group construction. Its theoretical value
derives from its connection with group extensions: if N < G, then G can be embedded
as a subgroup of the wreath product Nrtitcg(G/N) (see Theorem 18.9 below). Its
practical value is its amenability to calculation.
The wreath product is explicity represented as a semidirect product in which one
group acts on another simply by permuting its directs factors, and this often makes
it easy to check by direct calculation that groups constructed as sections of wreath
products have sought-after properties. Again we hold to our guiding principle of
presenting only those (in part very specialized) results which we will need later; for a
balanced and comprehensive survey of the theory of regular wreath products we refer
the reader to P.M. Neumann [1].
(18.1) Definition. Let X and G be finite groups, let fl be a set with cardinality n, and
assume that either fl is already a G-set, or else that a permutation representation
a: G -» Sym(fl)
is given, in which case fl is converted into a G-set in the usual way via equation (5.y):
Ю.Ч = m(»a)
for all cj e fl and g e G. Let
В = X xa S X)
well
denote the direct product of n copies of X. We shall sometimes denote the elements
of В as maps f: fl -► X (for example when fl = G, viewed as the right regular G-set),
and sometimes as n-tuples(xj,..., x„) when fl = {1,.... n}. We now define an action
of G on В as follows.
*18*) (х1,...,хп)9 = (х19-„...,хад-,).
Thus the ith coordinate of (xj,..., x„) appears as the (ip) th coordinate of (xj,..., xn)9,
and it is clear that g induces an automorphism (call it ag) on B. For g, h e G, we
have
18. The wreath product
63
(х„...,х„Г' = (х1(вЪг.,...>Хв(вЬг1)
— (Х(11г')в > > X(nh Чй1)
and so the map a: G -> Aut(B) is a homomorphism. (Note that if the elements of В
are written as maps f : Cl -> X, the action of g e G on В is defined by fe(a>) = /(cog-1)
for all co e fl.) It is clear from the definition that the image Xf of the direct component
Xt of В is Xig, in other words that g permutes the direct components of В according
to its permutation action on fl.
The semidirect product [B]G via a is called the wreath product of X with G with
respect to a (or with respect to fl if a G-set, rather than a permutation representation,
is given). It is denoted by X Qjo G (or by X % G). We will frequently identify and
G with their corresponding subgroups of A'rLa G. If Y < X, we set
^ = {(Ь.-,К):у,еГ}<В,
and, in particular, X" = B, which is called the base group of X G. If G is a permuta-
tion group (i.e. a subgroup of Sym(fl)), then the inclusion map is a natural permuta-
tion representation, and we denote the associated wreath product by X 1jna, G in this
case.
The elementary facts contained in the following lemma follow directly from the
definitions.
(18.2) Lemma. Let W = Xr\jaG.
(a) If Y < X, then Y11 is normalized by G, and if Y <! X, then Y’ < W.
(b) if H < G and p = a„, then X'H s X % H.
(c) IfY<X, then Y'G S YQj„ G < W.
(d) If Y < X, then W/Y^ = (А7У)Ч G.
The next lemma will help us to describe, among other things, the derived subgroup
of a wreath product (see (18.4)(d)).
(18.3) Lemma. Let 1 < n e N, and let X and G be subgroups of a finite group W such
the normal closure В = <X” > fc a direct product of the distinct conjugates
X, XK\ .... Xw" of X in W (here {1, w2,..., w„} is a transversal to NH.(X} in W), and
(ii) W = BG.
Let
м = {x,... X„w":X,e X and хгхг ...x„e X'}.
П(а) M is a subgroup of G and is generated by the set У = {x *x '-хе X and
2<i< n};
м
A. Prerequisites general group theory
(b) В' < M = [B, GJ;
(c) If X is non-abelian, then [B, G] > |X|;
(d) For 1 < i j < n let Nt and N; be non-central minimal normal subgroups of X"1
and X"1 respectively, and if n = 2, assume further that [Nt, (A'"')'] # 1 # [A/,-,
Then If and If are minimal normal subgroups of [B, G] and are non-isomorphic as
[B, G]-modules.
Proof, (a) Let S denote the subgroup generated by the set У. Since XfX' is abelian,
the condition:
XiX2....x„6X'
is independent of the order of the terms in the product, and it follows easily that M
is a subgroup of W containing S.
If a, b e X, then [a, b] = a^laW2b~,bW2(ab)((ab)~1)'"1 e Sf, and so S contains X'.
Let x,x2 ... x„e X' and set m = xLx22 ... x"", a typical element of M; further,
set у/ = xjx,'1 )w‘ for i = 2, ..., n, and note that у; e У. Then my2 ... y„ =
xtx2 ... x„ e X' < S. It follows that me S and hence that M = S.
(b) By definition of M we have (X-)"'1 < M and therefore
В' = X' x (X'fF2 x • • • x < M.
Let x, у 6 X. Because of Hypothesis (ii), there exists an element g e C such that
g = bw2 for some b e B. Since the subgroup [B, G] is normal in В and contains
x~1(x‘)“’2, it also contains x1(xf’)“2(x(x~f’)‘'2)>, which equals [x, y] because у central-
izes XW2. Therefore [B, G] contains X' and hence its normal closure B'. If w; = bg
(with be В and g e G), the equation
x_1xW| = x-1x9[x, b]₽
shows that M, which is generated by V, is contained in [B, G]B' = [B, GJ. The
equation also shows that [B, G] < MB' = M, and therefore Assertion (b) holds.
(c) This follows from the fact that [B. G] is subdirect in В by (a) and 1 # B' <
[B, G] by (b).
(d) Since Ni Z(XW‘), evidently we have Nt < (X"1)'. Thus Nt < В' < [B, G] by
Part (b). Let Cf = Ce(A/,). Since < Ct, and since [B, G] is subdirect in В by
Part (a), we have <" [B, G] = В = CiXWl, and so X*1' and [B, G] induce isomorphic
groups of automorphisms on If. Hence If is a minimal normal subgroup of [B, GJ.
Suppose that n > 3, and let {i, j, k} be a 3-element subset of {1,..., n}. Let y, be an
element of Xw> which does not centralize Nj, and let a be an element of G such that
y“ e X“\ Then y^yf is evidently an element of С[в С|(^)\С[В С|(^.). Thus and Nj
have different centralizers in [B, G] and are therefore certainly non-isomorphic as
[B, G]-modules. If n = 2, then [W;, (Х*'У] = 1, by hypothesis [N(, (A'"')'] # 1, and
the same conclusion holds. □
Next we describe some important subgroups of a wreath product.
18. The wreath product
65
(18.4) Proposition. Let X and G be finite groups and leta-C^t < > c
permutation representation of degree n >2 Let W Y^с ** transitive
group. Further, set W = * G and В = X\ the base
D = {(x,...,x):xeX},
the diagonal subgroup of B, and set
(18./?) M= {(x1,...,xB):xieXWxIx2...x„eX'}.
Then
(a) C„,(G) = D x Z(G) and N„.(G) = DxG
(b) [B, G] = M > B',
(c) <GH'> = MG, and
(d) W'= MG'.
Proof, (a) It is clear from the transitivity of a that CB(G) = D. Let c e C„,(G), and
write c = bg with be В and geG. For an arbitrary x e G we have bg = (bg)x = bxgx,
and therefore b~lbx = g(gxyl eBc.G=\. Therefore b e CB(G) = D and g e Z(G)’
and thus C1(,(G) < D x Z(G). Since the reverse inclusion is obvious, equality holds.
Furthermore, we have N^(G) = NW(G) r. BG = (N„.(G) n B)G = NJG)G = DG =
D x G.
(b) Since В is direct product of the subgroups {X9:geG}, where X =
{(x, 1,..., l):x e X}, the hypotheses of (18.3) are satisfied with the subgroup defined
by (18./?) corresponding to the M of that Lemma. Thus Assertion (b) here follows
from (18.3)(b).
(c) By (7.4)(h) we have (Gn) = G[G, W], and so, appealing to (7.4)(f), we con-
clude that <GH'> = G[G, GB] = G[G, G] [G, B] = GM by Part (b).
(d) Set X = MG'. Then X < W' by Part (b). On the other hand, since [G, B] =
M < X, we have W/X = GB/X = (GX/X) x (BX/X), and since В' < X by Part (b),
it follows that W/X is abelian. Thus W < X, and Assertion (d) follows. □
(18.5) Proposition. Let X and G be finite groups, let a: G -» Sym(n) be a transitive
permutation representation, and set IV = X rlj„ G.
(a) If Y is a minimal normal subgroup of X and Y Z(X), then Yb is a minimal
normal subgroup of W.
(b) Assume that XandG are soluble, that a is faithful, and that X is non-abelian and
primitive. Then W is primitive, and Soc(W') = SocfX)".
(c) Let p be a prime. Then every finite soluble group is a proper epimorphic image
of a primitive group whose socle is a p-group
Proof, (a) Let У* = У, x ••• x У„ < X" = X, x ••• x X„, and let 1 A N < P with
N < W. Then there exist elements y,, e Yt (1 < i < n) such that y,J'2... J„e N, and
without loss of generality we may assume that yi # 1. Since Y £ Z(X) there exists an
element x e X] such that у J / У i - Then the element n ~ (_y t y2 • • Ул) (Уз1У2 • -У»)
-f • -ii » v /-ч л/ «since У. N <J Xi and К *<Xi, it follows
yjyf is anon-tnvialelementof ii n/V. мпсе i_ i,
66
A. Prerequisites—general group theory
that У, < N. and then from the transitivity of a we obtain x • • • x Y„ < N. There-
fore У" = N.
(b) Let У = Soc(X). Then У = СЛ(У) by (15.8)(b), and so clearly У" = Сд^У").
Furthermore, since X is not abelian by hypothesis, we have У / Z(X), and therefore
У6 < W by Part (a). Let и e IT\ X". Then и = xg with x e X" and 1 # g e G. Since
a is a faithful representation, there exists an i e {1.n} such that ig / i, and if
1 56 у e Yh it follows that
yu = y*<> = y> e Yig * Y,,
whence y“ # y. Consequently C^fT11) < X", and so СЖ(У") = Слв(У") = У", Be-
cause X and G are soluble, so also is W, and therefore W is primitive by (15.8)(b).
(c) Let G be a finite group, and let X = E(q/pY the semidirect product of Zp with
Zg(< Aut(Zp)), where q is some prime dividing p — 1. Since the regular permutation
representation a is both transitive and faithful, the group X Qj,, G is primitive by Part
(b) and has G as a proper epimorphic image. □
Remark. Proposition 18.5 (b) is true under much weaker assumptions (for example,
the solubility of G can be dispensed with), but for later applications we only need this
soluble version, which has a shorter proof.
(18.6) Lemma. Let X and G be finite groups, and let a : G -» Sym(n) be a permutation
representation. For any me N there exists a monomorphism
t.(X"’)QjeG-(XQjeG)m
such that
(i) t((Xm)") = (X"y”, and
(ii) т((Х",)гЬо G) is subdirect in (X G)m.
Proof. It is obvious that the map t defined by
Т(((Х]], ..., Xm]),..., (X in, • * » Xmn ))> 9) = ((-xi 1, • • •, xin)9. - - •, (Ci, • •, xm„)s),
where хц e X and g e G, has the desired properties. □
Next we turn our attention to a particular case of the wreath product which has
some important special properties.
(18.7) Definition. If fl = G, the regular G-set, or, equivalently, if a: G -> Sym(G) is
the regular permutation representation (see (5.3)), we call the associated wreath
product the regular wreath product and denote it by
X%egG.
If /: G -»X is an element of the base group X", the G-action is defined by
18. The wreath product
r(h) = /(/.r1)
67
for an h, ge G; moreover, if we write Xй = X„6 GXh, then (Х„Г = X„9 for all h, g e G
Since the regular representation is both transitive and faithful, Propositions 18.4
and 18.5 are both valid for the regular wreath product.
(18.8) Lemma. Let X and G be non-trivial finite groups and let W = X Qj G
(a) If H < G, then X'H s (X|G"') foree H. "e ’
(b) If N < W and N n Xй = 1, then N = 1.
(c) Let N be a normal supplement to Xb in W. Then N contains the subgroup M of
Xй defined in (18./?). J
(d) If X is perfect and G is single-headed, then W is single-headed; if, further, G is
also perfect, then W is perfect.
Proof, (a) Let G = g,H be a left coset decomposition (with m = |G : H|), and
for he H set
li = X91(.x-xX9m„sX".
Then Xй = Х),бИ Th, and since H acts on the direct components Th according the
right regular representation, it follows that W — XbH s (X'”)rb„8 H.
(b) Let xgeN with x e Xй and geG. Since (TV, Xй] < N n Xй = 1, we have
X. = (X. )xs = (X, )* = X., and therefore g = 1. But then x e NnXй = 1, and it
follows that N = 1.
(c) In preparation, we make the following assertion:
(18.y) Let и be a given element of G, and let w = gf with 1 ¥ geG and f e Xй.
Then there exists an element he Xй such that h ‘wh = fog with fB(u) = 1.
To prove this, let T be a left transversal to (д') in G. Since g =£ 1, we can suppose that
и f T. Let o(g) = n, and define an element h e Xй recursively as follows:
h(t) = 1 for all t e T, and
W) = f(tg‘) 'h(tg‘-') for 1 < i < n - 1 and all t e T.
From this definition we have
(18.<5) Л(а) = /(аГ1Л(с0“1) for all a eG\T.
Now
h~lwh = IG'gfh = gh~sfh = g/i,
where
,/ofo) = (h(ag'l))-'f(a)h(a),
68
A. Prerequisites—general group theory
and so/0(a) = 1 for all a e G\T by (18.<5); in particular,/0(u) = 1. and Assertion (18.))
is proved.
Now let N < If'= X*N, and let M be the subgroup of X* defined in (18.//),
generated, according to(18.3)(a), by elements of the form m, where m(l) = m(u)~l for
someu e G\{ 1} andm(p) = 1 for all u e G\{1. u}. Since W = NX*,we have uf e N for
some / eX', and by (18.y) there exists a conjugate uf0 of uf such that /0(u) = 1. The
normal subgroup N therefore contains uf0 and hence also [u/0, tv] for all w e W. Let
I be the element of X* defined by T(l) = m(l) and /(u) = 1 for all 1 ve G. Then N
contains
[«/>, Л = ff'u~lrlufol = fo-lr“fol.
Since l~“(v) = 1 for all г A u and/0(u) = 1, the elements/, and / “ commute, and so
N contains /““/ = m. It follows that M < TV, as desired.
(d) Let N be a maximal normal subgroup of W such that X11 / N. Then W = NX*,
and so M < N by (c). But M contains (Х"У by (18.41(c) and (Л"“У = (X')" = by
hypothesis, and it follows that W < N, a contradiction. Therefore X* < N, and
consequently N = X*K, where К is the unique maximal normal subgroup of K. If
G = G', then by (18.4) we conclude that W’ = MG = X*G = W. □
We now come to the fundamental embedding theorem for regular wreath products,
which gives the construction special significance in the abstract theory of groups.
(18.9) Theorem. Let N be a normal subgroup of a finite group G. Then there exists a
monomorphism
p-. G W = NQjreg (G/N)
such that N* p(G) = W and N* p(G) = p(N).
Proof. Let {gt,..., } be a transversal to N in G. If x e G, then Ngtx = Ngt for some
j e {1,..., r), and therefore there exists a unique element nf(x) in N such that
= n,(x)07.
We denote the element j by ix since it clearly depends only on the coset x = Nx and
not on the coset representative x (note that NgtNx = Ng, because N < G). To
simplify notation we will denote the coset gt = Ngt by the symbol i. Then the map
i -»ix is the permutation induced by x on G/N in the right regular permutation
representation of G/N (= {1,....r}). With this notation we obtain nfxylg,^ =
9,(xy) = ЩМваУ = п,(х)пц(у)д^у, and since ixy = i(xy) = (ix)y, it follows that
(18.e)
n,(xy) = П1(х)пй(у).
Define a map p: G -► W by
p(x) = ((n,(x),.... n,(x)), x) e N Qj„g (G/N).
18. The wreath product
69
Then
/r(xy) = ((n,(xy),n,(xy)), xy)
— ((ni(x),..., n,(x))(nljt(y).n„(j')), xy) by (18.e)
= ((njx),.... nr(x))(n,(yX • •. и,(у)Г xy) by (18.a)
— g(x)g(y) by (4.y), the rule for multiplication in a semidirect product.
Hence p is a homomorphism. If x e Ker(p), then x = 1, and consequently x e /V; but
then 1 = njx) = gtxgt ‘.andsox = 1. Therefore pis a monomorphism. It is obvious
from the definition of p that p(G) covers G/N" and also that p(x) e №= if and only if
x e N; thus p(N) = N“ np(G). q
Next we describe some consequences of Theorem 18.9.
(18.10) Theorem. Let p be a prime, and let P be a group of order p". Then P is
isomorphic with a subgroup of the n-fold iterated wreath product
Proof. The wreath product in question is W„, where W„ is defined formally by the
inductive rule: W1 = Zp and 1jreg Z„ for i = 2,..., n. Ifn = 1, then P = Zp =
W,. Let Q be a maximal normal subgroup of P, and since |Q| = p"-1, assume
inductively that Q Q* < W'.j. By (18.9) the wreath product Qc^,zt Z^, contains a
subgroup isomorphic with P. But clearly
Q Qjreg Z„ S Q* Z„ < И/ rb„e Z„ = W„,
and therefore W„ has a subgroup isomorphic with P. □
(18.11) Proposition. If p e P and n e N, let W = Z„„-, 1jreg Z„ and В = (Z^.)’, the
base group.
(a) W contains a subnormal subgroup isomorphic with Z^.
(b) Let M = [B, Z„] and N = MZp, if w e N\M, then wp =1.
(c) W = BN, where В and N are normal subgroups of W of exponent p” when
n>2.
Proof. Statement (a) is clear from (18.9). The proof of (b) requires some calculation.
By suitable choice of notation we may clearly suppose that w -- g mwhere 4
and where m = (x„ x„ .... V-,) with x.x, ... V-. = 1 by (18.4)(b). Suppose
that for i > 1 we have already shown
(18.0 = S'W' Vi. •••- V- 2 •
70
A, Prerequisites—general group theory
Then we obtain
w'+l =g ‘(Xj,. Xg,-.)w'
= 9 "+,,(x„ -•,
= 9 ,i+”(x»- • • • x9Xi, • • •, . Xr-Xg,-.)
on substituting the value of g'w' given by (18.J)-
Since (18.0 holds for i = 1, by induction it holds for all i e KJ, and therefore, in
particular, we have w” = g~’’(I,..., 1) = 1.
(c) By (18.4)(c) we have N < W; moreover W = BZP = BZpN = BN. Since B, and
hence M obviously have exponent p"~‘, it follows from Part (b) that N also has
exponent p"'- □
(18.12) Definition. If <a> = Z„, a cyclic group of order n > 1, the map a a~l is
obviously an automorphism of <a> of order 1 or 2. Thus a group <b> = Z2 is a group
of operators for <a> under the action ab = <T ’, and we can form the semidirect
product [<a>] <b>. This group is called the dihedral group of order 2n, and we will
denote it by Dih(2n). {Note. Different notations for this group around in the litera-
ture.) It has the following well-known (and easily-verified) presentation with genera-
tors and relations;
Dih(2n) = <a, b; a" = b2 = (ab)2 = 1>.
We shall need the following technical result about wreath products with cyclic and
dihedral 2-groups in Chapter X.
(18.13) Lemma. Letj> I,and set Dj= Dih(2J+1) = (a, b; a2’ = b2 = (ab)2 = IfLet
a denote the transitive permutation representation of degree 2j on the cosets of <b>, let
G be an arbitrary finite group, and let W = G Г1Я Dr Then
(a) G^aby^G^D^,
(b) G*(by G2 x (G'2i'‘-l>rbKtZ2),
(c) Gh<ab> S G21 ' rLreg Z2, and
(d) GE<«> = GrLreeZ2J.
Proof, (a) Let N = (a2, ah'). Since (ab)2 = 1 and (a2)”1’ = (a2)-1, it is clear that N s
Dj-t. In particular, |О2: 7V| = 2 and N < If. The elements of N have the form a2' and
a2s+1b, and so neither b nor any conjugates of b are in N. If we SI = D;/<b>, the
Dj-set affording a, then the stabilizer (DJ,., is a conjugate of <b>. It follows that
Na = N r,(Dj)a = 1, and hence that the restriction SlN, which affords aN, is a union
of regular N-sets. But |S2| = |G; <b>| = 2J = |N|. Therefore is the regular N-set,
and is the regular permutation of N. Assertion (a) now follows.
(b) Let w = <b>x e fi. If a>b = co, then x~*bx e <b>, and hence x e CD.(b). Since
Czo/(b) = a21 ’, = z say, it follows that x e x (by and hence that w = (by or
18. The wreath product
71
<*>Z’ Thus the restriction Q<(, has 2 orbtls of length 1 and 2- - 1 (regular) orbits
of length 2, and Assertion (b) is now dear. ' ё
(c) If w = <b>xe Q the supposition that шаЬ = m implies that xabx~' = b and
hence that a xb x b e D,- = which is impossible. Therefore the restriction
mi с /ПЮП °f (regUlar’ °rbitS °f length 2’ and Assenion И now follows
(d) Since <a > is a subgroup of D} of index 2 and contains no conjugates of <b> the
argument of Part (a) shows that Q<o> is the regular <0>-set, and Assertion (d) is clear.
□
The wreath product provides the appropriate framework for describing the auto-
morphism group of a uniquely decomposable” direct product.
(18.14) Proposition. Let X be a directly indecomposable finite group such that
(|X/X’|, |Z(X)|) = 1. Then
Aut(X") = AutfX)^, Sym(n).
Proof. Let D = X”, the direct product of n copies of X. and let В denote the subgroup
of all fl e Aut(D) such that Xf = X( for all the direct components X,- of D. If fl e В
and (.Xj,.... x„) e D, the maps fit,.... Д, defined by
(xf’....,x„A-) = (*i,-,V
are clearly automorphisms of X,, .... X„ respectively, and it is straightforward to
check that the map
...,)?„)
is an isomorphism from В onto Aut(Xj) x ••• x Aut(X„). Now let a e Aut(D). By
(4.10) there exists a permutation л = ла e Sym(n) such that X? = Xilr. If л is the
automorphism of D defined by
Л (Xi, — • X.) -* (XjK-i, —, хпл i),
then clearly ял;1 e B. Consequently, а = рл„е В Sym(n), where Sym(n) is the image
of Sym(n) in Aut(D) under the map л -» л. Thus Aut(D) = В Sym(n) S [AutlXJ x
• x Aut(X„)] Sym(n) s AuKXjQi^, Sym(n). □
Finally, we describe a generalization of the wreath product which allows groups
to be constructed with a more complicated structure than the standard version. The
base group of this construction may be viewed as a generalization of the induced
module U° of a module V for a subgroup H of G; in place of U we have a (possibly
non-abelian) group X which has H as a group of operators.
(IS.,S) и». № (—«I.
H;!G..tlaie1HI>e.eroUpofoptra»rSrOTX"««:H-WL''T-l'" -'>
72
A. Prerequisites—general group theory
be a right transversal to H in G, denote the coset НЦ by the symbol i, and if Httg = Ht}
write ig = j for g e G. (With this G-action, the set G/H = {1,.... л} is just the “right
coset” G-set described in Lemma 5.10.) For each g e G there exists a unique element
hj(g) e H such that
(18.»/) tiS~l =
We now use equation (18.t/) and the homomorphism о to define an action of G as a
group of operators on the direct product В = X"; for (x,,..., x„) e В and g e G, the
action is defined by
(xj,..., x„y = ((x19-, )‘'“,tor -.Г»""”-').
In functional notation, an element of В is a function f from {1,..., n} to X, and
therefore fa is defined thus:
/’(<) =
for ie {1,..., n}. Forg,keG the Equation(18.i/)yields
= hi(Mtik = ^(к)к1к ,(д)1л^д-,,
and so
(18.0) hjgk) = hMh^dg).
Therefore
(/9)*(i) = (/9(i/c~l))",',‘,w = (/(i/t-10-1))‘’<‘--'<«'
becausetrffi;* '(^'‘MM^)-1) = 0((hi(k)hit-i(gyr‘) = <r(ht(gk)~l)by(18.0).Itfollows
that
(/»)-=/<»*>
for all g,keG and hence that G really is a group of operators for В (cf. (4.21)). The
semidirect product
[BJG
with respect to this action is called the twisted wreath product of X by G and is
denoted by the symbol
G.
Remarks. 1. If о: H -» Aut(X) is the trivial homomorphism defined by setting a(h)
equal to the identity automorphism of X for all h e H, then it is clear that X г0(ц,в| G =
19. Subdirect and central products
73
* % G the ordinary wreath product, where a is the transitive permutation represen
tation of G on the right cosets of H, and in this situation we will often use the notation
X%G
instead.
2. For simplicity we have defined the twisted wreath product with respect to a
transitive permutation representation. It is clear how the construction could be
extended to the non-transitive case, with various homomorphisms <г(: H - Aut(X)
corresponding to the stabilizers Ht of the different orbits of G on the direct compo-
nents of the base group.
19. Subdirect and central products
In this section we consider two constructions closely related to the direct product.
The first of these is a special type of subdirect subgroup.
(19.1) Theorem. Let G2, G2, and H be groups, let G, -» H be an epimorphism, and
write K, = KerfeJ (i = 1, 2). Let D denote the (external) direct product Gj x G2, let
Gj denote the subgroup Gt x 1, G2 the subgroup 1 x G2, and set
G = {(gi, g2) 3i e Gf and glel = g2e2}.
Then G is a subgroup of D, and there exist epimorphisms G -> Gt (i = 1,2) such that
(a) Ker(^j) = G r\G2 = K2 and Ker(<52) = Gn G, s Kt,
(b) Ker(<5j) Ker(<52) = x K2, and
(c) G/(K, xK2) = H.
Proof. It is clear that G is a subdirect subgroup of D (see (4.151(b)). Let iq denote the
projection nc D -» G; onto the ith component (see (4.15)(a)), and let <5; = (л,)с, the
restriction of л,-to G. Then Ker(<5 J = G n Кефц) = Gn G2 = {(L g2)’g2e2 — 0 =
{(L Зг): З2 G X2} = 1 x K2 S K2. Similarly Ker(<52) = GnGj = K, x 1 =
Since (Kj x l)n(l x K2) = 1, Assertion (b) is clear.
Finally, define a map e: G -♦ H x H by (g)e = (gbti:t. g62e2) for each 3 e G, and
note that e is obviously a homomorphism. If g = (Si- 82)6 G-then 3^iEi = 8if:i =
8282 = д62г2. Therefore the image of 1 is the diagonal subgroup {(h,h):he H} of
H x H; in particular Ge S H. If 3 = (Sn З2)G Ker(e), then 1 — 3<Vi — з.е,- or 1
1,2, hence g, e Kt, it follows that Ker(e) = K, x K2. and therefore Statement (c) now
follows from the isomorphism theorem.
(19 2) Definition. The subgroup G of G, x G2 constructed in Theorem 19.1 is called
a subdirect product of Gt and G2 with amalgamated factor group
type of this group G is not uniquely determined. It depends on the choice of the
epimorphisms and e2 Theorem 19.1.)
74
A. Prerequisites—general group theory
The second construction, being a certain quotient of a direct product, has the
flaiour of a dual of the first.
(19.3) Definition. A group G is called an (internal) central product of its subgroups
l’„ . .. U„ if
CPI: G = U, ... l/„, and
CP2: [Ц, Ц-] = 1 for all 1 < i A j < n.
It is clear from this definition that Ut < G for i = 1, ..., n and that Цг. Ц <
Z(Uj) n Z(Uj) whenever i # j.
(19.4) Lemma. Let Ut, ..., V„ be subgroups of a group G, let D = x ••• x V„
denote their external direct product, and set Ut = {(1,..., 1, iq, 1,..., 1): u, e Ц}. Then
the following statements are equivalent:
(a) G is a central product of Ui,U„;
(b) There exists an epimorphism e: D -» G such that L\i: = U, for i = 1,..., n.
Proof. (a)=>(b): If G = Ul ... V„ with [Ц, Ц-] = 1 for i j, then the map
e:(ut, u2,...,un)^>uIu2 ...u„
is obviously an epimorphism with the required properties.
(b)=>(a): Since e is onto, we have G = De = Ule...U„e= Ц...Ц. Moreover,
[Ц, Ц] = [Це, Це] = [Ц, Ц]е = 1 if i yt j. Therefore the defining properties CPI
and CP2 of a central product are satisfied. □
The next result indicates how to construct a group G which is a central product of
subgroups Ut,..., V„ of prescribed isomorphism type such that Ц n Ц (i A j) is a
specified subgroup of the centres of each of Ц,..., V„.
(19.5) Proposition. Let Vt,..., Vn be finite groups, and assume that A is an abelian
group for which there exist monomorphisms pp. A -► Z(V,)for i = l,...,n. Let D denote
the external direct product Vk x ••• x V„, let I< (=I<) denote the ith coordinate sub-
group of D, and set
N = {(alpt,...,a„li„):aieA,ala2...a„ = 1}.
Then N < D, VfC\N = 1, and with Ц = I]7V/N the quotient group G = D/N has the
following properties:
(i) G is a central product of the subgroups Vt,..., V„ and Ц =1) for i = 1,..., n;
(ii) For 1 < i # j < n we have Ц n Vj = Qj=1 Vk = AtN/N = A, where
Aj = {(1,..., 1, a[q, 1,..., 1): a e A} < Ц.
Proof. It is obvious that N < Z(I]) x x Z(V„) = Z(D), and therefore N <J G.
That I] r. N = 1 is also clear, and so s I's Ц. If we extend isomorphisms from
Vt to Vj to an isomorphism в: V, x • • • x V„ -► D = V, x • • • x V„ in the obvious way,
19. Subdireci and ceniral products
75
and if v: D -» D/N is the natural homomorphism, then the map i. = Gv is dearlv an
epimorphism from G, x • • x G„ onto G such that ₽ У
(1 x x 1 x Ц x I x x l)r: = G,(<G).
Therefore G is the central product of its subgroups G,,..., Gn by (9 4)
Set Z = AtN/N, and note that Z s 4,/(4, n N)~a" s a.' Let 1 < i <
Since N contains {(up,, 1,..., 1, а *p(, 1,..., I):ae4}, it follows that A.N/N =
A,N/N = Z, and therefore Z< G,. Conversely, let хеЦоЦ for i# j. Then
x = (1,..., 1, v,, 1, 1)N = (1,1, Vj, 1,l)/v for suitable ц- 6 Ц. and p-e V-.
Therefore 1 J
(1,..., 1, t>„ 1,..., l,t>7‘, 1,..., l)e N,
and since i 4 j, by definition of N we have v: = up, for some a e A. Consequently,
x e A,N/N = Z, and we have proved that G, n Ц < Z, whence equality holds and
the remaining conclusions are obvious. □
Notation. If the groups Ц,..., V„ in the statement of (19.5) have isomorphic centres,
and pi’. 4-»Z(I<) is an isomorphism for i = 1, ..., n, we will sometimes use the
notation V, у-- - у V„ for the group G = D/N constructed there, even though the
isomorphism type of G is not uniquely determined in general. In this case we call G
a central product of Ц, ..., V„ with amalgamated centres and bear in mind that
T, y • • • Y V„ stands for any one of a class of such central products.
(19.6) Lemma. Let G be a finite group which is a central product of subgroups G,,
..., U„, and let 7) 6 Sylp(GJ for i = 1,..., n. Then PlP2...Pne Sylp(G).
Proof. By (19.4) there exists an epimorphism e from D = G, x • • x U„ onto G with
GjE = Gf. Since P} x • x P„ is obviously a Sylow p-subgroup of D (it is a sub-
group of the right order), it follows from (6.4)(a) that P[P2. -P„ — PiePге...Р„е =
(P, x • x P„)e is a Sylow p-subgroup of De = G. □
(19.7) Lemma. Let G be a central product of subgroups G„ G„, and let
Z(Uf < Uifori= l,...,n.IfN = NtN1...N„then
G/N = (IfN/N) x x (G„N/N)S(l7i/Ni) x ••• x (14ДЧ.К
Proof. Let D denote the external direct product (G,/N,) x ••• x (G„/N„). Then the
map e: D -» G/N defined by
(u.N,,- .,u„Nn)c = u,u2...u„/V
is obviously a well-defined epimorphism. Let
и и и eN = N.N2...N„ and so u}e Ifrs NtN2. ,.N„ = N/G^J £
N^= £ consequently £ is an isomorphism. Since £ maps the ith coordmate
76 A. Prerequisites—general group theory
subgroup of D onto UjN/N, it follows that the product (Ut N/N)(V2N/N).. ,(U„N/N)
is also direct. □
Next we describe the terms of the upper and lower central series of a central product;
the result is hardly surprising.
(19.8) Proposition. Let G be a central product of subgroups Ut...., V„. Then for all
m > 1 we have
(a) Km(G) = П?=! W and
(b) Zm(G) = n?=1Zm(m
Proof, (a) We prove this statement by induction on m, noting that it is obviously
true when tn — 1. Appealing to the induction hypothesis, we have
KJG) = [Km.,(G\ G] = fl fl Ц|-
L‘-l >=1 J
By repeated appeal to (7.4) (f) this last expression equals
П ц] = fl [Kn-jtu ц] = П К».(Ц)
i,J=i i=l t'=l
since [Gf, G;] = 1 when i # j. Thus the induction step holds and the proof is
complete.
(b) We first check the case m = 1. Since Z(Vt) is centralized both by Gf and (because
the product is central) by G; when j i, it is clear that Z(Gf) < Z(G) for i = 1,.... n.
Let z = uIu2...u„e Z(G) with u, e G; for i = 1, ..., n. If x e Gf, we have
Uj... Uj_| u?ui+l. ,.u„ = zx = z = u,... u.-j UjUI+1... u„. It follows that u? — uh hence
that u, e Z(Gj), and therefore that Z(G) < Z(G1)...Z(G„). Thus Assertion (b) holds
when m = 1.
Now let m > 1, and set N = Z(G). Then for I > 1 we have
Zj(Gj) r. N = Z,(Gf) n П Z(G;) = Z(Gj)(Zj(Gj) n П Z(G;)) = Z(Gj)
7=1
since Ц r. HjVj Ц Z(Gy) by definition of a central product. Therefore
Zj(Gj)N/N s Zj(Gj)/(Zj(Gj) n N) = ZfU^/Z^),
and with the help of (19.7) it follows that
zm(G)/z(G) = zm_I(G/z(G)) s z„_j (x IVZW))
= X Zm_j(Gj/Z(Gj)) = X Zm(Gj)/Z(Gj)
20. Exlraspecial p-groups and their
allomorphism groups
77
X Z"(ViWN
= (n
Hence Zm(G) = []?=I ZJU& as required.
□
20. Extraspecial p-groups and their automorphism groups
We begin with some essential preliminary remarks about vector spaces endowed with
special bilinear forms.
(20.1) Symplectic spaces and their groups. Let И be a finite dimensional vector space
over a field K. A symplectic form on К is a bilinear form f: V x V — К satisfying
Ж 0 = 0
for all v e Ц and a vector space endowed with such a form is called a symplectic space.
If V is a subspace of a symplectic space V, we write
17х = {w e V: f(u, w) = 0 for all и e Uj,
and V is called non-degenerate if I71 = 0. If W < 17х, we write V _L W. A 2-dimensional
subspace c2> is called a hyperbolic plane if/(t\, v2) = 1. It is well known (see,
for example, [H] II, 9.6) that a non-degenerate I7 has hyperbolic planes H,,..., H,
such that
(2O.a) V = H, @ H2 @ • © H, and H, 1 Hj for 1 < iV j < t.
Thus V has even dimension 2t.
If V, and V2 are symplectic spaces with respect to forms ft and f2 respectively, a
linear map а: Ц -► V2 is called an isometry if it is non-singular and satisfies
fju, v) = /2(ua, va)
for all u. v e К. By (2O.a) a non-degenerate symplectic space V is determined up to
.sometry by Dimt( V). The group of isometries from V to itself is called the sympie^ic
group on V and is denoted by Sp(K). Since it depends only on К and 2t( - Dim* ИХ
we also denote it by Sp(2r, K) (or by Sp(2t, <j) when К is a finite field with q
elements).
Let К be a field of characteristic 2, let
— К be a map satisfying
(20.2) Quadratic forms and orthogonal groups.
f: V x I7 -► К be a bilinear form, and let q: V
78
A. Prerequisites—general group theory
(20-jS) q(au + bv) = a2q(u) + b2q(r) + abf(u, t>)
for all a, be К and u, i> e V. Such a map is called a quadratic form on V. On setting
a = b = 1 and и — v in (20./?), we at once obtain f(u, u) = 0 for all ue V, and so f is
a symplectic form on И If this form f is non-degenerate, we say that the quadratic
form q is поп-degenerate. The group of non-singular linear maps a: V -► И which
satisfy
q(ua) = q(v)
for all г e V is called the orthogonal group on T and denoted by O(V). Clearly
O(n<Sp(F).
(20.3) Definition. Let P be a p-group. We call P extraspecial if each of the subgroups
Ф(Р), P', and Z(P) has order p.
This definition obviously implies that Ф(Р) = P' = Z(P). Thus an extraspecial
p-group P has nilpotency class 2, the quotient P/Z(P) is elementary abelian, and for
all x, у e P we have (xy)‘ = x’y'[x, y]"'-11'2 for i > 1; in particular, (xy)p = xpyp when
p is an odd prime.
The basic connection between vector spaces endowed with forms and extraspecial
p-groups is described in the following lemma.
(20.4) Lemma. Let P be an extraspecial p-group, let Z(P) = <z>, and let V = P/Z(P),
viewed as a vector space over Fp. If x = xZ(P) and у = yZ(P) are in V, we have
[x, y] = z° for some О < a < p, and the map f : V x V -* (Fp defined by
f(x, у) = a
is a well-defined bilinear form with respect to which V is a non-degenerate symplectic
space. If p = 2, the map q: V -» (F2 defined by q(x) = b when x2 = zb is a non-degenerate
quadratic form on V associated with f. The order of P is p2,+1, and Pisa central product
of t extraspecial groups of order p3.
Proof. It is clear that the commutator [x, y] and the element x2 depend only on the
cosets and not on the coset representatives, and so f and q are well defined. Since P
has class two, we have [x,x2, y] = [xt, y] [x2, y] and [x, У1У2] = [x, У1] [x, y2],
and therefore f is bilinear. Now xe Kx«>[x, y] = 1 for all у ePoxeZ(P)«>
x = 0; furthermore, since [x, x] = 1 = z°, we have /(x, x) = 0 for all x e V. Therefore
f endows V with the structure of a non-degenerate symplectic space and
V=PfL--LH,.
If P; is the inverse image of the hyperbolic plane Я, under the natural homomorphism
P -► PIZ(P) = V, then Pj has 2 generators x and у such that [x, y] — z. Hence Pf is
an extraspecial group of order p3, and since Я, _L HJ for 1 < i j < t, it follows that
[Pi, Pj] = 1. Therefore P = Pt P2... P, is a central product of its subgroups P,,..., P,.
Extraspecial p-groups and their automorphism groups 79
Finally, if p = 2, we note that (xv)2 = vVlr ,л
hfd^^VT’?' ST (Wi| dearly holds 'vhen a = 0ПмЬе=0 hthtrefom
holds for all a, b e F2. and q is a quadratic form. therefore
!pfODdeni°o dSCr're a Classif,“tion of extraspecial groups, we need some notation
Let D and Q denote respectively the dihedral and quaternion groups of order 8 and
for an odd prime p set
(2O.y)
E = <x, у: x” = y«- = 1, [x,6 Z(£)>, and
F = <x, у: xpI = y” = 1, [x, y] = xpy
It is straightforward to verify that if P is an extraspecial p-group of order p3 then
P = D or Q when p = 2, and P s E or F when p > 2.
Let P = F v F. Then P is uniquely determined up to isomorphism, and we can
write P = F, F2 with [Ft, F2] = > and Ft = <x„ yj, where x, and у,- satisfy the stated
relations for F(i = 1, 2). Since Z(F) = (xp), by replacing x2 by suitable power we can
suppose that (X|X2)P = 1. Then H = (xlx2,yiy is a subgroup of P isomorphic
with E. In V = PIZ(P), the subspace H/Z(P) is a hyperbolic plane, and so there
exists L < P such that V = (H/Z(P)) _L(L/Z(P)), with L/Z(P) a hyperbolic plane.
Therefore L is extraspecial of order p3, and P is the central product of Hand L. If L
had exponent p, then P would have exponent p. Hence L = F, and we have shown
that F y F s E y F. It turns out that Q y Q = D y D, and from these two facts and
(20.4) it is easy to deduce the following description of an arbitrary extraspecial
group—for details we cite [H] III, 13.7 and 13.8.
(20.5) Theorem. Let p be a prime, and let P be an extraspecial group of order p2,+1.
Then exactly one of the following four cases arises:
(i) p 7 2, Exp(P) = p, and P is a central product of t copies of E;
(ii) p 7 2, Exp(P) = p2, and P is a central product of t — 1 copies of E with a copy
of F;
(iii) p = 2, and P is a central product of t copies of D;
(iv) p = 2, and P is a central product of t — 1 copies of D with a copy of Q.
In particular, an extraspecial group of odd order is uniquely determined by its order
and exponent. In Cases (i), (ii), and (iii) P possesses a maximal abelian subgroup of order
p'tl and exponent p (i.e. elementary): in Case (iv) all the maximal abelian subgroups of
P are isomorphic with (Z2)'~1 x Z4.
Our next objective is to determine the automorphism group of an extraspecial
p-group P. For odd p we will only give a full proof in the case where P has exponent
p. A first step in this case is the following description of the (to within isomorphism)
unique such group of a given order p ' .
(20.6) Proposition. Let p be an odd prime, let F(#0) be a symplectic vector space over
Fp with respect to a non-degenerate bilinear fonn J: V x V-* p,an et ™ '
Define a binary operation on the set V x Fp as follows: if (u, •), fe p) <”
80 A. Prerequisites—general group theory
(u, z)(p, p) = (u + p, 2 + p + |/(u, p)). Let P denote the set V x Fp together with this
binary operation.
(a) P is an extraspecial group of order p2,+1 and exponent p.
(b) If a belongs to the symplectic group Sp(l/), then the map a*: P -» P defined by
(u, >.)“* = (ua, 2) for ue V and A 6 Fp
is an automorphism of P centralizing Z(P) and inducing a on P/Z(P) S V.
(c) If P is an automorphism of P which centralizes Z(P), then the automorphism a
which P induces on P/Z(P) = V belongs to Sp(l/).
Proof, (a) It is an elementary calculation to check that P satisfies the group axioms
with (0,0) as the identity element and (у, 2)-1 = ( — v, — 2). Let Z = {(0, 2): A 6 Fp}.
Since [(u, 2), (p. p)] = (0,/(u, p)), it follows that P' = Z < Z(P). Since Z(P)/Z V1
and V2 = 0 because f is non-degenerate, it follows that Z(P) = Z. Finally we note
that (p, Z)p = (pv, pA) = (0,0), and therefore P has exponent p; hence P/P' is elemen-
tary, and Ф(Р) = P’.
(b) The map a* defined in the statement is obviously bijection. For (u, 2) and
(p, p) 6 P we have (u, 2)“*(r, д)“* = (ua, 2.)(pa, 2) = ((u + p)a, A + p + 2/(ua, *’«)) an(l
((u, z)(p, p))“" = ((u + p)a, A. + p + jf(u, p)). Since f(ua, pa) = f(u, p) for a 6 Sp(F), it
is clear that a* 6 Aut(P); moreover, by its very definition a* centralizes Z(P). The rest
of (b) is now obvious.
(c) Let и, p e V, and set x = (u, 0) and у = (p, 0). Since xf = (ua, 2) and yf = (pa, p)
for suitable A, p e Fp and since p centralizes Z(P), it follows that (0, f(u, p)) =
(0, f(u, p)/ = [x, уУ = [x/ y₽] = [(xa, 2), (pa, p)] = (0, /(ua, pa)), and therefore
aeSp(F). □
Remark. In (20.6)(b) we evidently have a*p* = (aj?)*; thus we have shown that
Aut(P) actually contains a subgroup isomorphic with Sp(F) which acts faithfully on
V = P/Z(P).
For the case p = 2 there exist two inequivalent non-degenerate quadratic forms
on an F2-space of dimension 2t. These correspond to the two isomorphism classes of
extraspecial 2-groups described in cases (iii) and (iv) of (20.5). (The quadratic form
associated with an extraspecial 2-group P is the map q: V( = P/Z(P)) -► Z(P) S F2
given by q(xZ(P)) = x2. An elementary account of quadratic forms over F2 can be
found in Lorenz [1].) The following result is the analogue of (20.6) for p = 2 and can
be proved in the same way.
(20.7) Proposition. Let V be a vector space of dimension 2t over F2, and let q: V -» F2
be a non-degenerate quadratic form. Define a binary operation on the set P = V x F2
by setting
(u, 2)(p, p) = (u + p, A + p + q(v))
for u, p e V and 2, p e F2.
20. Extraspecial p-groups and their automorphism groups 8I
(a) P is an extraspecial group of order 22"1
(b) ‘f^OlVyheorthogonalgrouppreservingqjhenthemap^-.P^Pdefi^by
(u, >.)“ = (ua, 2) forueV and Ze f.
is an automorphism of P centralizing Z(P) and inducing a on P/ZIP) V.
M, M aulo™orPl'ism of p wl"cl< centralizes Z(P), then the automorphism a
which P induces on P/Z(P)(^ E) belongs to 0(E).
We can now prove our main result.
(20.8) Theorem. Let p be a prime, and let P be an extraspecial group of order p2,+1
(t > 1). Let A = Aut(P), let В = CA(Z(P)) and C = C„(P/Z(P)). Then we have:
(a) C = Inn(P), an elementary abelian group of order p2';
(b) If p is odd and P has exponent p, then B/C = Sp(2t, p);
(c) If p — 2, then B/C = O(q}, the orthogonal group for the quadratic form q
associated with P;
(d) A = ВТ, the semidirect product of В with a cyclic group T of order (p - 1).
Proof, (a) Write Z(P) = Z = <z>, and let {xfZ,..., x2tZ} be a basis for P/Z. If a e C,
then (x;Z)’ = XjZ and so x“ = x(z( for some zf e Z: furthermore, a is uniquely deter-
mined by the sequence (z,,..., z21) because P = (x,,..., x2l). Therefore the number
of distinct a’s in C is bounded above by p2', the number of such sequences. However,
C obviously contains Inn(P) S P/Z(P), and \P/Z(P)\ = p2'. Hence C = Inn(P) s
P/Z(P).
(b)_For p e В let p be the automorphism induced by p on V = P/Z (defined by
(xZ)P = xpZ). Since p centralizes Z, it follows from (20.6) (c) that p e Sp(P), and so
the map P -* P is a homomorphism from В into Sp( P) with kernel C. Consequently
B/C is isomorphic with a subgroup of Sp(k'). But by (20.6) (b) every a 6 Sp(k') “lifts”
to an automorphism a* of P with a* = a. Therefore the bar map is onto, and
B/C=?Sp(F)sSp(2t,p).
(c) This follows from (20.7) just as (b) follows from (20.6).
(d) If p = 2, obviously A = B, and there is nothing to prove. Suppose that p > 2,
let X = <x> be a cyclic group of order p. and choose a e {2,..., p — 1} so that the
map a: x —> x° generates Aut(X). (It is easy to see that Aut(X) is cyclic of order p — 1;
cf. [H] I, 4.6 for example.) Since the elements x°, у satisfy the same relations as x, у
in the extraspecial groups E and F defined in (20.) j, the map sending x -► x and у -г у
extends to an automorphism a of P when P = E or F. In each case [x, y] — [x, y] ,
and so a induces on Z(P) an automorphism a of order p — 1. Let D — P1 x x Pc.
where each ft is either E or F, and let a act on D according to its action described
above on each direct component. In particular, if z — (Zi,...,zf)eZ(D), t en
z“ = (z? , zf) and consequently every subgroup of Z(D) is a-invanant. If P is now
an extraspecial group of order p2'+1, then P S D/N for some N < Z(D) by (19.5) and
(20 5)- since № = N, the automorphism a induces on D/N an automorp ism a,
which in turn induces on Z(D}/N (=Z(D/N) s Z„) an automorphism of order p — L
Identifying P with D/N, we see that Aui(P) therefore has a subgroup T - <«> such
82
A. Prerequisites—general group theory
that CT(Z(P)) = 1. Hence T n В = 1. However, В is the kernel of the homomorphism
from A -»Aut(Z), and so | A/BI < |Aut(Z)| = p l. Since | T| = o(a) = p — 1,
we conclude that A = ВТ. □
We state, without proof, the corresponding result for the case where p is odd and
P has exponent p2. Full details can be found in Winter [1].
(20.9) Theorem. When, in the notation of (20.8), the group P has odd exponent p2, the
group B/C is isomorphic with a semidirect product of a normal extraspecial group of
order p2'1 with Sp(2t — 2, p); in particular, |B/C| = p when t = 1.
We end this section by discussing some special cases, mainly numerical examples
that will be needed later. In dimension two, calculations are simplified by the follow-
ing useful fact.
(20.10) Lemma. Sp(2, K) = SL(2, K) for any field K.
Proof. Let V = <t>,, vf) be a hyperbolic plane with (f(v,, r,)) = ( |, and let
A = (ny) e GL(2, K) act on F thus: V”1
t^A = allvl + 0121’2
V2A = fl21rl + a22v2-
Then the following implications hold: A 6 Sp(2, K)o-f(vt A, v2A) = /(t’i, v2) =
1 oat 1a22 — a12a2I = Det(A) = 1 о A 6 SL(2, K). □
(20.11) Lemma. Let E be an extraspecial group of order 27 and exponent 3. Then
Aut(£) = [Inn(£)]GL(2, 3) and CGL(2 3I(Z(£)) = SL(2, 3). Furthermore, Aut(E) con-
tains an element of order 8 that inverts Z(E).
Procf. From (20.8) and (20.10) we know that Aut(£) = [Inn(£)]H, where
H = SL(2, 3)D and D = <<5> has order 2; moreover, CH(Z(£)) = SL(2, 3). (A comple-
ment H to Inn(£) in Aut(£) is here obtained as the normalizer of the quaternion
Sylow 2-subgroup of O3 2(Aut(£))). Since 6$ SL(2, 3), it induces on E/Z(E] a
linear map with determinant — 1, and since |GL(2, 3)/SL(2, 3)| = 2, it follows that
H = GL(2, 3). For the desired element of order 8 one can take for example
( J 6 GL(2, 3)\SL(2, 3). □
(20.12) Lemma. Let p be an odd prime, and let E be an extraspecial group of order p3
and exponent p. Then Aut(£) has a subgroup К = <a> x of order 4, where a and
/3 are defined with respect to the presentation of E in (20.y) as follows:
(a) x* = x-1, y® = у (whence a2 — 1);
(b) x1' = x,yfl = y-1 (whence ft2 = 1).
21. Automorphisms of abelian groups 83
Proof. Since the pairs (x>. y) and (x, у *) satisfy the same relations as (x, y) in E the
maps a and fi can be extended to automorphisms of E. It is clear a)3 and Ba each
coincide with the unique automorphism of E which inverts x and у. □
(20.13) Lemma. Let E be an extraspecial group of order 37 and exponent 3. Then
Aut(E) contains an element a of order 7 which operates irreducibly on E/Z(E) and
centralizes Z(E).
Proof By [Hl II, 9.13 the order of Sp(6, 3) is divisible by 7, and so contains an
element of order 7. Thus by (20.8) Aut(E) contains an element a of order 7 which
centralizes Z(E). Since 7 does not divide 3" - 1 when n < 6, it follows from B, 9.8
that a acts irreducibly on the elementary abelian group E/Z(E) of order 36. □
Let V be a symplectic К-space, decomposed as in (20.a) into a direct sum of t
orthogonal hyperplanes H, = <u,-, r,> with f(ut, t>() = 1. Let A = (atJ) e GL(t, K), and
set A-1 = (by). Direct calculation shows that the linear map a: V ->• V defined by
u-,a = Jj=1 aytq and vta = £'=1 b^Vj satisfies flwa, w'a) = f(w, w') for all tv, w' e V,
and therefore a 6 Sp(F'). This elementary observation is the key to showing that any
finite group G can be a faithful group of operators for an extraspecial group.
(20.14) Proposition. Let p be an odd prime and G a finite group. Then there exists
an extraspecial p-group of exponent p such that G = G for some G < Aut(E) with
[Z(E), G] = 1.
Proof. By (20.6) and the subsequent Remark, it suffices to show that for some t e N
the group Sp(2t, p) contains a subgroup isomorphic with the given group G when p
is an odd prime. By (5.9) there is a subgroup of Sym(|G|) isomorphic with G, and if
л 6 Sym(|G|) the map which sends л to the permutation matrix (<5„(f)f) is a mono-
morphism from Sym(|G|) into GL(|G|, p). Finally, if X* denotes the transpose of a
matrix X, the map r: GL(|G|, p) -»Sp(2|G|, p) defined by
Л4 0 A
Лт = \0 (zT1)*/
is clearly a monomorphism and, composed with the other maps, yields an embedding
of G into Sp(2t, p) with t = |G|.
21. Automorphisms of abelian groups
This is not a comprehensive treatment of the theme of the but merety a few
special results which are needed later: specifically, these dea w.*
automorphism group of a cyclic group and the structure of the Sylow subgroups
some finite linear groups.
84
A. Prerequisites—general group theory
(21.1) Theorem. Let G be a cyclic group of order n.
(a)Aut(G) is isomorphic with the group of units of the ring = //n/. The units of
7Ln hare the form a + nZ with 1 < a < n and (a, n) = 1; in particular, |Aut(G)| = <p(n),
where <p is the Euler function of number theory.
(b) If p is an odd prime and n = pm(m > T),thenMit(G) is cyclic of order p^fp — 1).
(c) If n = 2m(m > 3), then Aut(G) is an abelian group of type (2, 2™ 2) with
{— 1 + 2™Z, 5 + 2™Z} as a set of generators. Furthermore, Aut(Z2) = 1 and
Aut(Z4) s Z2.
Proofs can be found in [HJ: Part (a) in Satz 1,4.5 and Parts(b) and (c) in Satz 1, 13.9.
In Section 3 of Chapter XI we will need some special facts about the structure of
the Sylow subgroups of certain linear groups. The following arithmetical lemma will
be helpful in determining these facts.
(21.2) Lemma. Let и = pf a power of the prime p, and let q' be the highest power of
a prime q dividing и — 1. Assume that t > 1 and, if q = 2, that t >2. Let qs be the
highest power of q dividing an integer r. Then q‘+l is the highest power of q dividing
ur- 1.
Notation. If q is a prime and n an integer, we will write q“ || n to mean that q“ is the
highest power of q dividing n.
Proof. By hypothesis и = 1 + q'x and r = qsy with (q, xy) = 1. Therefore
ur — 1 = (1 + q'x)r — 1
= rq'x +
= qs+'xy + X <чх‘,
i=2
where af
We will prove that
(21.«)
9s+,+1l|ai
for i = 2,..., r,
and then the conclusion of the lemma will immediately follow.
First, let <7b||i! Then b = [i/q] + [i/q2] + (where [x] denotes the integral part
of x), and so
(21.0)
i + b < i + i/q + i/q2 + • • • + i/q"1 (for some m)
< i/(l - l/q) = qi/(q - 1).
To prove (21.a), we consider two cases:
85
21 Automorphisms of abelian groups
Case 1: q odd.
Since i > 2, we have
and therefore
91 = (2(q ~ 1) - (q - 2)).
9-1 (9 1) '
= 2i-(9-^)i<2i-l.
\9- 1/
In view of (21./?), it follows that b < i - 1 and hence that b < i - 2. and because
(j) = Ч‘У(Ч’У - 1) • (qsy - i + l)/(il), we conclude that qs~‘+2 divides But for
t > 1 and i > 2, we have it > i + t - 1, and therefore a, = Г jq" is divisible by
g(S-i+2)+(i+.-i) = gS+I« as claimed
Case 2: q = 2.
In this case qi/(q - 1) = 2i, and from (21./?) it follows that b < i - 1. Hence qs-i+1
divides But because i > 2 and by hypothesis t > 2, we have it > i + t, and
therefore (s — i + 1) + it > s + t + 1. Consequently qs+,+1 divides
= a„ and
again our claim is justified.
□
(21.3) Theorem (Weir [2], Carter and Fong [1]). Let p and q be distinct primes, and
let q‘ be the highest power of q dividing p — 1. Assume that t > 1 and, if q = 2, that
t >2 (i.e. that p = 1 (mod 4)).
(a) A Sylow q-subgroup W of the general linear group GL(q‘, p) of degree qk over
Fp is isomorphic with the к-fold wreath product
(• - (Zg> ^reg ) Г^гер • ) ^reg
in particular, | Wj = q1, where I = 1 + q + • + qk 1 + tq .
(b) Let S = IVn SL(q‘, p), a Sylow q-subgroup of the special linear group. Then S
is generated by a set of elements whose orders belong to {1, q,q'} if q is odd and to
Furthermore, there exists an element w 6 W with o(w) = q‘ = IW: S| such that W =
(S, w>.
Proof (a) First we compute the integer 1, where q‘ || |GL(q‘, p)|. By counting bases it
is straightforward to verify that
86 A. Prerequisites—general group theory
|GL(n, p')| = (pv - l)(pn/ - pz)... (pn/ - p'" n/).
Hence |GL(q‘, p)|p = (p — l)(p2 - ib.-lp*1 - 1). If qs||r, it follows from (21.2) that
qs+' is the highest power of q dividing pr — 1. Hence q‘ divides each factor of the
product
(p-l)(p2- l)...(p9k-l),
one additional power of q divides qk-1 factors, a further power of q divides just qk~2
factors, and so on. Hence
I = tqk + qkl + •• + q + 1,
and q1 is the order of a Sylow q-subgroup of GLlq*, p).
Let A be a cyclic group of order p. Since q' divides p - 1, by (21.l)(b) we have
Z„. < Aut(A), and we can form the semidirect product E = [A]Zgl, which is evidently
a primitive group with A = Soc(E)- Let X denote the к-fold iterated wreath product
X = (...(£4eEZ,)fb„g.--)4eEZ,.
By induction on к it is clear that the base group of X contains an elementary abelian
p-subgroup V = A" of rank qk, which is complemented in X by a subgroup К
isomorphic with
(...(Z^^Z^^.-J^Z,
(with к wreath product symbols). Since С£(Л) = 1, it follows easily that C£(V) = 1,
and hence that К is isomorphic with a subgroup of Aut(L) s GL(q‘, p). But an
easy induction argument shows that | К | = q‘, the order of a Sylow q-subgroup of
GL(q‘, p). Hence H7 = К by Sylow’s theorem.
(b) If к = 0, we have GL(q‘, p) = F* s Zp_t and SL(ql, p) = 1, in which case the
assertions of Part (b) are obviously true. We therefore proceed by induction on k,
assuming that they have already proved for к = 0,1,..., i. Let И7 and S, (< И7) denote
Sylow q-subgroups of GL(<f, p) and SL(q', p) respectively, and let I7 be an Fp-space
of dimension q‘ on which GL(q‘, p) operates naturally. Since SL is the kernel of the
epimorphism Det: GL -> f’,we have GL/SL s Zpi, and it follows that Hj/S, s ZQ1.
From the description of W in Part (a), we have Wi+J = H'rLi,cg ZQ, and Ц(1 may
be identified with the subgroup I7" of the base group of the wreath product
([L,]H<)TjregZ,.
First suppose that q is odd. In this case, a generator of Z4 acts on the regular module
F,Ze with determinant 1, and therefore the following subgroup T of Hj+ri
T = Qj„B Z,) {(*!,..., x,) 6 И7": П Det(xj on I7) = 1
acts with determinant 1 on Ij+n 'n other words, T < Sj+1. But by induction we can
21. Automorphisms of abelian groups
87
suppose that IV( contains an element x of order q' such that <Sb x> = Wt. Then S
is generated by SfQj^Z, together with elements of the form (x, 1’...,
which also have order q'. Since by our inductive assumption, the subgroup S, is
generated by elements of orders 1. q or q‘, it is clear that Si+1 is also similarly
generated, furthermore I’K+i = Sj+I ((x, 1,..., 1)\ and therefore the induction step is
complete.
Finally, suppose that q = 2, and let x denote the involution in Wo (s Z2,). Since
lVt S И/ОГЪ,СВ Z2, we can write = (IV0 x lV0)(y}, where у is an involution inter-
changing the direct components. Since у and (x, 1) each act with determinant - 1 on
V,. the group
T={(w,w *):we IV0}<(x, l)y>
acts with determinant 1 on Ц. Now T is clearly complemented in by a subgroup
<(w, 1)> of order 2‘, and so by order considerations we deduce that T — S,. Since
<(x, 1), y> S Z2 Qj„s Z2 = Dih(8), the element (x. 1 )y has order 4, and therefore Sv is
generated by elements of order 4 and 2‘.
Now let i > 1 and write IV1+1 = (Щ x H<)<z>, where z is an involution inter-
changing the direct factors. Since Dimfp(H) is even, Det(z on Ц+1) = 1, and in this
case the induction step proceeds exactly as in the previous case of q odd, yielding
generators of S(+1 of orders 2, 4 and 2', as well as an element (w„ 1) of order 2' such
that IF",, = Siu <(wj, 1)>. This completes the proof of Part (b). □
A proof of the following result can be found in [H] I, 14.9(b).
(21.4) Theorem. Let n > 3, and G be a nonabelian group of order 2"+l with a cyclic
normal subgroup <x> of order 2". Then G is isomorphic with one cf the following four
groups:
(1) The dihedral group
Dih(2,+1) = <x, у: x2" = y2 = 1, x” = x ’>;
(2) The generalized quaternion group
Quat(2,+1) = <v, у: v2"' = Г- y4 = 1, *’ = x 1 >;
(Of course, Quat(8) is simply called the quaternion group.}
(3) The quasidihedral group
<x,y:x2" = y2 = l,x’ = *2'
(4) The group
, 2" „2-1 X> = X2,’',+1>-
<x, у: x2 = У - Ь * A '
88
A. Prerequisites—general group theory
The subgroups of PSL(2, p) were classified by Dickson in 1911. A proof of Dickson’s
result, which includes most of the following lemma, can be found in Chapter II,
Section 8 of [Н].
(21.5) Lemma. Let p be a prime congruent to 3 modulo 4, and let 2' be the highest
power of 2 dividing p + 1. Let e denote a 2'T1 st root of unity in (F^, and set
(° 1
\1 e + e’/’
( ° Л ,
у = I I, and w
\-l 0/
Then x e GL(2, p), o(x) = 2'+1, o(y) = 4, and o(w) = 2. Furthermore:
(a) The subgroup T = <x, y> has order 2'+2 and is a Sylow 2-subgroup of GL(2, p);
(b) The derived group T' contains the element x2 of order 2‘;
(c) The subgroup Q = <x2, y) is a Sylow 2-subgroup of SL(2, p); it is a (generalized)
quaternion group and is generated by elements of order 4;
(d) T=<Q,w>.
Proof, (a) Since p = 3 (mod 4), we have 2||(p— 1) and 2'> 4, and because
|GL(2, p)| = p(p — l)2(p + 1), it follows that 2,+2 is the order of a Sylow 2-subgroup
of GL(2, p). Since 2'+1 ||p2 — 1, the multiplicative group of the field ftpi contains an
element e of order 2'+1; moreover, the element e + e’’ evidently belongs to the fixed
field of the automorphism x -»xp of F,,2. Thus x 6 GL(2, p). Let
u = ( 1)eGL(2, p2),
\e ep/
Using the fact that (— 1 — e2)(ep — e) 1 = e, we obtain
и *xu =
1 /—1—e2
e'-eV 1 + ep+1
0\
e'J’
-1 - e₽+1 \
1 + (ep)2}
e
0
an element of GL(2, p2) of order 2'+1. Hence o(x) = o(u“*xu) = 2'+1. Direct calcula-
tion gives o(y) = 4 and o(w) = 2.
Next set z = yx. An easy computation yields xz = x-1x2‘ and o(z) = 2, whence it
follows that <x, z> (= <x, y> = T) is a quasidihedral group of order 2,+2. Therefore
T 6 Syl2(GL(2, p)).
(b) Clearly T'contains x *xz = x 2x2‘, which clearly has order 2‘ since t > 2. Thus
Г = <x2>.
(c) Since Det(x2) = Det(y) = 1, we have <x2, y> < SL(2, p). But |T : <x2, y>| = 2
and 21|(p — 1) = |GL(2, p): SL(2, p)|, therefore <x2, y) 6 Syl2(SL(2, p)).
Further matrix calculations show that y2 = x2‘ and that у *x2y = x“2. Therefore
Q = <x2, y> is a (generalized) quaternion group generated by elements у and x2y,
both of which have order 4.
(d) Since Det(w) = — 1, we have w e T/Q, and since | T: QI = 2, clearly T = (Q, w>.
□
I
1
21. Automorphisms of abelian groups
We can now carry out a similar analysis to Theorem 21 3 for the case о = 2
p = 3 (mod 4) 1
(21 6) Theorem. (Carter and Fong [1].) Let p be a prime congruent to 3 (mod 4) and
,et 2 “Z h!9^ P°Wer °f 2 dividine P + 1 2). Let к be a positive integer, and
let T e Syl2(GL(2, p)).
(a) A Sylow 2-subgroup W of GL(2I‘, p) is isomorphic with the (k - Iff old wreath
product
In particular, 1fT| — 2', where I = 1 + 2 + • + 2‘-2 + (t + 2)2k-1.
(b) Let S = Wc. SL(2*, p), a Sylow 2-subgroup of SL(2‘, p). Then S is generated by
elements of order 2 or 4, and W contains an involution w such that W = S(w') and
Sn<w> = 1.
Proof. Because this proof runs along similar lines to that of Theorem 21.3, we give
it in brief. Since p = -1 (mod 4), we have p° = -1 (mod 4) for all odd values of a,
and therefore 21| p“ — 1 when a is odd. On the other hand, by hypothesis 2,+11| p2 — 1,
and so by (21.2) the highest power of 2 dividing p2r — 1 is 2s+,+1 when r = 2sy
with у odd. Hence the exponent of the highest power of 2 dividing |GL|p. =
(p- l)...(p2k- 1) is
2‘“* + (t + Пг*'1 + 2‘“2 + - Ч-2 + 1 =(t + 2)2‘“‘ + 2h 2 + -+2+ 1 = I.
Since | T| = 2,+2 by 21.5(a), it follows that 2' is also the order of the (k - l)-fold
wreath product (21.y), and Assertion (a) now follows.
(b) Let W, e Syl2(GL(2‘, p)) and S, = H<r1SL(2i,p). Let Ц be the F„-space of
dimension 2‘ on which GL(2‘, p) naturally operates. The assertions of Part (b) cer-
tainly holds if к = 1: for then И7, = T, and by (21.5) the group S, is a generalized
quaternion group of order 2I+1; in particular, 5 | is generated by elements of order 4
and Wt = <S, w> with w2 = 1.
Now let i > 1, and assume by induction that the assertions of Part (b) hold for Wt
and S,. Then by Part (a) we can write Wi+I = (WJ x WJ)<z>, where z is the involution
which interchanges the direct components. Since DimFp( Ц) is even, Det(r on Ц+1) ,
and so Si+1 is generated by 5,%, Z2 together with elements of the form (wh w, ),
where <S„ w, > = W,. By induction Si+1 is therefore generated by elements of order 2
and 4, and IL+1 = <Si+1, («>,, 1)>. Thus the induction step is complete. □
Chapter В
Prerequisites—representation theory
1. Tensor products
The construction known as the tensor product (of rings, algebras, modules) is an
indispensible piece of algebraic machinery. For the needs of representation theory it
provides a basis-free definition of an induced module, the framework for the process
of extending the ground field of a representation, and a binary operation on modules
corresponding to multiplication of characters. In this section all rings have a 1.
(1.1) Definitions. Let R be a ring.
(a) Let К be a right R-module. A subset {i>>}л is called a free basis if for every
right R-module W every map 6: ► W extends to a unique module homo-
morphism from V to W. (This is equivalent to the statement that each v e V has a
unique expression of the form
t> = 'T/i + + vlnr„
for some finite subset {Aj,..., Z„} £ A with rlt..., r„ e R.) An R-module is called free
if it has a free basis. (The right regular R-module RK has free basis 1, and every finitely
generated free R-module is a direct sum of finitely many copies of RK. If R is a field
K, then an R-module is simply a vector space over К and in this case every R-module
is free.)
(b) Let К be a right R-module, W a left R-module, and let A be an abelian group.
A map p: V x W -> A is called balanced if (a) it is bilinear and (b) it satisfies
(er, w)p = (i>, rw)p
for all v e К >v e W, and r e R.
(c) Let V and W be as in (b). An abelian group T is called a tensor product of V
and W over R if
TP1: there exists a balanced map 6: V x W -> T such that <lm(b)> = T, and
TP2: if S is an abelian group and p; V x W -> S is a balanced map,
then there exists a homomorphism a: T -» S such that da = p.
1. Tensor products
91
(Note that the “universal” Property TP2
unique up to group isomorphism.)
means that a tensor product, if it exists, is
To construct a tensor product let F denote the free Z-module with the Cartesian
product V x W as a free basts. Let D denote the subgroup of F generated bwhe
following elements: i>y me
(ft + p2, w) - ([>!, w) - (p2, w),
(l>, Wj + W2) — (p, W,) - (p, Wj),
(pr, w) - (p, rw)
with p, p„ р2ек w, w1; w2eW, and reR. Set T=F/D, and define the map
й: V x W - T thus: P
й: (p, w) -> (p, w) + D.
Then it is clear that й is a balanced map whose image generates 1, and it is straight-
forward to verify that, given a balanced p:V x W->S, then the map a: T -> S defined
by
a: (p, w) + D -► (p, w)p
is a well-defined group homomorphism satisfying Sa = p.
Notation. We will denote the group T by V ®R Wand the image (p, w)<5 by p ® w.
Thus (iq + p2) ® w = p, ® w + p2 ® w, p ® (wj + w2) = p ® Wj + c ® w2, and
vr ® w = p ® nv for all p, pt, p2 e Ц w, Wj, w2 e W. and reR. The elements of the
form p ® w in V ®R W, called pure tensors, generate V ®R IF as abelian group, but
are not, in general, linearly independent over Z and so do not form a Z-basis.
Although constructed out of Я-modules, the tensor product is merely an abelian
group; but with some extra structure, which we now describe, it can be made into an
R-module.
(1.2) Definition. Let R and S be rings. An (R. S}-biniodule is an abelian group M
which is at the same time a left R-module and a right S-module and further satisfies
r(ms) = (nn)s
for all r e R, s e S. and m e M.
(1.3) Examples, (a) If R is any ring (not necessarily commutative), then the associa-
tive law ensures that R is an (R, R)-bimodule. , . ., j™,™,
(b) When R is a commutative ring and M is a right «-module then M-becomes
an (R, R)-bimodule if we define the left R-action by rm = mr for all ,n e Л/ and reR.
92
В. Prerequisites—representation theory
(1.4) Lemma. Let V be a right R-module and W an (R, S)-biinodide. Then the tensor
product V ®R IV becomes a right S-module when an S-action is defined by
(l.cc) (u ® w)s = v ® ws
for all v e К w e IV, and seS. In particular, for a commutative ring R, the tensor
product V ®R W can always be viewed as an R-module.
Proof. Since the pure tensors in T — V ®R W are not necessarily linearly indepen-
dent, it is not clear that (l.a) yields a well-defined S-action on the whole of T. We will
show from the axioms that there is a unique, well-defined S-action t -> ts on T such
that (l.a) holds for all pure tensors t = v ® w and such that (t + t")s = ts + t's for all
t, t' e T and all s e S.
For a given s e S, let ns: V x И'-> 1' ®R IV denote the map defined by
(t>, W)7t, = V ® WS.
Then an easy calculation shows that tts is balanced, and so by the “universal” property
TP2 there exists a group homomorphism ps: T -> T with (p ® w)ps = (p, w)ns =
p ® ws. If t e T, we define ts = tps and readily check the stated properties. The
S-module axioms for T then follow easily. □
A proof of the following result can be found in [H] under V, 9.4.
(1.5) Lemma. Let Rbea ring, V a right R-module, and W a left R-module. Assume that
V = ф Ц and IV = ф Wj,
it, I jtJ
where Vt and are submodules. Then
V ®r ф kJ flj (as abelian groups).
i e l.jeJ
The next result follows easily from (1.5).
(1.6) Lemma. Let V be a right R-module and W a free left R-module with free basis
{wj.jeJ}.
(a) The abelian group V ®R W admits the decomposition
Ofi) т®«и' = @|/,
jej
where kj = {u ® Wj: v 6 V} s V*, the additive group of V, and if R is an algebra over
a field K, then (l.ji) is a vector space decomposition.
(b) If additionally V is free with basis {uf: i 61} and R is commutative, then V ®R W
is a free R-module with basis {p( ® w,-: i e I and j e J}; in particular, if V and W are
vector spaces over a field К with bases {t\} and {wj}, then V ®K W is a К-space with
basis {Pj ® Wj} and DimK(F®K IV) = DimK(F) DimJf(H/).
1. Tensor products
93
(1.7) Lemina (Associativity of the Tensor Product) Let R
a right R-module, В an (R, Sfbimodule and C an К n h ’ л / I? /‘I"9S- “ A be
unique T-tnodule isomorphism *S’ lbh,m(>d“le- Tf™ there exists a
B'-(A®RB)®SC -*A®R(B®SC)
such that ((a ® b)® c)p = a ® (b ® c) for all a e A, h e В, с e C.
Pr°A°fA deu ‘ к1' B C iS “(R’ r)-bimodule and that (А ®КВ)®,С
and A <*)r(B (x)s C) are both right T-modules.
Let с e C and define a map rc: A x В A ®R(B C) by (o, b)rc = a ® (b ® c).
Since tc is balanced with respect to R, there exists a homomorphism p-.A®KB->
A ®K(B ®s C) such that c K
(a ® b')pc = (a, b)tc = a®(b®c).
Now define a map a: (A ®R В) ® C — A ®R(B ®. C) by
(x, c)a = xpc
for x e A ®R B. Then a is balanced with respect to S, and consequently there exists
a map p: (A ®K B)®SC ->A ®H(B ®s C) such that (x ® c)p = (x, c)a = xpc; in par-
ticular, ((a ® b) ® c)p = (a ® b)pc = a® (b ® c). It is straightforward to check that
p is a homomorphism of T-modules and that it has an analogously-constructed
inverse. □
(1.8) Theorem. Let R be a ring, let M and M' be right R-modules, and N and N' left
R-modules. Further, let a: M -> M' and p'. N -> N’ be R-module homomorphisms. Then
there exists a unique p e Homz(M ®R N, M' ®R N') such that
(tn ® n)p = ma®np
for all me M,neN. We denote this pbya® p. If R is commutative, then a® Pis an
R-module homomorphism.
If, further, a' e Hom(M' - M") and f!' e Hom(W' - ЛГ), then (a ® P)(a ® P) =
aa' ® pp. .
This so-called “functorial property” of the tensor product is stated and proved as
Theorem V, 9.6 in [Н].
(1.9) Definitions, (a) Let A and В be respectively a x a' and b x b' matrices with
entries in some ring R. Then the tensor (or Kronecker) product А® В к the ob xab
matrix which is partioned into a x a' blocks of b x b' submatnces such that the
rap L« Л - <«,> b= tfe mi™ «« "H”1 “ ‘ "
trace, Tr(a), of a by
94 В. Prerequisites—representation theory
Tr(a) = Tr(A) = a,, + a22 + + a„„.
Since Tr(.4) = Тг(Р"’ЛЛ for any non-singular n x n matrix P, the definition is
independent of the choice of basis. An easy consequence of these definitions is the
following:
(1.10) Corollary. Let M and N be free modules over a ring R with free bases
{m, ma} and {nlt.... nb} respectively. If a: M -> M and are R-linear
maps, and A and В are the matrices of a and p with respect to these maps, then there
exists an ordering of the basis {m, ® n;} of M ®R N such that A® В is the matrix of
a ® p with respect to this basis. In particular, it follows that
Tr(a ® p) = Tr(a) Tr(j?).
Our most important application of tensor products will be in the context of
modules for group algebras (see (3.2) for the formal definition of a group algebra).
(1.11) Lemma. Let G be a group, К a field, and let M and N be KG-modules. Then
the К-space M ®K N becomes a KG-module if we define the G-action on pure tensors
thus:
(l.y) (m ® ri)g = mg ® ng
for all me M,ne N, and g eG, and extend it linearly to the whole space.
Proof. The statement that the К-space M is a KG-module is equivalent to the
statement that the map
ag: m -> mg
is a non-singular К-linear map satisfying agh = agah for all g,heG (see A, Definition
3.6). By (1.8) the К-linear map ae ® pg sends m ® n to mg ® ng ( = (m ® n)g) and
satisfies ag/l ® Pgh = (a9 ® Pg)(ah ® ph) for all g,he G. Thus Equation (l.y) defines the
structure of a KG-module on M ® N. □
If M is a КЯ-module and N is a KL-module, where H and L are groups, and if
G = H x L, then we can view M and N as KG-modules by inflation with
Ker(G on M) = 1 xL and Ker(G on N) = H x 1. From this viewpoint we can apply
(1.11) to deduce the following
(1.12) Corollary. If M and N are KH- and KL-modules respectively, then M ®K N is
a K(H x L)-module with the H x L-action defined on the pure tensors by
(m ® n)(h, I) = mh ® nl
for all he H and I e L.
"ИД»
95
2. Projective and injective modules
Rem«r/c Let M and /V be KG-modules. Then M ®K A is a module for G x G by
Д’' У / V'T dulag°nal SUbgr°Up G = {(9- 9): Й e G} of G x G, then
M N becomes a G-module by restriction. This is precisely the G-module defined
in (1.11).
2. Projective and injective modules
If /С is a field and if the characteristic of К does not divide the order of a group G,
then every submodule of a КG-module is a direct summand by Maschke’s theorem.
Modular representation theory is concerned with the other case when Char(R') does
divide |G|. Here it is only the projective modules which, as submodules, are guar-
anteed always to be direct summands. The dual notion, that of an injective module,
coincides with the concept of a projective module for modules over a group algebra
КС (see (3.2)). Throughout this section R will denote a ring with a 1, and all modules
are finitely generated and unital.
(2.1) Lemma. Let M be a (right) R-module.
(a) There exists a free R-module F and an epimorphism a: F -» M.
(b) If M is simple, the free module F in (a) may be chosen to be cyclic (viz. F = Rr,
the right regular R-module).
Proof, (a) Let {nij,..., ms} be a set of generators for M. Let F be a free R-module
with free basis ft, ..., fs. For each f e F, there exist uniquely determined elements
rj,..., rs of R such that f = ft rt + • • + f,r„; then define
fa = m,'i + • + msrs.
It is straightforward to check that for all f.f 'sF and reR one has
(/ + f')a =fa + fa and (fr)a = (fa)r.
Thus a is a module homomorphism and is clearly onto because m,,..., m, generate
M.
(b) If 0 # m e M, then mR is evidently a non-zero submodule of M; therefore
mR = M if M is simple. Thus M is cyclic in this case, and is clear from the proof of
Part (a) that then F be also may chosen to be cyclic.
for each homomorphism
homomorphism ф:Р->А
(2.2) Definition. An R-module P is called projective if
0:P->B and each epimorphism a: Л -»B, there exists a
such that фа = 6.
P
Ф / "
A-------~ В -------‘ 0
a
96
В. Prerequisites—representation theory
(2.3) Lemma. A free R-module is projective.
Proof. Let F be a free R-module with basis {f: i e I}, let 6 e HomR(f, B), and let
a: A -> В be an epimorphism. If h, = f(). then there exists an a, e A such that u-a =
for all i e I. Since F is free, the map sending each f to a, extends to an R-module
homomorphism ф: F A, and since фа and в agree on the basis {/j}, they are
identical R-module homomorphisms. □
(2.4) Proposition. Let M, be an R-module for each i e I, and let M = @it,t bf, viewed
as an R-module. Then M is projective if and only if is projective for all i e I.
Proof. To prove that the condition is necessary, suppose that M is projective. To
establish that M, is projective, we must complete the following commutative diagram:
? / e
A——> В ------------► 0
Let ttj denote the projection of M onto M,. Since M is projective, there exists
ф' 6 HomR(M, A) such that ф'а = nt6 e HomR(/W, B). Let ф denote the restriction
ф\м.. Then, for m e M„ we have
тфа = тф' a = mnf) = ml).
Thus фа = в, and so Mi is projective.
We now prove the sufficiency of the condition. We assume that each M, is projective
and aim to complete the following commutative diagram:
M
? / e
л"—В --------------> 0
Let 6t ~ в1м_. Then, since M, is projective, there exists e HomR(M,, A) such that
ф,а = 6j. Define a map ф: M -> A by Then it is clear that
ф e HomR(M, Л), and since (^1тИФа ~ = (Xim№ we obtain
фа. = 0, as desired. О
We can now prove two important characterizations of projective modules.
(2.5) Theorem. Any two of the following statements are equivalent:
(a) P is a projective R-module-,
97
2. Projective and injective modules
(b) If M is an R-module and К is a submodule of M with M/K ~ p ,t,
submodule P such that M = P © к (whereupon P s py ~ P’ M haS °
(с) P is isomorphic with a direct summand of a free R-module.
Proof. (a}~(bj Since itf/K s p, t^re exists an epimorphism a from M onto Pwith
Sy ma"“ on 7°^ * Ф 6 that
We show that
(2л)
M = Рф © К.
Let k e Рф n K, say k = хф with x e P. Then
0 = ka = хфа = xiP = x,
and so k = Оф = 0. Furthermore, if me M, we have
(m — таф)а = ma — (та)фа = 0
since фа = tP. Thus m — таф e Ker(a) = K, and since таф e Рф, it follows that M =
Рф + К. Thus (2.a) holds.
(h)=>(c): According to (2.1) there exists a free R-module F with a submodule К
such that F/K = P. Then by Property (b) there exists a submodule P' of F such that
F = P @K with P P. Thus Property (c) holds.
(c) => (a): Let F be a free module with F = P © T. Since F is projective by (2.3), it
follows from (2.4) that P is projective. □
With a given module one wants to associate a “smallest projective module. For
KG-modules this can be done as follows.
(2.6) Theorem. Let V be a module for a group algebra KG. Then there exists a
KG-module P(F) with the following properties:
(i) P(V ) is projective;
(ii) There exists an epimorphism from P(V) onto V;
(iii) If U is a projective module with V as an epimorphic image, then P(F)
isomorphic with a direct summand of V.
This theorem is proved in [H] VII, 16.8.
98
В Prerequisites—representation theory
(2.7) Definition. The KG-module P(K) described in Theorem 2.6 above is called the
projective envelope (also the projective cover) of И (By Property (iii) of that theorem
P(F) is uniquely determined up to isomorphism, and P(K) = V if and only if V is
projective.)
The concept of a “projective” module has a dual, obtained by reversing the arrows
in Definition 2.2.
(2.8) Definition. An R-module I is called injective if, for every homomorphism
в: A -»I and each monomorphism a: A -» B, there exists a homomorphism ф: В -> I
such that аф = в.
I
0 -----» A -------> В
a
The characteristic property of projective modules described in (2.6) (b) has an
analogue for injective modules.
(2.9) Theorem. An R-module I is injective if and only if, whenever an R-module A
contains a submodule Г = I, A = Г @ В for some R-submodule B.
Proof First suppose that I is injective, and let a: I -> Г (£ A) be with a an isomor-
phism. Let a = a о y, where у: Г -> A denotes the inclusion map. By definition there
exists a map ф: A -> Г such that a = аф.
P
\ *
0 -----> I ——► 'a
a
Therefore фц. = а 1аф\г = а~*(аф) = a~ia = i, the identity map on Г, and in
consequence, for a e A, we have
(a — аф)ф = аф — аффи- = аф — аф = 0.
Непсе а — аф е Ker(^) and so А = Г + Кег(ф). But since ф{1 = i, we have
Г n Ker(0) = 0, and therefore A = Г @ Ker(<J).
To prove the sufficiency, now suppose that the stated condition holds and that the
module-homomorphisms shown in the diagram are given.
2. Projeciive and injective modules
99
I
Fv
о ----— A--------. в
a
We must find a homomorphism ф: В - [ such that аф = fi
It is easy to verify that the subset
К = {(—ав, aa):aeA]
is a submodule of the direct sum I © B. Let W = (/ © B)/K, and define у: I W by
iy = (i, 0) + K.
We claim that Ker(y) = 0. For if (i, 0) = (-ав, aa) e K, then aa = 0, and since a is a
monomorphism, we have i = -06 = 0. Thus, if Г = Im(/J, the map у: I Г is an
isomorphism and has an inverse 0 :T ->1 defined by ((i, 0) + K)[i = i. By the stated
property, W has a submodule B' such that
(2-P)
W = Г © В’.
Let л denote the projection of W onto Г with respect to this direct decomposition,
define 6: В ->IT by
bb = (0, b) + K,
and finally set ф = brfl. Clearly ф e HomR(B, I), and if a e A, we have
ааф = aabr.p = ((0, aa) + K)n0
= ((ав, 0) + K)n0 (by definition of K)
= ((ав, 0) + K)(l = ав (because ав e I).
Hence аф = в, as desired.
For modules over a group algebra KG (more generally over a quasi-Frobenius
algebra cf. [HB], page 86, Remark 7.9) it turns out that the class of projective
modules coincides with the class of injective modules.
(2.10) Theorem. Let V be a module over a group algebra KG. Then V is injective if
and only if V is projectile.
A proof of this is given in
from this theorem that a
plementary submodule.
Theorem 7.8 of Chapter VII in [Н]. In view of(2.9) it follows
projective submodule of a KG-module always has a com-
100
В. Prerequisites—representation theory
(2.11) Definition. Let M be a module over a ring R. An injective envelope (or injective
hull] of M is an injective module I and a module monomorphism p: M -> I such that,
whenever p(M) < J < J, the submodule J is not injective.
A proof of the following theorem can be found in Curtis and Reiner [1], Theorem
57.13.
(2.12) Theorem (Eckmann and Schopf). Let M be an R-module.
(a) There exists an injective envelope (I, p) for M.
(b) If (T, p') is also an injective envelope for M, then there exists an isomorphism
в: l ->Г such that рв = p'.
(2.13) Corollary. Let M be a submodule of an injective R-module I, and let у be a
module-automorphism of M. Then I has an automorphism a such that a|M = y.
Proof. Let J be a submodule of 1 minimal subject to the requirements that
(i) M < J < I and (ii) J is injective. Let i: M -> J be the injection map. The (J, i) and
(J, yi) are evidently injective envelopes of M, and so by (2.12) there exists an auto-
morphism в of J such that 16 = yi. Therefore 0|M = y. Now by (2.9) there is a
submodule В of I such that I = J © B, and each x e I has a unique expression
x = у + b
with у e J and b e B. It is then straightforward to verify that the map a: I -> J defined
by
ха = ув + b
is an automorphism of I satisfying a|M — y. □
We end this section with a complementation theorem; the proof we give exploits
the fact that for group algebras projective modules are injective.
(2.14) Theorem. Let G be a group with a normal elementary abelian p-subgroup N. If
N is projective as an FpG-module, then N is complemented in G.
Proof. By Theorem 18.9 of Chapter A we can identify G with a subgroup of W =
/VQjreg G/N so that, if В denotes the base group of W, we have BnG = N and
BG = W. Since В is complemented by a subgroup H(s G/N) in W, the submodule
N of B, viewed as an FpH-module, is projective and hence by (2.10) injective. Thus N
is complemented in В by some Lf-submodule, M say, and the subgroup L = MH is
a complement to N in B. Consequently G = G n NL = N(G n L), and therefore
G n L is the desired complement to N in G. □
101
3. Modules and representations of K-algebras
3. Modules and representations of K-algebras
Throughout this section К will denote a (commutative) field.
(3.1) Definition. A K-algebra A is a vector space over К endowed with a further
(ob)c = a(bc); a(b + c) = ab + nc; (a + b)c = ac + be
and which satisfies
a(J.b) = A(ab)
for all a, b, c e A and A e K. A K-algebra, being a ring, will always have a multiplica-
tive identity, denoted by 1, and will therefore contain a subring {zl Kj iso-
morphic with К (which we will usually identify with K). By the dimension of A we
mean its dimension as a vector space over K.
Remarks. 1. If A is finite dimensional with basis {a,,..., a„}, then there exist ele-
ments уу e К such that
n
k=l
These multiplication constants y~ completely determine the structure of A. They have
to satisfy equations derived from fulfilment of the associative law of multiplication
and so they are not independent.
2. If A and В are K-algebras, they can be viewed as (K, K)-bimodules, and so the
tensor product A ®к В is defined as a К-space. It becomes a K-algebra if we define
multiplication on pure tensors by
(o ® b)(a' ® b') = aa‘® bb'
and extend bilinearly. An important special case of this construction is when В is an
extension field of K; in this case В is a (K, B)-bimodule and A ®к В can be viewed
as a right В-module; in other words, as vector space over B.
The most important example we shall meet is the group algebra.
(3 2) Definition. Let G be a finite group (written multiplicatively) and К a field The
underlying set of the group algebra KG consists of all formal linear combinations
V a 0 with scalars a e K. The set KG is then viewed as a vector space over К
wi’thf'{fr g e G} as a balis, and multiplication is defined on KG by extending the
multiplication on G bilinearly thus.
102
В. Prerequisites—representation theory
The associativity of this binary operation follows from the associativity of multi-
plication in G, and the rest of the algebra axioms are obvious from the definition.
With respect to the basis G the multiplication constants of KG are given by уД = 1
if gh = к and 0 otherwise. Clearly the group of units of К G contains G as a subgroup,
and DimK(KG) = |G|.
(3.3) Examples. We mention two further examples of K-algebras which are
important.
(a) Let V be a vector space over K. Then the set EndK(E) of all К-linear maps
a: V -> V has a natural K-algebra structure. If a, fl e Endj.(E) and 2, p e K, then
aA + g[l and aft are defined thus:
n(2a + gft) = A (t>a) + g(vft), and v(aft) = (va)ft.
(b) The set Jtlyt, K) of all n x n matrices over К evidently has the structure of a
K-algebra if we define
(AA + pB)y = Aay + gby
with usual matrix multiplication (AB)y = aikbkJ as the binary operation. (Here
A — (°i,)> В — (by)6 ~ff(n, K).) We obtain the well-known algebra isomorphism from
EndK(E) to Jf(n, K) by choosing a basis {n1,of V andmappingae EndK(E)
to (a,;) e .//(n, K), where
Ц-а = X aUvj-
i=l
If A is a K-algebra, we will consider only right Л-modules (in the sense of Definition
A, 3.6) which are finite dimensional as vector spaces over K. (Since К s Л, an
Л-module is a К-module by restriction.) According to A, 2.1, the isomorphism
theorems hold for right Л-modules; so also does the theorem of Jordan and Holder
(A, 3.2). We recall that an Л-module M is called simple, or irreducible, if M has exactly
two submodules, namely 0 and M( A 0). A semisimple (or completely reducible) module
is a direct sum of simple modules according to A, 4.5, and by A, 4.6 such modules
are characterized by the property that every submodule possesses a complementary
submodule. Moreover, by A, 11.5 (Maschke’s theorem) a KG-module is always
semisimple when Char(K) does not divide |G|. In general, an Л-module can always
be decomposed as a direct sum of indecomposable submodules because of the
assumption of finite dimension; moreover, by the theorem of Krull, Remak, and
Schmidt such decompositions have the weak uniqueness property formulated in
A, 4.9. Corresponding definitions for left Л-modules lead to corresponding theorems,
but from now on “Л-module” without qualification will always mean "right A-module
of finite dimension over K".
103
3. Modules and representations of K-algebras
(3.4) Definitions. Let A be a K-algebra. An /-module which is a direct sum of
painvise-isomorphic simp e submodules is called homogeneous. If Mis an 4-Zufc
and N a simple /-module, then the sum of all the submodules of M which are
isomorphic with N is called the homogeneous component of M belonging to N.
By A, 4.4 a homogeneous component is semisimple and by the Jordan-Holder
theorem it is certainly homogeneous. On the other hand, if a module M is semisimple
then K ’
М = М1© -@Л1,
with each Mt simple, and so the homogeneous component of M belonging to N is
just ф {Mt: N\; in particular, every semisimple module is a direct sum of its
homogeneous components.
It follows easily from the equivalence: (b) <=> (c) of A, 4.6 that a submodule of a
semisimple module is again semisimple; the next result shows that submodules respect
the decomposition into homogeneous components.
(3.5) Lemma. Let M = Mi © • • • © M,be a decomposition of a semisimple A-module
M into its homogeneous components M:. If U is a submodule of M, then
U = (UnM1)©--©(UnM,),
and {U n M,: i = 1,..., t} are the homogeneous components of U.
Proof. Let Nt denote the simple module to which M, belongs and Ut the (possibly
zero) homogeneous component of U belonging to A' (i = 1,..., t). If IK is a simple
submodule of U, then W = Nt for some i. and so
U = Ui®-@U,.
Since Ut is a sum of simple submodules isomorphic with A„ we have Ц < M;. On
the other hand, U n M, is also a sum of copies of A, by the Jordan-Holder theorem,
and therefore U n Mf < l/f. Hence Ц = U ri Mj for f = 1,..., t. О
It is clear that quotient modules, as well as submodules, of semisimple modules
are again semisimple. For later purposes we also need the fact that the class of
semisimple modules is “residually closed .
(3 6) Lemma. Let M be an А-module with submodules such that M/M2 is
semisimple for i = 1....t. Then М/(П'=1 MJ is semisimple.
Proof. By induction on t we may suppose that t = 2 and without loss of g^erah У
that Mj nM2 = 0. Since M/M, is semisimple there exists a _
(M, + in M/М.. Obviously IT=M/M2 is semisimple. Therefore M —
W © M2 is semisimple.
104
В. Prerequisites—representation theory
(3.7) Definitions. Let M be an Л-module.
(a) The socle Soc(M) of M is the sum of all the simple submodules of M.
(b) The radical Rad(M) of M is defined dually to be the intersection of all the
maximal submodules of M. The quotient M/Rad(M) is called the head of M.
Remarks. 1. By A, 4.6 the socle of M is the largest semisimple submodule of M.
2. By (3.6) the radical of M is the smallest submodule with semisimple quotient
module; in other words, M/Rad(M) is the “largest” semisimple quotient module of M.
3. By analogy with groups the submodule Rad(M) might be dubbed the “Frattini
submodule” of M; in fact, some authors denote it by Ф(М). Also it is sometimes
denoted by J(M) because of its relation with the Jacobson radical (see section 4).
We will now describe the well-known connection between linear representations,
matrix representations and modules. In fact, they are all equivalent, being simply
different ways of viewing and recording the same information.
(3.8) Definitions. Let Л be a finite dimensional K-algebra. A linear representation
(respectively matrix representation) of A is an algebra homomorphism from A into
EndK(F) (respectively into J((n, Ю) for some vector space V of finite dimension n,
called the degree of the representation. Of course, an algebra homomorphism is
simply a К-linear map в which satisfies 0(ab) — 0(a)0(b) for all a, b e A and 6(1) = 1.
Let M be an Л-module, let a e Л, and let 0a: M -> M denote the map defined thus:
(3.a) m —> ma
for all m e M. The axioms RM1 and RM3 of A, 3.6 ensure that 0„ e End(M). More-
over, RM2 and RM3 guarantee that 0Xa+lib = Л0а + p0b an^ — QA> and hence that
the map
6: a -> 0„
is a linear representation of Л, non-zero because 0b is the identity map on M.
Conversely, given a finite dimensional К-space M and a representation в: a -» 6„
from A to EndK(M) such that 6[ is the identity map on M, the Л-action defined by
Equation 3.a converts M into an Л-module.
The connection between linear and matrix representations is obtained by choosing
a К-basis {nq,..., m„} for M. If a e EndK(M), then the equations
(З./i) = X (i = 1,..., n)
y=i
associate with a an n x n matrix Ло = (o;j) with entries in К and give rise to an algebra
isomorphism a -> Aa from EndK(M) onto .ff(n, K), which of course depends on the
choice of basis for M. The existence of this isomorphism allows us to move freely
between linear and matrix representations with scope to simplify proofs by judicious
choice of bases. Furthermore, given a matrix representation a of Л of degree n such
3. Modules and
representations of K-algebras
105
that aa — (fly), we can make an arbitrary
an A -module by defining
vector space M of dimension n over К into
mia = £ aijm)
,n)
and extending the action linearly to the whole of M.
ensure that Axiom RM4 of A, 3.6 is satisfied.)
For group algebras there is more to be said.
(Of course, we need la = to
(3.9) Definitions. Let G be a group, К a field, and V a К-space (of dimension n say).
A linear (respectively matrix) representation of G is a group-homomorphism from G
into GL(F) (respectively GL(n, K)). [The symbols GL(F) and GL(n, K) denote
respectively the groups of non-singular linear transformations of V and non-singular
(invertible) n x n matrices over KJ
The map a -»A = (o,7) defined by (3,/f) yields, on restriction, a group-isomorphism
from GL(F) onto Gl.fn, K). Here the important fact is that linear representations of
G and KG are essentially the same: The restriction of an algebra-representation в of
KG to G is obviously a group representation (provided that 10 is the identity).
Conversely, if 0: G -»GL(F) is just a group-representation, then it can be extended
linearly to KG thus:
f Z Z
\gcG / geG
and в becomes an algebra-homomorphism from KG to EndK(F), that is to say, a
representation of the algebra KG. Although the two ideas are interchangeable, it is
important to keep them separate because, for example, Ker(0) is an ideal of KG and
is usually quite different from Ker(fi), which is a normal subgroup of G; even when
the group-representation в is faithful (i.e. when Ker(0) — {lG j), it does not follow that
в is faithful. Ker(0) is also denoted by Ker(G on V).
(3.10) Definitions (Inflation and deflation of modules). Let N < G, and let M be a
K(G/N)-module. If we define a G-action on M by setting mg equal to m(gN) whenever
m e M and g e G, it is evident that M becomes a KG-module, we say by inflation. The
reverse process can be carried out whenever M is a KG-module and N is a normal
subgroup of G contained in Ker(G on M) = {g 6 G: mg = m for all m e М]. In this
case, when we define a G/N-action on M by
m(gN) = mg
M, the К-space M becomes a К(G/Л')-module by deflation.
deflated module with respect to the same normal subgroup
for all g e G and m e
Obviously, inflating a
leaves it unchanged, and vice versa.
Because the early history of representation
matrix representations, the subject acquired a
theory was mainly concerned with
terminology different from the one
106
В. Prerequisites—representation theory
developed for modules in the context of the general theory of algebraic structures.
Thus an irreducible representation is one whose associated module is simple, a
completely reducible representation one whose module is semisimple. Two matrix
representations are equivalent if their associated modules are isomorphic. (We recall
that if A is a K-algebra and if M, N are right Л-modules, then a module homomor-
phism is a К-linear map6:M->N such that (ma}0 = (mO}a for all a e A.) Thus matrix
representations 6, ф of KG (or of G) are equivalent if and only if they have the same
degree, и say, and there exists a fixed invertible n x n matrix X such that
в(а} = Х~'ф(а}Х
for all a e KG (or, equivalently, for all g e G).
If M and TV are isomorphic right KG-modules, obviously Ker(G on M) =
Ker(G on N).
(3.11) Remarks. As we saw in A, .3.4, A, 4.19 and A, 4.20, an abelian p-chief factor of
a group G may be viewed as a simple Fp G-module. This is one reason why representa-
tion theory, especially over finite fields, is so important in the study of finite soluble
groups. Another reason is the facility it offers for constructing soluble groups with
prescribed properties: Let M be an FpH-module for some finite group H. By regarding
M as a (multiplicatively-written) elementary abelian p-group on which H acts as a
group of operators via
mh = mh (the module action)
for m e M and he H, we can form the semidirect product G = [М]Я and thereby
obtain a group which has M as a normal subgroup, Я as a complementary subgroup,
and such that hlmh is the element of M denoted in module notation by mh.
Moreover, we have
Кег(Я on M) = CH(M) (as subgroups of G),
and if M is a simple module, then M becomes a minimal normal subgroup of G; if,
additionally, M is faithful for Я, then G is a primitive group with Я as a stabilizer
(see Chapter A, Section 15). This method of constructing a soluble group G out
of a soluble group Я and an FpG-module M is used frequently in the sequel in
combination with the techniques of representation theory. An important fact in this
context is the following
(.3.12) Proposition. Let К be a field of characterisitic p > 0, and let M be a simple
KG-module.
(a) If G is a p-group, then M = Kc, the trivial simple KG-module;
(h) More generally, Op(G) < Ker(G on M).
Proof. Since Char(K) = p, the prime subfield is Fp = {0, 1K, 2- 1K,..., (p — !)• 1K}.
Let 0 A m e M, and let Mo denote the Fp-subspace of M generated by the finite set
3. Modules and representations of К-algebras 107
{mff: g e GJ. Since 1 < Dim,p(M0) < oo, the subspace Mo is a G-invariant set eontain-
mg p elements for some a > 1. If G is a p-group, by A, 5.4 we can find a non-zero
'"°m^nhiU J a" 9 G C Then "’°K is a simP,c submodule of M
isomorphic with Kc, and so M = m0K, as asserted in (a)
To prove Assertion (b), let R = Op(G), and let A be a simple submodule of the
restriction MR. By Assertion (a) we have N Kc, and therefore CUR) =
{m e M. mx - in for all x e R} is a non-zero subspace of M. If x e R and g e G then
gx = x g for some x' e R; thus if m e CM(R), we conclude that (mg)x = m(x'g) ’= mg
and hence that CM(R) is a submodule of M. Therefore, since M is simple M = C (Ri-
jn other words, R < Ker(G on M). *□
The next elementary result is often used implicitly.
(3.13) Remark. Let M — © • • © Л1„ be a direct sum of simple KG-modules M .
Then
Ker(G on M) = p Ker(G on M,).
In particular, Op(G) < Ker(G on M) when Char(K) = p > 0.
Proof. If m e M, then m can be uniquely expressed in the form m = m, + •• + m„
with each nq in Mt. Thus mg = mom, + ••• + m„ = m,g + ••• + m„g<=>m; = n^g
for all i = 1,..., n. This proves the stated equation, and the final assertion then follows
from (3.12). □
Our next result relates the radical of a КН-module M to the Frattini subgroup of
the semidirect product [M]H described in (3.11).
(3.14) Lemma. Let H be a group, and let M bean ^H-module for some prime p. Then
Rad(M) = Ф([Л/]Н)п Л1.
Proof. Let Mls ..., M, denote the set of maximal submodules of M, regarded as
//-invariant subgroups of [Af] H. Then MtH <• [Л/J/L and therefore
Ф(ГМ1Н) n M < П (M,H nM) = П Mi = Rad(Af).
i=l *=J
On the other hand, if M f L <• MH. then МЦМпЦ is a chief factor of MH by
A, 8.4. Since M is abelian, MnL is therefore a maximal proper H-mvanant sub
group of M, in other words a maximal submodule of M in module termmologje
Since Ф(МН)п M is evidently the intersection of the subgroups MnL as L r
over the maximal subgroups of MH which do not contain M, we conclude that
Rad(M) < Ф(МН) n M.
108
В. Prerequisites—representation theory
By (2.1) (b) every simple module is a quotient of a cyclic free module. There is a special
terminology for cyclic free modules.
(3.15) Definition. If R is a ring with a 1, the cyclic free right R-module, denoted by
Rr, is also called the (right) regular module, particularly when R = KG, a group
algebra. In this case the representation afforded by the regular module is called the
(right) regular representation of KG (or of G).
(3.16) Examples. Let К be a field and G a group.
(a) Let M -- (KG)kg be the regular KG-module.
(i) With respect to the “natural” basis {g: g e G} of KG, an element x of G is
represented by a |G| x |G| permutation matrix whose (g, h)-entry ag h is given by
1
0
if h = gx
otherwise.
(ii) The subspaces A and В of M defined thus:
are submodules of M such that M/A ~ В = KG.
(iii) If К = Fp, then the semidirect product [M] G is isomorphic with Zp Qj„g G. We
shall describe some special properties of M in this case in (11.1).
(b) Let D be a division algebra over K, and let R = .//(n, D), the K-algebra of all
n x n matrices with entries in D. Let E, denote the subspace of ith column vectors
(comprising the matrices in R which are zero off the ith column). Then it is straight-
forward to verify that each E; is a simple submodule of the regular module RK, that
E, = Ej for 1 < i, j < n, and that
Rk — Ei © ” ’ © En
is therefore semisimple.
The ring EndKG(M) of endomorphisms of a KG-module M is clearly a subalgebra
of the K-algebra EndK(M) of К-linear transformations of M, and if M is simple, it is
a skew field (a division algebra over K) by Schur’s lemma (A, 4.8). Such skew fields
arise in the fundamental theorem of Wedderburn, which is stated in the next section,
Section 4. If К is finite, evidently the skew field EndKG(M) is also finite, and then
another theorem of Wedderburn comes into play. We will present a previously
unpublished and completely elementary proof due to Helmut Bender. We are grateful
to him for permission to publish it here. (Recall that a division ring is a ring in which
the multiplicative semigroup of non-zero elements forms a group.)
(3.17) Theorem (Wedderburn). A finite division ring is a field. (In other words, multi-
plication is commutative).
3. Modules and representations of K-algebras ]09
Proof. Let D be a counterexample of minimal order. Since a multiplicatively-closed
subset of a finite group is a subgroup, a subring of a finite division ring is again a
division ring. Therefore D is a non-commutative division ring, all of whose proper
subrings are fields. If F is a proper subring of D, the ring axioms imply that D is a
vector space over F. Therefore, in particular, D is a vector space over its centre
Z = {z e D: zd = dz for all d e D}.
Set n = Dimz(D) and q = |Z|. Since {0, 1} c Z, we know that q > 2.
If d e D\Z, then the centralizer CD(d) of d, being a proper subring of D, is a field.
In fact, Cp(d) is a maximal subring of D because every proper subring containing
CD(d) is commutative and therefore centralizes d. Since CD(d) contains Z and d, we
have Z <- Cp(d), hence Z is not a maximal subring of D, and it follows that every
maximal subring S of D has the form S = CD(d) for any element d in S\Z. Thus if//
denotes the set of all maximal subrings of D, it follows that the subsets S\Z (S e //j
form a partition of D\Z, and therefore
(3-T)
?"-4 = |D\Z|= £ |S\Z|.
Se V
Let S e //, let a = Dimz(S), and for any R £ D let R* denote the set R\{0}. Noting
that S* is a subgroup of the multiplicative group D*, we set m = |ND.(S*):S*| and
observe that the number of conjugates of the form d~'Sd (deD*) of S in D is
|D*: Nd.(S*)| = (q" - \)/m(q“ — 1); hence the contribution of these conjugates to the
right-hand side of Equation 3.y is
(3-<5)
(<-W-D
- 1)
Set H = JVD.(S*) > S* and from our earlier observation recall that CH(d) = S* for
all d e S\Z. Hence the Я-orbits (by conjugation) on S\Z each contain |H:S*| = m
elements, and therefore m divides |S\Z| = q“ — q. Consequently
(3.c)
the contribution to the right-hand sum in (3.y) of all the S in ./ with
Dimz(S) = a is an integral multiple, s say, of (q" — l)/(q° — !)•
If every Se У had the same dimension over Z, then we should have q" 1
s(on _ |)/(u“ - ]) and could then conclude that q" - 1 divides (q - 1 )(<?“ - D since
the highest common factor of q" - q and if - 1 is q - L But this is impossible
because a < n and this implies that q" - 1 > (q - !)(«“ ’) There ore
(3.f) |{Dimz(S):Se У}| > 2.
If m = 1 for some S e .Z an easy calculation, using the fact that q 2^2 and>a^2,
shows that the integer labelled (3.<5), which denotes the s.ze of a D -conjugacy class,
is greater than (qn — q)/2- Therefore if the set
ПО В. Prerequisites—representation theory
</0 = {Sey’:WD.(S*) = S*}
is non-empty, we conclude that
(3.r;) D* acts transitively by conjugation on
It follows from (3.f) and (3,t/) that the set У\УВ's non-empty. Let A e У\У0, set
H = NdAA). and let h e H\A* (non-empty by the choice of Л). Then by our earlier
observation that A = CD(d) whenever d e A\Z, we evidently have CA(H) = Z. Since
the set £l2.0h‘A is obviously a subring of D properly containing the maximal subring
A. we have
(3.0) D = A@hA@-@h'~'A,
where r = Dirn^D). (Here r is obviously the degree of the minimum polynomial of
Л over Л.) Let В denote the maximal subring CD(h). Since CA(h) = Z, it follows at once
from (3.0) that
B( = CD(h)) = Z@hZ@-@hr~1Z;
in particular, Dimz(B) = r.
Next we show that hA* generates H/A*. Let x e H, and write x = with
di e A. Let d e A\Z. Since H normalizes Л \Z, there exists an element d' in A\Z such
that dx = xd', and we obtain
£ = dx = xd' = h‘did'.
i=O i=O
Since A is commutative, the fact that {1, h,..., hr~l} forms an Л-basis ofD then yields:
h~'dh‘ = d' whenever d; 0.
If there exist 0<j<k<r — 1 such that dj 0 dk, we conclude that ’ e CD(d) =
Л, contradicting the fact that 1, h,..., hrt are linearly independent over A. Therefore
at most one dt in the expression for x is non-zero, and it follows that
H = 4*uM*u-urU*
and, in particular, that
m = |ND»(-4*):-4*l = r = Dim4(D) = n/a.
where a = DimzM). Furthermore, h' e A* and r is the smallest positive integer with
this property. But h is an arbitrary element of H\A* and can chosen so that hA* has
prime order in H/A*. Therefore r is a prime.
4. The structure of a group algebra
111
We now suppose that B( = C0(/i)) belongs to ,<Z\.<Z0 and derive a contradtetion
The argument of the preceding paragraph shows that Dime(£>) is also a prime But
Dtmz(B) ., and so Dime(D) = Dimz(D)/Dimz(B) = n/r = a. Thus n is the product
of two primes i and a. Since every element of if lies strictly between Z and D it follows
from (3.C) that n cannot be the square of a prime. Hence the primes a and r are distinct
and in view of (3.e), we can deduce from (3.y) that
(3./)
for suitable s, t e N. Now n = ar and (q" - q, qn - 1) = q _ 1; therefore
q°r - 1 divides (q - l)(q“ - l)(q- - 1).
However, if a > 2, r > 2 and ar > 6. then ar > a + r + 1, and from this it follows
easily that qar — 1 > (q — 1 )(qn — l)(q' — 1), which yields the desired contradiction.
Hence Be.'/0.
We are now in the position to obtain a final contradiction. Let Л, be an arbitrary
element of let ht e ND.(4f )\4f, and let Bt = CD(/i,), the analogue of B. By the
previous argument we conclude that Bj e Ef0 and thus from (З.4) that Bi is conjugate
to B. In particular, Dimz(B,) = r, and as before we obtain Dimz(/1 J = n/r = a. There-
fore {Dimz(S): S e У ) = (a, r}, and again an equation of the form (3.z) holds. As before,
the value of a must be distinct from the prime r, hence a > 2, r > 2 and ar > 6, and
the arithmetic of the previous paragraph yields the final contradiction. □
If M is a simple KG-module, by Schur’s Lemma (A, 4.8) the endomorphism ring
EndKC(M) is a division ring; therefore, when К and G are finite, it is a field by the
preceding theorem. The final result in this section provides another sufficient condi-
tion for EndKC(M) to be a field.
(3.18) Theorem. Let M be a simple KG-module. If К is algebraically closed, then
EndKG(M) s K.
Proof. Let О ф a e EndKC(M). Since К is algebraically closed, there is an eigenvector
m( Ф 0) such that ma = Лт for some Л = л„ e K. Thus, for x e KG we have (mx)a —
(m«)x = Лтх, and since M = mKG, we conclude that a = f,i, where 1 is the identity
map on M. Thus the map a - is an isomorphism from EndAC(M) onto K. □
4. The structure of a group algebra
Throughout this section К will denote a field and all K-algebras will be finite
dimensional.
(4.1) Definition. If A is a K-algebra, the intersection of the kernels of all the irreduci-
ble representations of A is called the Jacobson radical of A and is denoted by J( ).
112
В. Prerequisites—representation theory
Thus
7(Л) = (ae A: Ma = 0 for all simple Л-modules M}.
Clearly 7(Л) is an ideal of Л, and by [H] V, 2.2 it is characterized as the intersection
of all the maximal right ideals of Л; in particular therefore, J(A/J(A)) = 0.
Notation. If N is a subset of an Л-module M and В a subset of Л, then NB denotes
the submodule of M generated by the elements {rib: ne N, be B}.
(4.2) Proposition ([НВ] VII, 1.6). Let A be a K-algebra and M an А-module. Then M
is semisimple if and only if MJ(A] = 0. Thus
(a) Rad(/Vf) = MJ(A), and
(b) Soc(M) = {me M: mJ(A) = 0}.
(4.3) Definition. Let Л be a K-algebra.
(a) The algebra A is called semisimple if its right regular module Aa is semisimple.
(Since AJ(A) = J(A), it follows from this definition and (4.2) that A/J(A) is the
“largest” semisimple quotient algebra of A and that A is semisimple if and only if
7(Л) = 0.)
(b) Let V be an Л-module, and set V, = V(J(A))' for i = 0,1,.... The series
V= V0>V1>V2>--
is called the (lower) Loewy series of V. The quotient module Vt_,/!< is called the ith
Loewy layer of V; it is the “largest” semisimple quotient of И(_,.
The structure of semisimple algebras is well understood.
(4.4) Wedderburn’s Theorem ([H] V, 4.4 and 4.5). Let A be a semisimple K-algebra.
(a) For i = I,..., к there exist division algebras over К such that
>=1
where each Л,- is a 2-sided ideal of A isomorphic with ^f(nh Df), the algebra of all nt x nt
matrices with entries in D,.
(b) There exist exactly к pairwise non-isomorphic simple А-modules V1,...,Vt. When
suitably numbered, these satisfy Кег(Л on Ц) = Л^, and then, in particular, Vt is
faithful for Ai.
(c) Each Di is anti-isomorphic with Нотл(1<, I<); furthermore, DimK(I<) =
n, DimK(D,), and О|тА(Л) = £J=1 nf DimA.(D,).
Remarks. 1. If К is finite, by (3.17) each O, is a field. This is also true when A = KG
for any field К of characteristic p > 0 (see [НВ] VII, 1.10).
2. If К is algebraically closed, then each D, is isomorphic with К by (3.18).
4. The structure of a group algebra j j j
3. In the special case Л = .//(n, D) for some division algebra D over К we
® t/' = /"и . nB /1'mOdU'C Can be de«Wosed thus: Ал =
Г ® Ф F(n copies), where V is isomorphic with any one of the simple submodules
of A comprising column vectors. In the general case, since Л (У- Л) = 0 the
restriction of the right Л-module (A,b to the subalgebra A, determines'the structure
of (Л,)^. Thus
(-4<L = Ц Ф'' ‘ Ф Vi (nt copies).
where Кег(Я on Ц) — and (Ц)л is isomorphic with the simple “column
vector” submodules when .//(«,, D,) is identified with Л,. In particular, the sub-
modules (Ai)a are the homogeneous components of Ал.
4. For a general K-algebra A, Wedderburn’s theorem can be applied to the largest
semisimple quotient algebra A/J(A) (see Theorem 4.6 below).
We now look more closely at group algebras. If Char(K) is zero, or if ChariK) =
p > 0 and p f |G|, it follows from Maschke’s theorem that the right regular module
(KG)kg is semisimple. Therefore KG is a semisimple algebra, and J(KG) = 0 in this
case. If, on the other hand, Char(K) = p > 0 and p divides | G|, because for a given
g e G there are |G| pairs (x, y) with xy = g, the element z = £e6c0 °f KG satisfies
z2 = 0. Since z e Z(KG), we conclude that the subset В = zKG is a non-zero 2-sided
ideal of KG satisfying B2 = 0. Let J be any right ideal of KG such that J" = 0 for
some n e N, and let F be a simple KG-module. Since VJ is evidently a submodule of
V, either VJ = V or VJ = 0. If VJ = V. then V = VJ = VJ2 = •• = VJ" = 0, a
contradiction. Hence VJ = 0 and J < J(KG). In particular, J(KG), and
therefore KG is not semisimple. Thus, with the convention that zero does not divide
any natural number, we have proved the following.
(4.5) Theorem. A group algebra KG is semisimple if and only if Char(K) does not
divide |G|.
Next we focus on the structure of the regular KG-module in the general case when
possibly Char(K) divides | G|. By the Krull-Remak-Schmidt theorem (A, 4.9) we can
write
(4.a)
(KG)a.g = P1® -®P„
where P, ..., P are indecomposable submodules which, to within isomorphism and
a permutation of the suffices, are unique. It follows from (2.6) that each P, is projective
and that any projective indecomposable module is isomorphic WIth one of these. By
Wedderburn’s Theorem 4.4 we have a decomposition
(4.Д)
(KG/J(KG))kg = ЦФ-ФЦ-
into a direct sum of simple modules Ц, and by Lemma 2.1 (b) and the
theorem every simple KG-module is isomorphic with at least one of the Ifs. Our
114
В Prerequisites—representation theory
next result shows that the direct decomposition of (4./?) “lifts” to a decomposition of
(KG)kg of the form (4.a), and, in particular, that t = t'.
(4.6) Theorem ([НВ] VII, 10.3). Given a decomposition (4./3) of KG/J(KG), there
exists a decomposition
(KG)kg = P,@ - @P,
with Pi indecomposable and (P; + J(KG))/J(KG) = If for i = 1, ..., t. Furthermore,
Pi n J (KG) = PiJ(KG), and so the head Pi/PjJ(KG) of Pt is isomorphic with If and is
therefore simple. In particular, each simple KG-module is isomorphic with the head of
some indecomposable projective module.
The next theorem states that a projective KG-module is uniquely determined up
to isomorphism by its head, whence the number of isomorphism types of indecom-
posable projective modules for a group algebra equals the number of its simple
modules.
(4.7) Theorem ([НВ] VII, 10.9). If P and P' are projective KG-modules, then
P/PJ(G) S P'/P'J(KG) if and only if P S P'.
Let Vt, ..., Vk be a complete set of representatives of the classes of simple KG-
modules, and let Pi denote the indecomposable projective KG-module whose head
is isomorphic with V,. If n; is the multiplicity of V, in the decomposition (4.fi), it follows
from (4.6) and (4.7) that
(4-7) (KG}KC = ©niPi,
i=l
when n,P( denotes the direct sum of n; copies of Pf.
(4.8) Theorem. Let M be a KG-module. Then there exists a projective KG-module P
(unique up to isomorphism) such that
(a) there exists an epimorphism e: P -» M, and
(b) M and P have isomorphic heads.
Proof. Let us decompose the head of M thus:
M/MJ(KG) = G, ® • ® U„
with U, simple, and choose projective modules P; such that Pi/PiJ(KG) s Gf (ac-
cording to (4.7)). Then set P = Pt © • © Pr, clearly a projective module. Since
P/PJ(KG) s (P;/P,J(KG)) s M/MJ(KG), there exists a homomorphism 0:
P - M/MJ(KG).
4. The structure of a group algebra
115
P
e
e
M M/MJ(KG)
By definition of a projective module, there exists a homomorphism e:P->M such
that the above diagram commutes, and c is onto because e(P) supplements the radical
MJ(KG) in M (since в is onto). q
(4.9) Remark. It follows from (4.6) that any projective module P* whose head has a
summand isomorphic with U, © • © Ur has a copy of P as a summand. Therefore
the module P described in (4.8) is the projective envelope of M (see Definition 2.11).
Group algebras belong to a special class of algebras called symmetric algebras,
which have some interesting properties; we mention two of these.
(4.10) Theorem. Let A be a symmetric algebra (in particular, a group algebra).
(а) ([НВ] VII, 7.8). An А-module is projective if and only if it is injective.
(b) ([НВ] VII, 11.6). If V is a projective А-module, then I//Rad(l/) s Soc(Iz); in
particular, each indecomposable projective module has a unique minimal submodule, and
this is isomorphic with its head.
Remarks, (a) Theorem 4.10(a) holds for the larger class of quasi-Frobenius algebras.
(b) A consequence of (4.8) and (4.10) is that a projective (or equivalently injective)
KG-module is uniquely determined by its socle.
Not surprisingly, the restriction of a module to a Sylow p-subgroup determines
whether it is projective in characteristic p.
(4.11) Theorem ([НВ] VII, 7.14). Let Char(K) = p > 0, let P e Syl,(G), and let V be
a KG-module. Then V is projective if and only if Vr is projective.
If P is a p-group and Char(K) = p, by (3.12) (a) the trivial module KP is the unique
simple KP-module. It follows from (4.4) that KP/J(KP) S KP, and hence from (4.6)
that (KP)KP is indecomposable. Thus we have:
(4.12) Theorem. If P is a p-group and Char(K) = p. then a KP-module is projective
if and only if it is free.
It follows from (4.12) that if M is a projective KP-module, then AT is a sum of copies
of the regular KP-module; in particular, DimK(M) is a multiple of |P|, and from (4.11)
we can deduce the following.
(4.13) Corollary (Dickson). If Char(K) = p > 0 and M is a projective KG-module,
then |G|p divides DimK(M).
116
В. Prerequisites—representation theory
We state without proof the following deep result of Chouinard.
(4.14) Theorem (Chouinard [1]). If Char(K) = p > 0, a KG-module is projective if
and only if it is projective (or, equivalently, free) on restriction to each elementary
abelian p-subgroup of G.
So far we have investigated the structure of KG as a right (regular) KG-module,
which is equivalent to studying the right ideals of the ring KG. Next we look at its
2-sided ideals. (This is equivalent to studying the module structure of KG viewed as
a right K(G x G)-module via the action x(gt, g2) = gj'xg2 for all л e KG and gt,
g2 e G.) We start with a general K-algebra.
(4.15) Theorem ([НВ] VII, 12.1). Let A be a K-algebra. Then A admits a decomposition
(4.5) A = Bj © • • ® Bb
into indecomposable 2-sided ideals B, (called the block ideals of Л). If A = At ф A2
with A, and A2 ideals of A, then At and A2 are sums of two disjoint subsets of
{B,,..., Bb}; in particular, the decomposition (4.5) is unique up to a permutation of the
suffices.
If 1 = et + • • • + eb is the unique expression for the multiplicative identity 1 of A
with et e B(, then ef is called the block idempotent of Bh and
(i) e, e Z(A),
(ii) Bj = CjA = Ae„ and
(iii) ejCj = SjjCj for 1 < i, j < b.
Furthermore, if V is an А-module, then V = Veb ф • • • © Veb is a submodule direct
decomposition.
Wedderburn’s Theorem 4.2 tells us that A/J(A) = A j © • • © Ak, where each Л; is
a complete matrix algebra over a division algebra Д; in particular, Л,- is a simple
K-algebra and so is certainly indecomposable. By comparison with Theorem 4.6, it
might be hoped that this decomposition of А/ЛА) could be lifted to Л; in other words,
that under the natural homomorphism A -» A/J(A) the indecomposable summands
Bj of A map onto the simple Л/s. Unfortunately this is not the case in general, not
even for group algebras. However, there is a clear connection between the Л.-’s and
the B.’s, and to describe this we introduce two related ideas.
(4.16) Definitions. Let KG be a group algebra, and let P,, Pk be a complete set
of representatives of the classes of indecomposable projective modules; further, set
Pj = Pj/PjJ(KG), so that P„ ..., Pk is a complete set of simple KG-modules by (4.6).
Then
(a) For 1 s i, j < к define the integer cy to be the multiplicity of Pj as a factor in
a composition series of Pr (By the Jordan-Holder theorem cy depends only on i and
j.) The k x к-matrix (c,J is called the Cartan matrix of KG.
(b) Define a relation ~ (called the block equivalence relation) on {Pj,..., Pk} by:
Pt ~ Pj if and only if there exist Qk,..., Qs in {P,,..., Pk} such that
4. The structure of a
group algebra
117
(i) P, = Qi and Pj = Qs, and
(n) Qi and Qi+1 have a common composition factor for each i = 1 2 s - 1
It is clear that ~ really is an equivalence relation, and so we obtain ’a partition
= 0,u-u$,
of the indecomposable projective modules into block equivalence classes dit.
The block equivalence relation determines the decomposition of KG into minimal
2-sided ideals.
(4.17) Theorem ([НВ] VII, 12.4). Let KG = P, ф - фP, be a direct decomposi-
tion of the right regular KG-module into indecomposable projective modules, and let
..., Яъ denote the block equivalence classes defined above. For 1 <i<b let B:
denote the sum of all submodules Pj in the given decomposition that are isomorphic with
a module in Sf. Then
KG = В1Ф-ФВ,,
is the decomposition of KG into a direct sum of indecomposable 2-sided ideals.
If et is the idempotent of the ith block ideal Bf (see the statement of (4.15)), then <?,
is the identity of B,, and so Ре, = P for all indecomposable projective modules P in
dSj. We now extend the meaning of
(4.18) Definition. A KG-module V is said to belong to the ith block (written V e dij)
if 14?, = V, where ef is the identity element in the ith block ideal.
If V e (if and 1 < jV i < b, then Vcj = Veiej = 0 by (4.15); thus a non-zero module
belongs to at most one block. It follows from the final assertion of (4.15) that every
indecomposable module belongs to some block, and, in particular, every simple
module belongs to some block. If i’ e V e ift,, then r = ue, for some и e К and
therefore ve, = ue; = uei = v (whence и = r). Consequently every submodule and
quotient module of a module in the ith block is again in the ith block: in particular,
if V e di,, then every composition factor belongs to Д. On the other hand, if U
is a simple module in and if P is the indecomposable projective module with
P/Rad(P) s U, then (Pcf + Rad(P))/Rad(P) = Ge, = U. Hence Pc, + Rad(P) = P,
and it follows from the definition of Rad(P) that Ре, = P; thus Pe^. We have
therefore shown that a simple module belongs to a given block if and only if it is
isomorphic with a composition factor of some indecomposable projective module in
that block. i .г
Let M be an KG-module with M/Rad(M) = G, Ф • • • © G, with the Ut simple If
Pi is the indecomposable projective module with Pi/Rad(Pj) S Gj, we saw in (4.8) that
M is an epimorphic image of the projective module P-Л © + ,eac
belongs to a fixed block .4?, then each P, is in di (and then obviously P e St). Thus we
have proved the following.
118
В. Prerequisites—representation theory
(4.19) Lemma. Let M be a KG-module such that all the composition factors of its head
M/Rad(Al) belong to the same block dS. Then there exist indecomposable projective
modules P,....Pr in dS such that M e Q(Pj © ©?,); in particular M e dL
The integer c,7, defined as the multiplicity of /’/Rad(f’) as a composition factor of
PL is zero if P, and Pj belong to different blocks by definition of the block equivalence
relation. Thus the indecomposable projective modules {f}} may be so numbered that
the Cartan matrix C = (cy) has the form
Pf’s will effect a further non-trivial decomposition Cj =
where each submatrix Cj is indecomposable in the sense that no renumbering of the
°\ . .
I. If a simple
4'2 /
module U is projective, it is the only indecomposable projective module in its block
(which then comprises just direct sums of copies of U) and the corresponding
submatrix of C is (1). Thus KG is semisimple if and only if C = Ik (where к is the
number of simple KG-modules).
(4.20) Definitions, (a) The indecomposable projective KG-module P with P/Rad(P) =
KG (the trivial simple module) is called the principal indecomposable module and
denoted by P1(KG).
(b) The block containing PfKG) is called the first (or principal) block and is
denoted by .
For p-soluble groups the structure of the first block is fairly well understood. For
example, when we have defined an induced module in Section 6, we shall be able to
give a precise description of the principal indecomposable module of a p-soluble
group. But for the moment we confine ourselves to two fundamental results. The first
implies that a p-soluble group G has only one block if and only if Or.(Gj = 1; the
second gives two criteria for a simple module to belong to the first block.
(4.21) Theorem (Cossey, Fong, and Gaschiitz—see [НВ] VII, 13.5). Let G be a
p-soluble group, and let Char(K) = p. Then KG is a directly indecomposable algebra
if and only if Or (G) = 1.
(4.22) Theorem (Fong and Gaschiitz [1]—see [НВ] VII, 13.7). Let G be a p-soluble
group, and let Char(K) = p. Any two of the following statements about a simple
KG-module V are equivalent:
(a) V is in the first block',
(b) Op.(G) < Ker(G on F);
(c) O„ „(G) < Ker(G on V).
4. The structure of a group algebra
119
. iFi"a”y’. 7 State Brauer’s theorem about “kernels” associated with the principal
block and then give characterizations derived from it of the classes of p-nilpotent
groups and p -groups. 1
(4.23) Theorem (Brauer; see [НВ] VII, 14.8). Let К be a field of characteristic p > 0
and let G be a finite group.
(a) If U is a simple module in the first block of KG and if P(U) denotes the
indecomposable projective module with head U, then
Ker(G on P(U)) = Op (G).
(b) The intersection Q Ker(G on G), taken over the simple modules in the first block
of KG, is equal to Op.p(G).
If G is a p-nilpotent group, it follows from (4.22) that G ( = Op. „(G)) is in the kernel
of the simple modules in the first block of KG. Thus the trivial module KG is the only
simple module in the first block, and. in particular, the composition factors of the
principal indecomposable module Pt are all isomorphic with KG. If V is a simple
KG-module, it follows that the composition factors of Pt ® U are all isomorphic with
U, and since Pt® U is projective, it contains a summand isomorphic with the
indecomposable projective module P(G). In particular, all the composition factors
of P(U) are isomorphic with U, and we conclude from (4.17) that each block of KG
contains a unique simple module. Conversely, if KG is the only module in the first
block, then G = Ker(G on KG) = OPfP(G) by (4.23) (b), and so G is p-nilpotent. Thus
we have proved the following characterization.
(4.24) Theorem ([НВ] VII, 14.9). Let К be a field of characteristic p > 0. Then any
two of the following statements about a finite group G are equivalent:
(a) G is p-nilpotent;
(b) The trivial module KG is the only simple module in the principal block;
(c) Each block of KG contains only one simple module.
Finally, we derive the promised characterization of the class of p'-groups.
(4.25) Theorem. Let К be a field of characteristic p > 0. Then the finite group G has
order prime to p if and only if the principal indecomposable KG-module is simple.
Proof. If G is a p'-group, all indecomposable KG-modules are simple by Maschke s
theorem. Conversely, if P(KG) is simple, by (4.23) (a) we have
G = Ker(G on KG) = Ker(G on P(KG)) = O„(G\,
in other words, G is a p'-group. □
120
В. Prerequisites—representation theory
5. Changing the field of a representation
Norarion. Throughout this section G will denote a finite group, К will be a field, and
L will be an extension field of K. (Thus К is simply a subfield of L.)
Let p: G -> GL(n, K) be a matrix representation of G over K. Since GL(n, K) is
obviously a subgroup of GL(n, L), we can regard p as a matrix representation over
any extension field L of K. If p is irreducible or indecomposable over K, it may not
remain so over L. (For example,
defines an irreducible representation of a cyclic group <</) of order 3 over F2 (or IR),
and is equivalent to the reducible representation
_ /co 0 \
O'* n 2 I
\0 o) )
over F4 (or C), where co e F4\F2 (or co = exp(2m/3)).) In this section we are mainly
concerned with what happens to modules and representations when we regard them
as written in larger (and sometimes over smaller) fields.
(5.1) Definitions (Extending the field of an algebra and a module). Let A be a K-
algebra and M an /-module.
(a) By (1.3) the tensor product A ®K L is an L-space, and it becomes an L-algebra
if we define
(a ® 2)(a' ® A') = aa' ® 22'
for all a, a’ e A and 2, 2' e L, and extend linearly. (If {a,}(6/ is a К-basis of A, then
{a; ® l}je/ is an L-basis of A. and the multiplication constants of both algebras with
respect to these bases are the same.) We will denote A ®K L by AL.
(b) By (1.3) the tensor product M ®K L is an L-space, and it is straightforward to
check that it becomes an A ®K L-module upon defining
(m ® 2)(a ® p) = ma ® 2p
for all m 6 M,ae A, 2, pe L. We will denote M ®K L by ML.
In the special case where A = KG, a group algebra, it is clear that A, = LG. If M
is a KG-module, the action of G on ML is obtained by viewing L as a trivial G-module
and using Equation l.y to define the G-action. Thus
(m ® 2)<y = mg ® 2
for all m 6 M, 2 6 L, and g E G.ll.W = {nq,..., mJ is a К-basis for M, then o/) ® 1 =
1 1 ............... I I IUH х
5. Changing the field of a representation (21
{"’* ® 1 • ’ ’ J is7L-basis for ML (whence the L-dimension of ML equals the
К-dimension of M), and since for g e G we have L 4
(m, ® 1 )g = £ g I) when = £
J J
the matrix representations of G afforded by M and M, with respect to bases US
and ® 1 are identical. Thus, in terms of matrix representations, this procedure
of tensoring a module with an extension field corresponds to regarding the
matrix entries (in fact in K) as elements of L. In view of this, the following result is
obvious:
(5.2) Lemma. For any KG-module V we have Ker(G on V) = Ker(G on VLj.
(We will sharpen this lemma at the end of this section.)
The next theorem states that the Jacobson radical of a group algebra is preserved
under field extensions, and from this it follows that the semisimplicity of a module
is also preserved.
(5.3) Theorem. Let L be an extension of a field K.
(а) ([НВ] VII, 1.5(a)) J (KG ®K L) = J(KG) ®K L.
(b) ([НВ] VII, 1.8). Let V be a KG-module. Then V is semisimple if and only if VL
is semisimple.
The following result is useful when studying the reducibility of modules and
representations under field extensions.
(5.4) Lemma. Let V and W be KG-modules and L an extension of K. Then
HomLC(VL, WL) S HomKC(V, W) ® L.
(This is an isomorphism of L-spaces in general and is an isom vrphism of L-algebras
when V = IV.)
Proof. Follow the proof of Hilfsatz 11.9 in Chapter V of [H] making appropriate
modifications when V W. □
(5.5) Definition. Let A be a K-algebra (as usual, of finite dimension over K). An
irreducible Л-module V is said to be absolutely irreducible if End,,(И S K. (Because
“absolute irreducibility” is the well-established term for this concept, we will often
use “irreducible module” in preference to “simple module in this section)
The following characterizations explain the real significance of absolute
irreducibility.
(5.6) Theorem. Each of the following statements about an irreducible KG-module V
implies each of the others:
122 В. Prerequisites—representation theory
(a) V is absolutely irreducible',
(bl V ®K L is irreducible for all extension fields Lof K;
(c) If К denotes the algebraic closure of K, then V ®к К is an irreducible KG-
module.
Proof. (a)=>(b): By (5.4) we have HomLC( VL, VL) s К ®, L = L.IfVL is not irreduc-
ible. then VL has non-zero submodules Wt and IV2 such that VL = W, ® Il2 by
(5.3)(b). Ifc, denotes the identity map in Hom/c(H<, 14() (identified with the obvious
subalgebra of HomLC(PL, I',)), then EjE2 = 0. Since this contradicts the fact that
HomLC( VL. VL} is a field and has no zero divisors, we conclude that VL is irreducible.
Since it is obvious that (b) => (c), it remains to show that (c) => (a): Set V = У/. If V
is irreducible, then Hom£C(K И = К by (3.18), and so Ноткс(И У)®к К has
К-dimension 1 by (5.4). Consequently Ноткс(Ц У) has K-dimension 1; in other
words, Ноткс(У, P) = K, and therefore P is absolutely irreducible. □
We will obviously be interested in fields large enough to make all the simple modules
absolutely irreducible.
(5.7) Definition. Let A be a K-algebra. Then К is said to be a splitting field for A if
the division rings Dk,...,Dk appearing in the Wedderburn decomposition
A/J(A) s .//(nJ, D,) ® • • • ® ,/I(nk, Dk)
(cf. (4.4)) are all isomorphic with K.
If A = KG, we call such a К a splitting field for the group G.
(5.8) Theorem. Let G be a finite group. A field К is a splitting field for KG if and
only if every irreducible KG-module is absolutely irreducible.
Proof. Let Pj,..., Pk be a complete set of representatives for the isomorphism classes
of KG-modules. Since by definition of the Jacobson radical k'J(KG) = 0 for i =
1. .... k, we can regard each Ц as a simple module for the semisimple algebra
KG/J(KG) and obtain from (4.4) the unique Wedderburn decomposition
KG/J(KG) s © ^(nf, D.)
i=l
with Dj anti-isomorphic with Ноткс(Ц, Ц) for i = 1, ..., k. Since У, is absolutely
irreducible if and only if Ноткс(У;, f<) = K, the conclusion of the theorem now
follows. □
From (5.6) and (5.8) we deduce the following.
Corollary 5.9. An algebraically closed field is a splitting field for any finite group.
5. Changing the field of a representation
123
F°r brevity we will henceforth use the expression “« complete set of irreducible
KG-modules to mean a complete set of representatives of the isomorphism classes
of such modules. Not surprisingly, when К is a splitting field for G, a complete set of
irreducible G-modules over K, when extended, forms a complete set over any exten-
sion L of K.
(5.10) Theorem ([НВ] VII, 2.4). Let L be an extension of K, let Jt = {Ц,.. Ц} be
a set of KG-modules, and set Jt k = {(V,)^,..., (14),}.
(a) If К is a splitting field for G and Л a complete set of irreducible KG-modules,
then J(L is a complete set of irreducible LG-modules.
(b) If L is algebraically closed and Jtk is a complete set of irreducible LG-modules,
then К is a splitting field for G and Jt is a complete set of irreducible KG-modules.
For a given field К and finite group G, it is natural to look for an extension of К
which is a splitting field for G. By (5.9) we know that the algebraic closure К of К is
a splitting field for G. Let Rt,..., Rk be a complete set of irreducible matrix represen-
tations for G over К (in the sense that each irreducible matrix representation of G
over К is equivalent to a unique Л,). Let {a,,..., am} denote all those elements of К
which appear as entries in all the matrices B,(g) as g runs through G and i = 1,...,
k, and let {a,....a„} denote the set of all the roots of all the minimum polynomials
of the elements at, ..., am over K. Since each a, is algebraic over K, the field
L = K(at,..., a„) is a finite extension of K, and if L is a separable extension of K,
then it is an elementary result of Galois theory that L is a Galois extension of K, in
other words, that the group AutK(L) of field automorphisms of L fixing К elementwise
has К as its fixed field. Since R,,..., Rk are matrix representations over L which
remain irreducible over L = K, we conclude from (5.10)(b) that L is a splitting field
for G. Fields whose finite extensions are always separable are sometimes called perfect,
and it is well known that finite fields and fields of characteristic zero have this
property. Thus we have shown the following:
(5.11) Proposition. Let Gbea group, and let К be a perfect field, in particular a finite
field or a field of characteristic zero. Then there exists a finite Galois extension L of
К which is a splitting field for G.
In view of this result, we will now concentrate on the behaviour of extensions VL
of KG-modules V, where L is a finite Galois extension of K, with Galois group Г say.
If R: G -»GLfii, L) is a matrix representation of G, and if у 6 Г = AutK(L), define a
map R*: G -»GL(n, L) by
«'(.</) = K-)
when R(g) = (аЛ Since у preserves addition and multiplication in L, it is clear that
K>(0/i) = R4g)Ri(h) for all g, h e G, and therefore R! is a matrix representation о .
We call R> the Galois conjugate of R under у and now formulate the equivalent concept
for modules.
124
В. Prerequisites—representation theory
(5.12) Definition. Let L be a Galois extension of К with Galois group Г, and let V
be an LG-module. For each у 6 Г we define an LG-module Vy, associated with Ц as
follows:
(a) Regard V as an abelian group, and let Fy denote a copy of V with v -+1? as
the group isomorphism.
(b) View Vy as an L-module by defining
A(uY) = (Uy-1 )r)y
for all A 6 L (elements of Г act on L by right multiplication). It is straightforward to
verify that for all vy, wy e Ky and А, g 6 L we have
(Л + m)ky = At>y + gvy,
2(t>y + wY) = Avy + Awy,
(Ag)vy = 2(Mry),
lLry = t>y,
and so Vy is a vector space over L. We make Vy into an LG-module by defining
(t>y)9 = (rg)y
for all e И g 6 G, and checking that (At>y + gwy)g = Avyg + gwyg, vy(gh) = (t>yg)h,
and vy lc = t>y for all A, ge L, v, w, e F, and geG. Direct calculation shows that if V
affords the matrix representation R. then I/y affords the Galois-conjugate represen-
tation Ry, and so we call Vy the conjugate LG-module to V under у e AutK(L).
It is clear from this definition that F<yi| s (Ку)й, and that U FY if and only if
V S Uy~' for y, 6 e Г. Therefore Г = AutK(L) acts as a group of permutations on the
isomorphism classes of LG-modules. Moreover, it is obvious from the matrix for-
mulation of conjugate representations that V is irreducible if and only if Fy is
irreducible, and so the classes of irreducible KG-modules also form a Г-set. It is also
important to make explicit the obvious fact that
(5.a) Ker(G on V) = Ker(G on Fy)
for any LG-module V and any у 6 AutK(L).
If V is an LG-module, it can obviously be viewed as a KG-module by restriction;
then, of course, its dimension increases because DimK(F) = |L: K| DimL(F). To
signal this viewpoint, let Fo denote V regarded as a KG-module. (Note that Fo and
V have the same underlying set and are, indeed, the same abelian group.)
(5.13) Lemma ([НВ] VII, 1.16(d)). Let L be a Galois extension of K, let V be an
irreducible LG-module, and let Fo denote V regarded as a KG-module by restriction.
Then Vo is homogeneous.
5. Changing the field of a representation
125
We can now enlarge Ц, to (V0)L = F0®K L, an LG-module which contains the
original LG-module V as a summand. (Although the underlying set of К ® L mav
wThaverea5**1’ d’menS1°n 35 an L‘sPace equals that of Ц,.) According to (5.13)
(5./J)
V0 = W®-@W
for some irreducible KG-module W. Since V is a summand of (V0)L, it is a summand
of WL by the theorem of Krull-Remak-Schmidt.
(5.14) Lemma ([НВ] VII, 1.18(a)). Let Lbe a Galois extension of K, and let V be an
irreducible LG-module. Then there exists an irreducible KG-module W such that V is
a summand of WL, and W is uniquely determined up to isomorphism.
With the hypotheses of (5.14) it turns out that
(5-7)
i-(1®xLs©P
>et
(see [НВ] VII, 1.16(a)); the number of summands on the right-hand side is |Г| =
|L: К I, the dimension of L as a К-space. If V is irreducible, each of the homogeneous
components of the right-hand side of (5.y) has | Д| summands, where
Д = Ди = {уеГ:У’== V},
the stabilizer of V in Г (and a subgroup of Г).
We saw in (5.3)(b) that if V is semisimple, then so also is VL. If V is irreducible,
more can be said.
(5.15) Theorem. Let L be an extension of a field K, and let W be an irreducible
KG-module. In view of Theorem 5.3 (b) write
(5.5)
wL = ©••© wr,
where Wlt..., Wr are irreducible LG-modules.
(a) If Char(K) > 0, then IV" whenever i * j.
(b) Assume that L is a Galois extension of К (of any characteristic), and set
Г = Aut.(L). Then the distinct isomorphismtypes among the modules Wt,..., WJorm
a single Г-orbit and the homogeneous components in (5.5) all have the same composition
length; in particular, the modules W„ ..., Wr all have the same L-dimension, namely
DimK(IV)/r. Furthermore, r divides |Г| = |L; K|. , ,
(c) If additionally Char(K) > 0 in (b), then {Wt, ...,Wr}isa single Г-orbit and each
homogeneous component in (5.<5) is irreducible.
Proof. Statement (a) is just Lemma 1.15 in Chapter VII of
implicit in Theorem 1.18 (b) of the same chapter. The key to the proof of (b)
126
В. Prerequisites—representation theory
following: if V is an irreducible summand of WL, and if I'o denotes V viewed as a
KG-module (by restriction), then by (5./3) and (5.y) we have
(5.1.) © И' £ (Ц>)/. £ WL © ® WL (s summands).
yer
In particular, substituting the expression (5.5) for WL in (5.t.) and equating the numbers
of irreducible summands on both sides, we obtain |Г| = rs.
Finally, we observe that Statement (c) follows at once from (a) and (b), □
Remark. The Galois group Г = AutK(L) can be viewed naturally as a group of
operators on WL = W ®K L by
(w ® Л)у = n> ® Ay
for w 6 W, Z 6 L, and у 6 Г. If V is an LG-submodule of WL, then Vy = Vy, and so
when Char(K) > 0, the operator group Г actually permutes the summands Wt in (5.5)
because the decomposition is unique by (5.15)(a).
We pursue this situation further. Assume that the hypotheses of (5.15)(b) hold
and that Char(K) > 0. If {y,,.... y,} is a transversal in Г to the stabilizer A of V in Г,
then
и,, s © v“
i=l
for some irreducible submodule V of WL.
Since r = | Г: A| and rs = | Г | (where s is defined in (5.;.)), it follows that s = | Д | =
\L: L4|, where L& denotes the fixed field of A. In order to identify this fixed field, we
need the concept of a character of a module (or representation).
(5.16) Definition. Let V be a KG-module, and let R: G -»GL(n, K) be the matrix
representation afforded by V with respect to a given basis. The character x = Xv
V (or of R) is the map x: G -» К defined by
X(fl) = Trace(K(g)),
where the trace of an n x n matrix A = (ay) is at, + • • • + a„„. Since for any non-
singular matrix P, we have Тгасе(Р_,АР) = Trace(A), the definition of x is inde-
pendent of the choice of basis of V. Furthermore, x(x 1 gx) = x(fl) for all x,geG, and
so the value of x on a conjugacy class of G is constant (in other words, x is a class
function).
Although character theory is an extensive and important subject and provides
powerful methods for studying the deeper structure of finite groups, we will not go
into it here. (Isaacs’ book [1] is an excellent account of the fundamentals of the subject
and is comprehensive in its treatment of the applications to soluble groups.) For our
immediate purposes we simply need the following.
5. Changing the field of a representation
127
(5.1?) Proposition ([НВ] VII, 1.11). Let К be a field cf characteristic p > 0
Q denote the set of p -elements of a finite group G.LetV,,..., И be a complete set of
irreducible KG-modules, and let / /, be their ihiirnrim Th > P /
' /i---,Zi"e ineir characters. Then, as elements in the
К-space of class functions from Q to K, the set {Z1,..., Zl} is linearly independenl, (jn
fact, it forms a basis of this space if К is a splitting field for G -see [НВ] VII, 3.9.)
If we return to the situation where L is a Galois extension of К with Galois group
Г and V is an irreducible LG-module with character y, it follows from (5.17) that
1ZS 1’’ for some у 6 Г if and only if the character yf afforded by Vy (and defined
thus: yflg) = /(.</)}' for all g e G) is equal to Z. Thus, if we set
Kz = K({x(s):0 e G}),
it follows that Kx is a subfield of L&, the fixed field of the stabilizer & of V. Now
let
£ = {ye Г:;1 fixes Kr elementwise}.
Then Д £ X. If possible, choose о e X\Д. Then V” Ц and therefore z° X by (5.17).
But this means that y(g)o y(g) for some geG, and so a does not fix Kz. We deduce
that Д = X and hence that Кг = L&. If Char(K) > 0, it follows from the discussion
preceding (5.16) that the number s of summands appearing on the right-hand side of
(5.e) equals |L: Kz|. We state this formally.
(5.18) Lemma. Let Lbe a Galois extension of a field К of positive characteristic, let
W be an irreducible KG-module, and let V be an irreducible summand of WL with
character y. Then DimA(lV) = |K(Z): K| Dim, (T). (Full details of this result and the
preceding analysis can be found in Chapter VII, Theorem 1.16 of [HB].)
If К = Кг, it follows from (5.18) that WL is irreducible. Therefore, since a finite
extension of a finite field is always a Galois extension, we have the following theorem
of Brauer’s.
(5.19) Theorem. Let L be a finite field, V an irreducible LG-module with character Z,
and К a subfield of L which contains {/(<;): g e G}. Then there exists a KG-module W
such that WL — V.
In terms of matrix representations, this means that an irreducible matrix represen-
tation R: G ->GL(n, L) is equivalent to an irreducible representation with entries in
the field obtained by adjoining to all the traces of all the matrices R(g) (g e G).
Another closely related and important theorem of Brauer’s in this context is a
characterization of the smallest splitting field for a group in positive characteristic.
(5.20) Theorem ([НВ] VII, 2.6). Let L he an algebraically closed,field of
p > 0. and let A. • •. A be the characters of a complete set of irreductble LG-modules.
Let
128
В. Prerequisites representation theory
К = ' 9 e G, i = l,...,k).
Then К is the unique smallest splitting field for G.
Let G be a group of exponent p“m with p i m. If f is the order of p (modulo m) and
if q = pf the IF, contains all mth roots of unity over Fp. Since the characters are
determined by their values on the p'-elements of G, and since the matrix of a
p'-element x can be diagonalized (with diagonal entries a„ satisfying a°}x> = 1) when
the field contains a primitive o(xjth root of unity, it follows that 0,(x) is a sum of mth
roots of unity. Therefore the field К defined in the statement of (5.19) is contained in
F,, and we have the follow ing theorem of Brauer.
(5.21) Corollary. Let Gbea group of exponent n, let n = p“m with (p, m) — 1, and let
q = р* = 1 (mod m). The F, is a splitting field for G and all its subgroups.
In characteristic zero one also obtains a splitting field for a group by adjoining to
the prime field a suitable root of unity, as in (5.21). This theorem is also due to Brauer.
(5.22) Theorem ([H] V, 19.11). In characteristic zero Q(e2’"/") is a splitting field for
a group of exponent n.
Over a splitting field the irreducible modules for a direct product of groups can be
fully described in terms of the irreducible modules for its components.
(5.23) Theorem ([НВ] VII, 9.14). Let К be a splitting field for two groups G and H,
and let {V,,..., Vk} and {,..., W,} be complete sets of irreducible modules for К G
and KH respectively. Then
{У,®к И<:1= l,...,Jtand j = 1,...,/}
(each regarded as a K(G x Hfmodule according to (1.12)) is a complete set of irreduc-
ible K(G x Hfmodules.
Remark. This theorem fails when the hypothesis that К is a splitting field is dropped.
For example, if V is the irreducible F2Z3-module of dimension 2, then V ® V is not
irreducible as an F2(Z3 x Z3)-module (see Section 9, Theorem 9.8).
We bring this section to a close by stating without proof the following theorem of
Deuring and Noether, which is useful for deriving information about a KG-module
V when something is known about V,..
(5.24) Theorem ([НВ] VII, 1.21). Let L be an extension of afield K, and let V and W
be KG-modules such that W ®K L is isomorphic with a direct summand of V ®K L.
Then W is isomorphic with a direct summand of V. In particular, if W ®K L s V ®K L,
then W S V.
(5.25) Corollary. If V L is a regular LG-module, then V is a regular KG-module.
г
I
1
a
6. Induced modules ,29
(5.26) Corollary. Let L be an extension of a field K.
(a) If V is a simple KG-module and if U is a non-zero section of V, then
Ker(G on V) = Ker(G on V).
(b) If V is a simple LG-module, and if W is a non-zero submodule of Ц, (which denotes
V viewed as a KG-module), then Ker(G on W) = Ker(G on V).
Proof, (a) Since VL is semisimple by (5.3)(b), we may suppose without loss of gene-
rality that V is a submodule of VL. Set N = Ker(G on U). Then the trivial module
LK occurs as a submodule of the restriction (PL)N of VL to hi and hence as a direct
summand since (1-, )N is semisimple by Clifford’s theorem. It then follows from (5.24)
that the trivial module KN occurs as a summand of and so C,.(N) 0. Since hi < G,
the N-submodule CV(N) is G-invariant, and because V is simple, we conclude that
CV(N) = К in other words that hi < Ker(G on V). Since obviously Ker(G on V) < N,
we have justified Part (a).
(b) Let hi = Ker(G on W). Then W < CFo(hi), and therefore CY(bI) 0 since Po
and V have the same underlying sets. It follows from the simplicity of V, as in Part
(a), that N < Ker(G on V). Again, the reverse inclusion is obvious, and therefore
equality holds. □
6. Induced modules
Throughout this section К will denote a field.
(6.1) Definition. Let H be a subgroup of a group G, and let |J"=1 Hr, denote the
partition of G into right cosets of H. Since the elements of G form a basis of KG, we
have the following decomposition
KG = © KHr,
i=l
of K„(KG) into a direct sum of free left КН-modules. Since KG is also a right
KG-module, we may view it as a (KH, KG)-bimodule, and then, for any right
КН-module Ц the tensor product
Pc = P®kh KG
is a right KG-module according to (1.4). We call this the induced module (of V from
H up to G) and reserve for it the notation V . By (1.6) we have
I
(6.a) Fo = @(j/®rf),
K Since I''® rand Г are clearly isomorphic as K-spaces,
where V ® r, = {t> ® r,: veVf since v is r, on
it follows that
130 В. Prerequisites—representation theory
DimK(f'G) = n DimJP) = |G : H\DimK(P).
The action of G on l/G, according to (1.4), is determined as follows. Since {n,..., r„)
is a transversal to H in G, for each g e G there exists a unique element h^g) e H such
that r,0 = /i,(g)rle for some permutation i -> ig of the set {1.n}. Then
(6./() (t>® rjg = v® ijg = v ® h,(g)ris = vh^g) ® rig
for all v e V and 1 < i < n, and, in particular, the elements g of G permute the
subspaces V ® r; of VG in the same way that they permute the cosets of H (viz.
Hr, -► Hrig). If Vaffords the representation R: H -> GL(f'), the induced representation
Rg is defined to be the representation afforded by VG.
The following is obvious from the definition of an induced module.
(6.2) Theorem. Let H be a subgroup of G, and let f\ and V2 be КН-modules. Then
(Ц Ф V2)G = Pf ф Pf.
The main purpose of the next result is to show that induction of modules is a
transitive operation.
(6.3) Proposition. Let H be a subgroup of a group G.
(a) As right G-modules KH ®KH KG and KG are isomorphic, in particular, since
K{1} = К is the trivial module for the identity subgroup of G, we have KG = KG.
(b) If V is a КН-module and H < L < G, then ((P)L)G = Vе.
Proof, (a) The map p from KG to KH ®KH KG defined by p: x -> 1 ® x (x e KG) is
obviously a monomorphism of right KG-modules. Since y®x = 1 ® yx for all
у e KH and x e KG, it follows that p is onto.
(b) By definition we have
(VL)G =(V®KHKL) ®kL KG
= V ®kH (KL ®KL KG) (by (1.7))
= V ®кп KG (by Part (a))
= Vе. □
Let Pbea KH-modulefor a subgroup Hof G, and let r e {ly,..., r„}, a right transver-
sal to H in G. Then the subspace V ® r of the induced module V°( — ф”=1 P ® r,) is
obviously a K(r‘Hr)-module because, for all v e V and he H, we have
(t> ® r)(r lhr) = v ® hr = vh ® r.
Evidently, if T = Ker(H on V), then r-1 Tr = Ker(r“*Hr on P® r). Moreover, if an
6. Induced modules
131
element g in G fixes V ® rt, that is to say if (V ® rfg = v ® r„ then
some h,(a) e H, and so g e r^Hr,. Since T < H, it is easy to see that
r,(l = hSg)r: for
П r, 1 Tr, = Corec(T),
i=l
and so we have proved the following.
(6.4) Proposition. Let H < G, let Vbea KH-module, and let T = Ker(H on V). Then
Ker(G on Vе) = Core0(7).
The next result is the analogue for modules of the well-known Frobenius recipro-
city for characters. Because the modules in question need not be semisimple, there
are two forms, each dual to the other.
(6.5) The first Nakayama reciprocity theorem. Let H <G, let У be a KH-module,
and let W be a KG-module. Then
HomKG(Vc, IF) s HomK„(K IF„),
an isomorphism as vector spaces over K.
Proof. The map q: F->(FG)H defined by г>/ = г® I for all veV is clearly a
КН-monomorphism. For each fi e HomKG(Vе, W) we therefore obtain an e
HomKC(F, W„). We now define a map
ф: HomKC(FG, IF) —»HomKH(K 1F„)
by ф: fl -> i//i, and begin by observing that ф is obviously a К-linear map. We show
that
(1) ф is injective: If i//? = 0. then for all ч e F, g e G we have
0 = (vqfllg = ((t> ® DP)g = (f ® g)P-
Since the tensors г ® g span VG. we have ft — 0.
(2) ф is surjective: Let a be a given element of Hom(F, И„). We define a correspond-
ing у e HomKG(FG, IF) by setting
(p ® a)y = (t’a)a
for all v e И a e KG. (The universal property of tensor products, as applied at several
points in Section 1, ensures that у is well defined.) Since
((с ® a)g}y = (<’ ® aglf = (va№ = «'’ ® a^9
132
В. Prerequisites—representation theory
for all v e F, a e KG, g e G, the claim that у is a KG-homomorphism is justified.
However,
n(y^) = = (v ® l)y = va
for all v e V, and so уф = a. Thus (2) holds and ф is the desired isomorphism. □
Instead of proving the second reciprocity theorem from first principles, we use an
indirect approach via the dual module, a concept which is important in other
contexts.
(6.6) Definition. If V is a KG-module, then the vector space F* = HomA(K K)
becomes a right KG-module if we define the G-action by
for all u e V, f e V*, and g e G. So viewed, V* is called the dual module to V.
(6.7) Lemma. Let H be a subgroup of a group G.
(a) Let V, W be KG-modules. Then
HomKG(F, W) S HomKO(fF*, F*)
as vector spaces over K.
(b) If U is a КН-module, then (U*)c S (VG)* as KG-modules.
Proof (a) For each a 6 HomKG(F, W) we define a map a*: W* -> F* by
= (‘’«)f
for all v e F and / e W*. Clearly a* is К-linear. For all t> e F, f e W*, g e G we have
t’(Cfc)a*) = (va}(fg) = ((ta)g'}f
= ((vg~')a)f = (vg~l}(fa*)
=
Therefore a* e HomKC(IF*, F*).
The map a -► a* is obviously К-linear, and if a* = 0, then (va)f = 0 for all v e F
and / e W*; hence va e Ker(/) for all/ e W*, and therefore a — 0. Consequently the
map a -> a* is a monomorphism of K-spaces.
It is straightforward to verify that the map r: F -► F** defined by
/(t>r) = t?/ (t>eF,/eF*)
is a KG-isomorphism. Hence F^= F** and similarly IF^ №**, and therefore from
6. Induced modules
133
the previous paragraph we conclude that
DimK(HomAC(iy*, V*)) < Йтк(Ноти(к”, И7**)) = DimK(HomKC(K W)).
Thus a -> a* is an isomorphism, as required.
(b) Let {r,, be a right transversal to H in G. We define a К-linear man
(x ®((i fj ®V')=f ил»
for all uL e U and / e U*. (We remark that the elements u„ .u„ in the expression
и = X?=i u, ® r; for an element и of Vе are uniquely determined by u; similarly the
elements fj e U* are uniquely determined in the second sum.)
We show first that V» is a monomorphism. If (£ifi ® D# = 0> then for a given
j e {1,..., n} we have Ujf; = 0 for all и} e V (setting xf = 0 for all i + j) and hence
fj = 0. Therefore Ker(^) = 0.
Next we show that ф is surjective. Let f e Then clearly the map /j: U -» К
defined by
ujfj = Ц ® 'i)Z (“j 6
belongs to U', and we have
(u, ®')) X fj ® rj) = “Ж = for j = 1,..., n.
Since elements of the form Uj ® r^span it follows that / = (Xj/i ® and
therefore ф is a К-space isomorphism.
It remains to show that ф respects the G-action on the two modules. Let g e G and
write lyg — htrie with h,, e H for i — 1, • •. >i- Then
=x u4>tfhf=X
Thus we have shown that ф is a KG-isomorphism from (V* 1° to (VCY
134
В. Prerequisites—representaiion theory
(6.8) The second Nakayama reciprocity theorem. Let H < G, let V be a KH-module,
and let И-’ be a KG-module. Then
HomKG(W, V°) S HomK„(%, V),
an isomorphism of K-spaces.
Proof. Appealing to (6.5) and both parts of (6.7), we conclude that
HomKG(IF, VG) Z HomKG((FG)*, И"*) Z Ноткс((Г*)с, IV*)
S HomK„(F*, (W*)H) S HomJ£„(W'„, V). □
(6.9) Definition. Let W and V be KG-modules with W simple. Ina given composition
series of V, the number »ни( И of factors isomorphic with W is called the multiplicity
of W in V. (By the Jordan-Holder theorem this is a genuine invariant of V.) If V is
semisimple, this number can also be described as the composition length of the
homogeneous component of V for W.
(6.10) Lemma. Let W and V be KG-modules, assume that W is simple, and let I
(respectively m) be the multiplicity of W in f'/Radft) (respectively Soc(F)). Then
(a) I DimK(EndKC(W)) — DimK(HomKG(L If7)), and
(b) m DimK(EndKG(W)) = DitnK(HomKG(IV, f')).
(We remark that when К is algebraically closed, the dimension of EndKC( W) is 1 by
(3-18).)
Proof, (a) Since W is simple, L/Ker(ct) = 0 or W for all a e Нот(Ц IV); thus all
simple submodules of F/Rad(F) that are not isomorphic to W lie in Ker(a). Since
F/Rad(P) is semisimple, it follows that
HomKG(K W) Z HomKG(fV © • • • © W, W) z © HomKG(W, W).
•=1
(b) As in (a) we have
HomKG(IT, V)zHomKO(W, W© © W) z © HomKG(lP, W).
•=1
By taking К-dimensions we obtain the stated results. □
Terminology. For any module V its head is defined to be
Head(F) = F/Rad(F).
(6.11) Theorem. Let H be a subgroup of G, let V be a simple KH-module and V a
simple KG-module. Let dv — DimK(EndKH(l/)) and dv = DimK(EndKC(F)). Then
6. Induced modules
135
(a) d,.m,,(Head(UG)) = dt,mt,(Soc( P„)), and
(b) drm,.(Soc(GG)) = г/цт^НеаШЦ,)).
Proof, (a) With the help of (6.5) and (6.10) we have
4mk.(Head(GG)) = DimK(HomKG(CG, V)) (by (6.10))
= DimK(Hom,,„((J, f',,)) (by (6.5))
= 4mt,(Soc(K„)) (by (6.10)).
Statement (b) follows similarly from (6.8) and (6.10). □
By (2.6) a module is projective if and only if it is a direct summand of a free module.
Let H be a subgroup of G, let И be a КН-module, and assume that V is projective.
Then И is a direct summand of a free КН-module, which by (6.3) (a) has the form TH
for some T = KH ® © KH. It follows from (6.2) that Vе is a direct summand of
(T")c = TG, which is free by (6.3)(a). Thus l/G is also projective. Conversely, suppose
that VG is projective. Since H, acting by right multiplication, permutes among them-
selves the right cosets of H different from H, from (6.ct) we obtain the decomposition
(VG)„ = (C®1)„©( f r®rf)
\i=2 /Н
and it follows that V (~(C® 1)H) is isomorphic with a direct summand of (C0),,.
Since the restriction to KH of a free KG-module is obviously a free КН-module, we
have now proved the following.
(6.12) Proposition. A КН-module V is projective if and only if the KG-module VG is
projective.
The next result describes an important connection between inducing modules and
forming tensor products.
(6.13) Lemma ([НВ] VII, 4.15 (a)). Let H be a subgroup of G, let U be a KH-module
and V a KG-module. Then
UG ®K Vs(U®K Fh)g
By taking H = 1 and U s K„ © • © K„ in (6.13), we can deduce the foUowing.
(6.14) Proposition. Let F and V be KG-modules. If F is free, then F®K Vis free.
An analogous result holds with “projective” in place of “free in (6.14).
(6.15) Proposition ([НВ] VII, 7.19(c)). Let P and V be KG-modules. If P is projective,
then P ®к V is projective.
136
В. Prerequisites—representation theory
In (4.20) we defined the principal indecomposable KG-module P, = P,(KG) to be
the indecomposable summand of the regular module with P1/Rad(PI) S KG; it is the
projective envelope of the trivial module. If К has characteristic p > 0, and if G has
a Hall p'-subgroup H (in particular, if G is p-soluble), then is precisely the module
induced from the trivial KH-module.
(6.16) Theorem ([НВ] VII, 10.12) Let К be a field of characteristic p > 0, and let U
be a p'-subgroup of a group G. Then the principal indecomposable module P,(KG) is
isomorphic with a direct summand of (K„)G, and if H e Hallp.(G), then Pt ~ (K/;)G. In
any case Dim^fPi) < | G: H |.
By A, 13.8 (a) a p-chief factor V of a group G is centralized by Op. P(G). Thus, viewed
as a simple G-module, V belongs to the first block by (4.22). In fact, more can be
said when G is p-soluble: the following theorem of Green and Hill shows that p-chief
factors are isomorphic with composition factors of P, (FpG), the latter being in general
a proper subset of the simple modules in the first block.
(6.17) Theorem (Green and Hill [1]—see[HB] VII, 15.8). Let H be a Hall p'-subgroup
of a p-soluble group G, and set S — NC(H). Then each p-chief factor of G, viewed as
an fFpG-module, is isomorphic with a composition factor of (KS)G, which is in turn
isomorphic with a quotient of the module (KH)G = Pf^G). In particular, p-chief factors
belong to the first block.
Even more can be said for the complemented p-chief factors: the following theorem
of Gaschutz shows that these arise as composition factors of the second Loewy layer
of the principal indecomposable module.
(6.18) Theorem (Gaschiitz—see [НВ] VIII, 15.5). Let V be a p-chief factor of a
p-soluble group G, and regard V as a simple FpG-module. Let К = Ker(G on V), and
let L be the smallest normal subgroup of G such that K/L is isomorphic with a sum of
copies of V. Let R — Rad(P1(IF/,G)) and R* = Rad(R). Then V occurs as a summand of
R/R* with multiplicity m > 0 if and only if V is a complemented chief factor of G and
m is the composition length of K/L (as an FpG-module).
Brandis [1] has recently given an interesting proof of Gaschiitz’s Theorem 6.18. Let
A denote the direct sum of the complemented p-chief factors in a given chief series of
G. Using the idea of a twisted homomorphism, he constructs an intermediate module
J and epimorphism 6 and e such that
Rad(P,(FpG)) A J 4 A.
Let Wp be the intersection of all maximal subgroups of G containing a given Hall
p'-subgroup of G (see Theorem V, 5.15 below—such subgroups Wp are sometimes
called p-prefrattini subgroups). Then the kernel of 6 turns out to be the induced module
(Rad(Pj(IFpk^,))G, and so extra information is obtained about Pj(FpG). In addition,
these methods yield a new approach to Theorem 6.17 of Green and Hill.
6. Induced modules
137
An important tool in the study of induced representations is the following technical
theorem of Mackey. In order to formulate it, we need to recall the notation of a
double coset.
(6.19) Definition. Let X, Y be subgroups of a group G. For an element g in G a set
of the form
Xgl' = {xgy: x e X, у e У}
is called an (X, Yydouble coset of G.
If XgYc^XhY * 0, there exist x,, x2 e X and у ,,y2 e У such that x.gjy = x2hy2.
But then XgY = Х(х11х2)/1(у2у1’)У = XhY since X and У are subgroups. There-
fore G is partitioned thus:
(6.y) G = (J Xg,Y
1=1
by its double cosets. A set {g1(..., gm] £ G such that (6.y) holds and Xg, У ф Xg}Y
whenever i j is called a full set of (X, Y)-double coset representatives of G.
(6.20) Mackey’s Theorem. Let X and Y be subgroups of a group G, and let {g,,..., gm]
be a full set of (X, Yydouble coset representatives of G. If V is a KX-module, then
(FG)r = ф((У ® gi)xslnr)r-
•=i
[Here V ® gt is viewed as a K(Xe‘ n y)-module via the action
(t> ® g,)x9' = vx ® gf
for all x9- e X9‘ n У]
Proof If G = U?=1 Xr, is the right coset decomposition of G ty X, then
кс = фу®о
according to (6.a). The set {Xr>: r, e Xg, У} forms on orbit under the action of У by
right multiplication on the set {Xr„ .... Xr„} for each i = 1,.... m. Thus, it Щ -
@{У ® ry Г; e XgtY], it follows that
(FG)r = ф Ж-
1=1
Let у, у e Y. Then Xg,y = Xg,y if and only if уу ч e X9- n У If
138
В. Prerequisites—representation theory
is the decomposition of Y into right (Xе- n Y)-cosets, it follows that
Xg.Y = @ XgiSl.
1=1
Consequently
W, = ф V ® g,s, = ф (V ® gjs,.
1=1 i=i
But, as we remarked after the statement of the theorem, V ® gt can be viewed
naturally as a K(Xe- n Y) module, and then, according to (б.я),
@(f'® 9,)S( S((f'® □
i=i
(6.21) Special cases of Mackey’s theorem, (a) In (6.20) let X = Y = N s G. Then
NgN = Ng for all g e G, and the (N, 7V)-double coset decomposition coincides
with the right coset decomposition of G by N. Therefore, if V is a K/V-module, we
have
(VC)K=®V®9b
1=1
where V ® g-t is a KJV-module via
(t> ® gjn = 1’(д1ПдГ' }®gt-
(b) If G = X Y, there is just one (X, Y)-double coset, namely G itself. Then Mackey’s
theorem becomes
(YG)y = (f'yrl.)‘'.
(c) If G = XY and X n Y = 1, then
(FG)y =ЦИ1))Г S KY® --®KY,
a direct sum of DimK(F) copies of the regular К Y-module.
(d) If G = X x Y, then
(6.<5) I'CS1'®RT,
where the G-action on the tensor product is the action for a direct product described
in (1.12). This fact is most easily seen by returning to the definition of an induced
module, in particular to the description in (б.я); since Y is a transversal to X in G, for
x e X and у e Y we have
7. Clifford’s theorems
139
vx®yy
for all v e И у e У, and (6.<5) is clear.
- 1° 1CB°?c1UIde 'b'S section we return briefly to the twisted wreath product, defined
in A. 18.15. Let H be a subgroup of a group G, and let {r,.r„} be a right trans-
versal to H in G. Assume that H is a group of a operators for a group X that is to
say, there is a homomorphism <r: H -> Aut(X) from which a H-action on X is defined
by setting
x" = the jmage of x un(jer
for all x e X and h e H. If X is viewed as a “non-commutative H-module” with this
action (and if multiplicative instead of additive notation is used), we can construct
the corresponding “induced module” XG to be the direct product
B = (X®r1)x(X®r2)x-(X®r„)
of copies X ® ij of X, with a G-action which permutes the direct components
according to the rule
X ® I/ -> (x ® r,)9 = x1'1 ® r19
when r,g — htrlg for g e G and 1 < i < n. The group multiplication for the component
X ® i\ is defined by (x ® r,)(y ® r() = xy ® r, for all x, у e X. It is routine to check
that, with the stated action, G is a group of operators for В and that the semidirecl
product [B]G is isomorphic with the twisted wreath product A'rL(// G. Because of
the close similarity between the two constructions, many facts about induced modules
are also true for the base group of a twisted wreath product; for example, the formula
for the kernel of G on В is given by (6.4), and the analogue of Mackey’s theorem also
holds. Whether a result carries over to the non-commutative situation is usually
immediately clear from the proof of the commutative case. More detailed information
on this subject can be found on p. 228 of Bryce and Cossey [8].
7. Clifford’s theorems
If V is a KG-module and N is a normal subgroup of G, what can be said about the
restricted K/V-module f'ft? Thanks to work of Clifford in the 193O’s, a fairly complete
answer can be given to this question in the case V is irreducible. Our first elementary
observation is that VN is semisimple (cf. A, 4.13(c) for a multiplicative version of this
result).
(7.1) Lemma. Let V be a simple KG-module. let N < G, and let W be a simple
submodule of VN. Then the subset Wg = {w0: w e Ж} of V is a simple submodule of
fA., and
140
В. Prerequisites—representation theory
V= £ Wg.
geG
In particular, VN is a semisimple KN-module.
Proof. First note that, for a given g e G, we have
(7.a) (wg)n = w(gng')g = (wn')g = w'g e Wg
for all w e W and n e N since n' = gng~’ e N. Hence Wg is an N-submodule. Clearly
the map w -»wg is a К-linear isomorphism from W to Wg, and (7.a) shows that V is
an N-submodule of W if and only if Ug is an N-submodule of Wg. Thus Wg is simple.
Since the subset £9eC Wg >s obviously a non-zero G-submodule of V, it coincides
with V when V is simple. Finally we cite A, 4.6 to conclude that If, as a sum of simple
modules, is semisimple. □
We now make a formal definition to describe the way in which the subspace Wg of
V in the above lemma is an N-module.
(7.2) Definition (Conjugate modules and stabilizers). Let N < G, and let W be a
KN-module.
(a) Let W be a copy of the К-space W, and let w denote the image of w (e W) under
some fixed К-linear isomorphism from W to W. For a fixed element g e G define an
N-action on W by
(7(?) wn = w(g~'ng)
for all w e W and n e N. It is easy to verify that W becomes a KN-module under this
action. We call it the conjugate of W by g and denote it by W9. [If the element tv of
W9 is denoted by we, then w9n = (wn®)9 for all w e W. Evidently W9 1 is isomorphic
with the N-module Wg of (7.1).]
(b) The stabilizer /<,(И ) of Win G is defined thus:
IC(IT) = {g e G: W9 IT as KN-modules}.
Clearly /(.(И7) is a subgroup of G. It is sometimes called the inertia subgroup of W.
Remarks. 1. Let g eN, and define a map 0: W9 -» W by w0 = wg~'. Then by (7.Д)
we have
(w0)n = wg = (wg~1ng)g 1 = (wg^ng^ = (wri)0
for all w e W and ne N. Thus 0 is an isomorphism of KN-modules, and it follows
that N < Ic(lC).
2. If R: N -> GL(s, K) is the representation of N afforded by W, then the repre-
sentation R9 afforded by the conjugate module W9 is given by
7. Clifford’s theorems
141
RSW = R(g‘ng).
Moreover, IC(PT) = {g e G. Re is equivalent to R}, and so for each g e lr(W) there
exists a non-singular s x s matrix, P(g) say, such that
P(g) ‘ R(n)P(g} = R»(n)
for all n e N.
The first important result in the theory developed by Clifford is the following.
(7.3) Theorem (Clifford [1]). Let N be a normal subgroup of a finite group G, let К
be a field, and let V be a simple KG-module. Let
(7-Г) VN= lj ®---® К
be the decomposition of the KN-module VN into its homogeneous components Ц (cf.
Definition 3.4). Let i e {1,..., t}.
(a) For each geG, there exists an i'e t] such that Цд = Under the action
i -»ig = i', the set {1,..., f} becomes a transitive G-set, and so G acts as a transitive
permutation group by right multiplication on the homogeneous components V;.
(b) Let W, be a simple submodule of (i= l,...,t). Then the stabilizer T, =
{g e G: Vtg = Ц} of Vt in this permutation representation coincides with /(,(И;); in
particular, | G: fG(W:)| = t, the number of homogeneous components, and the subgroups
{fc(W<):' = 1, • • • > t} form a conjugacy class of G.
(c) if we regard V^asaK T,-module, then the induced module if' is isomorphic with V.
(d) Each homogeneous component viewed as a Trmodule, is simple.
Proof. We first remark that a decomposition of the form (7.y) exists because by (7.1)
the KN-module VN is semisimple.
Let geG, let be a simple submodule of PJ, and let P-. be the homogeneous com-
ponent containing the simple submodule W,g (£ Pf(9 '). Since Ц £ 6 И', we
have Vtg S Щд ® • •• ф W,g, and hence Цд £ P<-. Now (IP'yto1 = Wh and so a
similar argument shows that Pj.g-1 £ И; it then follows that Vc = (V;.g l)g £ V,g, and
therefore fg = PJ.. Thus G permutes the homogeneous components among them-
selves, and since the sum of components in a G-orbit is obviously a KG-submodule
of К the simplicity of V implies that G acts transitively. This proves Assertion (a).
If v.g = If then W<g is one of the simple submodules of l<; since these are all
isomorphic with Wt, it follows that Wf 1 s Wi9 s W, and hence g e fG(H<). Thus
f < IG(W). On the other hand, if g e lG(Wf then W# =s W<, and so Wjg < P<n K<7-
Since PJn Pj = 0 when i * j, it follows that f< = V,g-, therefore fG(W<) < Tt, and
Assertion (b) is proved.
Since Ц7, £ Ц by definition of f, we may certainly regard Vt as a К /.-module, e
r,,..., r, be a transversal to 7J in G. Then
(7.<5) V = (р;)Г1®(И)г2®"Ф(^)г<> and
(7,£) yf = p;®^ Ф vi®rc
142
В. Prerequisites—representation theory
Because the sum on the right-hand side of Equation (7.Й) is direct, each v in V can
be expressed uniquely in the form
V = t?! Г] + V2r2 + ''' + V,r,
with u,, v2,.... v, e Vt. The map
V -> l>! ® Г, + V2 ® Г2 + ''' + V, ® Г,
is clearly a linear isomorphism from V to Vf and since Vfog = Vjtj(g)rJS when гуд =
tj(g)rje rA^e 7} for 1 < j < t, comparison with the G-action on the induced
module l/' described in (6./?) shows that V and l'G are isomorphic as KG-modules.
Thus Assertion (c) is true and implies, in particular, that k}c is simple. If U is a
K7,-submodule of Ц, then UG is a submodule of I}6 of dimension tDimK(G). Hence
U = 0 or U = If in other words, Ц is simple, as claimed in Assertion (d). □
Remarks. 1. Since V:g = I}-, we have W,g = W,.. Furthermore, if Ц is a direct sum of
e copies of И}, then clearly Ц. is a direct sum of e copies of WL.. This common
composition length e of the homogeneous KN-modules Ц,..., V, is sometimes called
the ramification number of V with respect to N.
2. If К = C, we can formulate Theorem 7.3 in terms of ordinary characters, as
follows: Let у be an irreducible character of G, and let ф be an irreducible constituent
of the restriction уд of у to N. Then
Хл = e £ Ф9’
geJ
where J is a transversal in G to the inertia subgroup
1а(Ф)= {деС:ф9 = ф}.
Let us now change the viewpoint by supposing that we are given only a simple
KN-module W (for some N < G, as usual) and that we wish to find some simple
KG-module V such that VN has a submodule (and hence by (7.1) a direct summand)
isomorphic with IK (In this situation we say that “ I lies over И7” and that “ W lies
under F”.) Certainly there always exists a least one such V: for by Nakayama’s
Reciprocity Theorem 6.5 we know that W is isomorphic with a submodule of the
restriction to N of any simple quotient of W°. The next result reduces the problem
of finding all simple KG-modules lying over W of finding all simple modules of the
stabilizer of W lying over W.
(7.4) Proposition. Let К be a field, let N be a normal subgroup of a group G, and let
W be a simple KN-module. Let T = lG(W) = {q e G: W9 s IT}.
(a) Let U be a simple К T-module such that UN has a submodule isomorphic with W.
Then the induced module V = U° is a simple KG-module, and !$, has a submodule
isomorphic with W. Furthermore, every simple KG-module that lies over W can be
obtained in this way.
7. Clifford’s theorems
143
(b) Let T S H < G, and let Y be a simple KH-module such
isomorphic with W. Then Y° is a simple KG-module.
that YN has a submodule
Proof, (a) First observe has only one homogeneous component by (7.3)(b) Let
M be a simple factor module of V = UG. By Nakayama’s first reciprocity theorem
(6.5), we have HomKr(L, MT) HomKC(t/G, M) * 0. Thus MT has a submodule
isomorphic with U, and so by the initial observation MK has a homogeneous com-
ponent, M, say, containing U„ as a submodule. By (7.3)(a) and (b) the K-dimension
of M is DimK(M,)|G : /G(Hj| > DimK(l/)|G: T| = DimK(F). Therefore V = M and
F is simple; furthermore, И7 i s a submodule of which in turn is a submodule of
PN. The final assertion of Part(a) follows directly from (7.3)(c).
(b) Let Y,. = LI .... where U is the homogeneous component whose composition
factors are isomorphic with W. By (7.3) (c) and (d) the subspace L may be viewed as
a simple К T-module such that UH s Y. Thus by the transitivity of induction Y° S
(L'")6 s fja js simple by part (a). q
From (7.3) and (7.4) we obtain at once the so-called Clifford correspondence, which
may be formulated as follows.
(7.5) Corollary. In the notation of (7.4), induction of modules is a bijection from the
set of (isomorphism classes of) simple К T-modules which lie over IV to the set of simple
KG-modules which lie over W. Any G-conjugate of W gives rise to the same set of
KG-modules under this correspondence.
(In the language of characters (when К = C), the Clifford correspondence can be
expressed as follows: Let N < G, let ф e Irr(N), and set T = 10(ф}. Then the map
ф -> фе is a bijection from Irr(T|ф) to Irr(G|<S), where, for example, Irr(7 l<M denotes
the set of irreducible characters ф of T such that ф is a constituent of ф^.)
In view of (7.5) it will obviously be helpful to have more information about the
stabilizer T = lc(W) of a simple KN-module IV when N is a normal subgroup of a
group G. The first elementary observation is the following.
(7.6) Lemma, Let N < Gand let WbeaKN-module. Thenlc(W) < Nc(Ker(Non И7)).
Proof. For any g in G we have
n e Ker(N on H7)o »'« = »’ for a" **’G W
O wg~'(gng 1) = wg~l for all w e W
gng ' e Ker(N on IVе).
then Ker(/V on We) =
kernels, and therefore g
□
Hence Ker(/V on IT) = </ ' Ker(N on IVе )g. If gelG(IV),
Ker(/V on И7) because isomorphic modules have the same
normalizes Ker(/V on И').
144
В. Prerequisites—representation theory
The following consequence is sometimes helpful in constructing simple modules.
(7.7) Corollary. Let N < G and N < H < G. Let U be a simple КH-module such that
If is homogeneous. If Nc(Ker(N on L')) < Я, then Ua is simple.
Proof. Let If = W ® • • • ф И< where W is a simple KN-module. Then H < /г,( И-j
by (7.3). On the other hand, since Ker(N on U) = Ker(N on И-j, by (7.6) we have
/c(U-j Nc(Ker(N on I/)). Therefore H = lc( И7), and so Uc is simple by (7.4). □
When N is abelian, somewhat more can be said about the stabilizer of a simple
KN-module, but we shall postpone our discussion of this case until we deal with the
representations of abelian groups in Section 9. We give one more elementary con-
sequence of Clifford’s Theorem 7.3.
(7.8) Lemma. Let L be a KG-module, let N < G, and assume that
(i) the homogeneous components of VN are simple, and
(ii) G permutes them transitively.
Then V is simple.
Proof. Let W be a (simple) homogeneous component of If, and let T = IC(W). By
(7.3) (b) we can view PL as a simple К T-module, and so Wc (induced from T) is simple
by (7.4). Clearly Assumption (ii) implies that IVе' = V. □
Let PL be a simple KN-module for some N < G, and set T = ЛДИ7). In order to apply
the Clifford correspondence we need to know which К T-modules lie above IV, and
Theorem 7.3 tells us nothing about these. In other words. Theorem 7.4 has no content
when T = G, that is to say when PN is homogeneous for each simple KG-module V
above W.
When /g(PP') = G, we say that IV is invariant in G. In order to be able to say more
about this situation we need the notation of a projective representation. Although
this concept can be formulated in the language of modules, it is easier to understand
and manipulate when described in terms of matrix representations. It should be borne
in mind that the use of the word “projective” in this context has no connection with
its meaning in Section 2; this is another reason for avoiding modules here.
(7.9) Definition. Let G be a finite group. A mapping P: G -> GL(n, K) is called a
projective representation of degree n over К if
(7-i) P(gh) = y(g, h)P(g) P(h),
where 0 / y(g, h) e К for all g, h e G.
A projective representation is irreducible if it leaves no proper non-zero subspace
fixed in its natural action on V(n, K), the vector space of n-tuples over K.
At several points in the proof of the next theorem we need the condition of
“absolute irreducibility” (defined in (5.5)); it will be helpful to express a consequence
of this property in the language of matrices.
7. Clifford’s theorems
145
(7.10) Lemma. Let W be an absolutely irreducible KN-module affording
representation S of N of degree s. Let В be an s x s matrix over К such that
a matrix
(7t/)
BS(n) = S(n)B
for aline N (such a В is sometimes called an interwining matrix of the representation)
Then В = bls for some beK, where ls denotes the s x s identity matrix.
Proof Let , w,} be the basis of W which affords the representation S. If
В = (by), the map
s
P- wi -»E fcytvj
extends to a К-linear map [i: W -» W, and then equation (7.g) implies that j? is a
KN-endomorphism. Hence, by definition of absolute irreducibility, the action of Д is
scalar multiplication by an element b of K; thus В — bls. □
The following second main theorem of Clifford shows how to decompose a represen-
tation R of G when its restriction to a normal subgroup is afforded by a homogeneous
module. It is an indispensible tool for analysing and applying the representation
theory of soluble groups.
(7.11) Theorem (Clifford [1]). Let N be a normal subgroup of a group G. Let V be a
simple KG-module such that VN is a direct sum of r copies of an absolutely irreducible
KN-module W. If R denotes the matrix representation of G afforded by V, then there
exist irreducible projective representations Pt and P2 of G such that
(i) R(g) = pi(fl) ® fz(0) for all geG,
(ii) (Rt )Л' is the (ordinary) representation of N afforded by W, and
(iii) P2 has degree r and P2(n) = Ir for all n e N; in particular, P2 may be viewed as
a projective representation of G/N.
Finally, if P2 is an ordinary representation, then so is P2.
Proof. Let S denote the matrix representation of N afforded by V/ with respect to a
basis JB of W, and let & denote the union of r copies of dB, one chosen in each of the
summands in the direct decomposition
= ire-eir.
Since VN is homogeneous, we have IG(W) = G by (7.3)(b); therefore S is equivalent to
the conjugate representation S9 for all g e G.
With respect to the basis dS of V we obtain
(7.0)
(S(n)
0
0
0 .. 0
S(n) ... 0
0 ... S(n).
146
В Prerequisites—representation theory
for all н e /V; this is an r x r block matrix in which each block is an s x s submatrix,
s being the К-dimension of W. With respect to the same partition of the matrix into
blocks, for each g e G we can write
(7./)
Rit(g) К12(9) Ku(0)'
«21(0) «22(0) «2Д0)
«Г1(0) «г2(0) •• «rr(0)>
where each 7?y(g) is an s x s submatrix. Since R is a representation of G, we have
R(n)R(g) = R(g)R(g~'ng) for all n e N, g e G. On substituting (7.0) and (7.z) in this
equation and equating blocks, we obtain
(7.x)
S(n)«o(0) = Ri2(0)S(g“'ng) = Ry(0)S9(n)
for all n e N, g e G, and 1 < i, j < r.
Let geG. Because S and S’ are equivalent representations, there exists a non-
singular s x s matrix, call it РДд), such that
(7.2) S(n)Pl(g) = Pl(g)S’(n)
for all n e N. First we remark that if g e N, then S(n)S(g) = S(g)S(g-1)S(n)S(g) =
S(g)Ss(n); therefore we can choose Pt(g) to be S(g) when g e N, in other words, we
have (Pi )N = S. Second we observe that the substitution of gng~' for n in (7.2) yields
SWPJg)’1 = P,(ff)'Se~'(n);
hence we may suppose that the equation PJg-1) = Pjg)-1 holds for all g e G. It
follows directly from (7.x) and (7.2) that the matrix В = Ry(g)Pt(g) 1 satisfies S(n)B =
BS(n) for all n e N, and since by hypothesis W is absolutely irreducible, it follows
from (7.10) that there is an element by(g) in К for which В = btj(g)Is. Thus
(7/<) «y(0) = М0)Л(0),
and if we define P2(g) to be the r x r matrix (by(g)), we have
(7v) «(0)= P,(0)®P2(0)
by the definition of the tensor product of matrices in (1.9)(a). If n e N, then Ry(n) =
Si>S(n) by (7.0), and because we have chosen PJzi) = S(n), it follows that by(n) = <5;J;
therefore
P2(n) = Ir
for all ne N.
7. Clifford’s theorems
147
We show next that P, is a projective representation of G. Let q, he G. Reoeated
application of (7.A) shows that the non-singular matrix C = P.tqhr'P ia\p (M cnm
mutes with S(n) for all n e N (here we use the fact that Pi(gh}‘ = />
so again by (7.10) we have C = y(g, h)l for some non-zero y(g, h) e K. Thus P, satisfies
requirement (7.f) for a projective representation and has already been chosen to
satisfy Assertion (ii) of the theorem.
Now we can prove that P2 is also a projective representation. Let о h e G From
(7.v) we obtain
P1 (ff)P.(h) ® y(g, h)P2(gh) = y(g, h)Pt(g)Pt(h) ® P2(gh}
= P^gh) ® P2(gh)
= P(gh)
= P(g)P(h)
= PI(g)Pl(h)®P2(g)P2(h).
Since R(gh) is non-singular, the matrix P, (g)P, (h) has a non-zero entry; thus, equating
the blocks corresponding to this entry in the matrices at each end of this sequence
of equations, we conclude that
(7-4) PAgh) = y(g,h)'P2(g)P2(h),
and therefore P2 is a projective representation. In view of (7.c) we have now justified
Assertion (iii). Moreover, if P1 is an ordinary representation, then y(g, h) = 1 for all
g, he G, and so P2 is an ordinary representation by (7.<£). To complete the proof, it
only remains to point out that if either P, or P2 were reducible, then R = Pt ® P2
would also be reducible. □
In applying the above theorem, there are obvious advantages in being able to
choose the projective representations Pt and P2 to be ordinary representations; for
example, in proofs of results about representations where one argues by induction
on their degrees. One situation where this can be achieved is when the KN-module
W is extendible, in other words, when there exists a KG-module W* such that
W* = Ж An obvious necessary condition for extendibility is that W should be
G-invariant (1С(Ж) = G), but this is by no means sufficient (for example, the
1-dimensional faithful module for Z(£>), D = Dih(8), is invariant but certainly
not extendible). The following theorem gives a sufficient condition for extendibility
which is sometimes useful.
(7.12) Theorem. Let N be a normal subgroup of a group G, and assume that N has a
complement H in G. Let К be an arbitrary field, and let W be an absolutely irreducible
KN-module, affording the matrix representation S. Assume further that
148
В. Prerequisites—representation theory
(i) И' is G-invariant (that is to say, S is equivalent to Se for all g e G), and
(ii) DimK(H') and |H| are coprune.
Then there exists a KG-module IT* such that Wf й IL Furthermore, W* can he so
chosen that the representation S* which it affords satisfies Det(S*(h)) = 1 for all he H,
and then it is unique (up to KG-isomorphism).
Proof. (This is based on the proof(Case 1) of Satz V, 17.12 in Huppert [5]. Since our
hypotheses are different from Huppert’s, we give the proof in full, modulo some
elementary facts about cohomology which can be found in Sections 16 and 17 of
Chapter I of Huppert’s book.)
Let s = DimK(H'), the degree of the representation S, and let h e H. By Hypothesis
(i) there exists a non-singular s x s matrix D(h) such that
(7.o) S'V') = D(h)~lS(ri)D(h)
for all ne N. Since, by definition, Se(n) = S(n~lgri) for all n e N, g e G, it follows that
for i > 1 we have Sl,'(n) = D(h)~‘S(n)D(h)‘, and so, if m = o(h), we can deduce from
(7.10) that D(hf = a!s. Set p = Det(£)(h)). Then pm = Det(z/S) = zs. By hypothesis
(m, s) = 1, and so there exist integers a and b such that am + bs = — 1. Consequently
p~‘ = it“iih = (/“p1"?, and setting v = /“pk, we obtain Det(v£>(h)) = vsp = 1. Thus,
replacing D(h) by vD(h) in (7.o), we may suppose without loss of generality that
Det(£)(h)) = 1 for all h e H and also that D(l) = /,.
Let ne N and h2, h2 e H. In view of the equation
D(hih2)~1S(n')D(hlh2) = S(nl',hl) = D(h2r1D(h1')~'S(n)D(hi)D(h2),
it follows again from (7.10) that
(7.rt) D(h2h2) = c(h,, h2)D(ht)D(h2)
for a suitable c(h2, h2) e Ky, the multiplicative group of K. Moreover, the associative
law
(D(ht)D(h2))D(h3) = DthJlDthJDfh,))
for matrix multiplication yields
c(h,, h2)c(h,fi2, h3) = c(h„ h2h3)c(h2, h3).
Thus c is a 2-cocycle from Z2(H, Kf where K* is regarded as a trivial //-module
(2-cocycles are described in Huppert [5] at the beginning of Section 17 of Chapter I).
On taking determinants in Equation 7.Л, we get c(h2, h2f = 1 for all ht, h2 e H and
therefore cs e B2(H, K'}, the subgroup of 2-coboundaries. However, by Satz I, 16.19
of Huppert [5], the exponent of the group H2(H,K’) = Z2(H, K'f/B2(H, K’) di-
vides |H|, which is coprime with s by hypothesis. Hence ceB2(H, K”), and by
definition there exists a 1-cochain d e Cl(H, K") with
7. Clifford’s theorems
149
c(hI, h2) = d(hi)d(h2)d(hlh2)~'
for all hj, h2 e H.
We now define a function S*: G -> GL(s, K) thus: for g e G, let n e N and h e H be
the unique elements such that g = nh, and set
S*(g) = S(n)d(h)D(h).
We can then deduce that
S*(hih2) = d^hJDththJ = dlMzIdMiMMiWi)
= d(hy)D(hJd(h2)D(h2) = S*(ht)S*(h2).
Furthermore, appealing to (7.o) and the fact that 0(1) = ls, we also have
S*(h)~lS*(n)S*(h) = d(h)~'D(h)~lS(n)d(h)D(h) = S‘(n) = S(h lnh) = S*(h~lnh),
and it follows that S* is a homomorphism, that is to say, a representation of G. Thus,
with a G-action defined by
wg = wS*(g) (w g W, g e G),
the N-module W becomes a G-module, W* say, with the property that WJ = W.
It remains to prove the existence and uniqueness of a representation whose matrices
have determinant 1 on H. Let T denote the composition Det о S*, evidently a
К-representation of G of degree 1, and let L = Kcrn(7’)n H. The group H/L is
isomorphic with the image of TL, which, as a finite subgroup of the multiplicative
group of a field, is cyclic. Let H/L = /haL~), let Т(й0) = к e K, and let I = \H : L|.
Since G L, we have к1 = T(hg) = 1, and there exists an integer r such that (k')! =
к 1 since (I, s) = 1 by hypothesis. Let R denote the representation of G of degree 1
which satisfies Ker(R) = NL and R(fi0) = кг. Then
Det о (S* ® R)(li0) = Det(S*(/i0)® R(M) = «"Tlhg) = 1,
and since <fi0, L> = H, we have Det((S* ® R)(h)) = 1 for all heH. Accordingly, we
can replace S* by S* ® R and take for W* the KG-module associated with the
representation S* ® R of G.
It remains to demonstrate the uniqueness of kF* so chosen. Let Wr* denote a
second extension of W from N to G such that the representation it affords, S** say,
also satisfies Det(S**(h)) = 1 for all heH. It follows from the final statement of
Theorem 7.11 that
S* = S**®Pt,
where P, is an ordinary representation of G with N in this kernel. Since S* and S**
150 В. Prerequisites—representation theory
have the same degree, Pt has degree 1. Let h e H. Then P,(h) = 2, and if m = o(h), it
follows that zm = 1. But
1 = Det(S*(h)) = 2s Det(S**(h)) = 2s,
and since (m, s) = 1 by hypothesis, we conclude that A = 1. Thus P,(G) = P^NH) =
(1), and S* = S**, as desired. □
Remarks. 1. Suppose that N is a normal Hall subgroup of G. Then the existence of
a complement H to TV in G is ensured by the Schur-Zassenhaus Theorem A, 11.3. If,
in addition, TV is p-soluble and К has characteristic either 0 or p, then the degrees of
the absolutely irreducible KTV-representations divide |TV| (see Theorem 7.14 below).
Thus the hypotheses of Theorem 7.12 are always satisfied when (|G: TV), |TV|) = 1 and
W is a G-invariant, absolutely irreducible КTV-module.
2. Becker [1] has shown that the hypothesis in Theorem 7.12 that W is absolutely
irreducible can be relaxed; it is sufficient to assume that W is simple.
An important fact about an irreducible projective representation of a group G is
that its degree divides |G|.
(7.13) Theorem. Let К be a field whose characteristic does not divide the order of a
finite group G. Assume that К is a splitting field for the subgroups of G. Then the
degree of an irreducible projective representation of G divides the order of G.
This theorem appears as Statement (c) in Hauptsatz 24.3 of Chapter V of [H] under
the hypothesis that К is an algebraically-closed field of characteristic zero. The proof
can be modified to cover our hypotheses by appealing to Satz 12.11 of the same
chapter.
The next theorem is an application of (7.11) and (7.13); it will be needed at several
points in the sequel.
(7.14) Theorem. Let К be a splitting field for the subgroups of a group G, and let V
be a simple KG-module.
(a) Let N be a normal subgroup of G, and assume that Char(K) does not divide
|G : N|. If W is a simple summand of then Dim^lf ) divides |G : JV| DimK(W).
(b) Assume that К has characteristic p>0 and that G is p-soluble. Then Dim^fl7)
divides |G|.
Proof, (a) We argue by induction on Dim^L). If О«тк(И = 1, the conclusion is
clear. Let U be the homogeneous component of V containing W, and set T = Ic( W).
Recall from Theorem 7.3 that Dimk(f') = |G : 7jDimK(G) and that V is a simple
KT’-module. If T < G, then DimK(G) < DimK(p); hence by induction DimK(G) di-
vides |T; TV| DimK(W), and the desired conclusion follows.
On the other hand, if 7' = G, we can apply (7.11) to deduce that the represeniation
afforded by V is a tensor product of two projective representations Pt and P2. where
Pi has degree Dim^lT) and P2 is a projective representation of G/N, whose degree
1- Clifford's theorems
151
divides |G/7V| by (7.13). Since DimJF) is the product of the degrees of P, and P, the
conclusion of Part (a) holds also in this case 2’
fr and'nnt6 ?b°reKd ЬУ 1П1иСЙОП °" |C|- LCt M be a maximal subgroup
of G, and note that, because G is p-soluble, the index |G; M\ is either equal to p or
else is a p -number. '
Case 1: |G: M\ = p.
If lzM is inhomogeneous, Clifford’s Theorem 7.3 implies that V s UG where V is a
homogeneous component of FM. In this case M = IG(V) and V is simple; therefore by
induction DimJ (J) divides |M|, and so Dim Jlz) = p DimJL') divides p|M| = |G|.
On the other hand, if f'M is homogeneous, we can deduce from (7.11) that its
composition length is the degree, r say, of an irreducible projective representation of
G/M But such a projective representation may be viewed as an ordinary
irreducible representation of a central extension of Zp, and since centre-by-cyclic
groups are abelian, it follows that r = 1. (For an alternative argument yielding this
conclusion, see Proposition 8.3.) Hence l-'„ is simple, and by induction DimJV)
divides | M |.
Case 2: pj|G:M|.
Let V be a simple summand of f'M. Then DimJlZ) divides | M| by induction. In this
case Part (a) is applicable and implies that DimJL) divides |G: M| DimJL'), thus
yielding the desired conclusion. □
Remark. We shall see in Lemma 9.2 that an absolutely irreducible module for an
abelian group has dimension 1. Therefore by taking the normal subgroup N in
Theorem 7.14(a) to be abelian, we obtain Ito’s celebrated theorem that the degrees
of the irreducible representations of a finite group (over an algebraically closed field
of “coprime characteristic”) divide the index of any abelian normal subgroup. How-
ever, it should be pointed out that in [H] Huppert uses a weak version of Ito’s
theorem (Satz V, 12.6) to prove his Satz V, 12.11, which we have cited for the proof
of our Theorem 7.13; this result, in turn, is used in the proof of Theorem 7.14(a).
Our next goal is a theorem of Fong giving the dimension of a projective indecom-
posable module over an algebraically closed field of characteristic p > 0. The proof
of Fong’s result requires not only the theorems of Clifford but also a deep theorem
of J.A. Green’s about the indecomposability of certain induced modules. Since the
machinery needed for the proof of Green’s theorem goes beyond the limits we have
set for this short account of representation theory, we will simply state the result
without proof and direct the reader to [HB] for the full details. A KG-module V is
said to be absolutely indecomposable if the radical of HomKC(F, И has codimension
one Thus in particular, an absolutely irreducible module is absolutely indecom-
posable. We recall that a field is perfect if it has no finite inseparable extensions; finite
fields and fields of characteristic zero are well known to be perfect.
(7.15) Theorem (Green [I]; see also [НВ] VII, 16.2). Let К be a perfect field gf
characteristic p > 0, and let N be a subnormal subgroup of p-power index in a group
152
В. Prerequisites—representation theory
G. If a KN-module W is absolutely indecomposable, then so also is the induced
KG-module WG.
With the help of (7.15) and the theorems of Clifford one can prove the following.
(7.16) Theorem (Fong [1]; see also [НВ] VIII, 16.9). Let К be an algebraically-
closed field of characteristic p > 0, and let G be a (p-)soluble group. If V is a simple
KG-module and P its projective envelope, then DimK(P) = | G|p DimK( F)p. (We recall
from (4.8) that P is the indecomposable projective module with P/Rad(P) S F.)
Although the proof of this result given in [HB] requires the hypothesis that G is
soluble, it holds more generally for p-soluble groups as Fong shows in his original
paper. With the help of this theorem we obtain the following description of an
indecomposable projective module when its simple head has p'-dimension.
(7.17) Theorem. Let К be an algebraically-closed field of characteristic p > 0, let G
be a p-soluble group, and let H be a Hall p'-subgroup of G. Let P(F) denote the
projective envelope of a simple KG-module F. If p does not divide DimK(F), then
P(F)s(F„)c.
Proof. First note that by hypothesis and (7.16) we have
DimK(P(F)) = |G|p DimK(F).
,Pi ® F
P(F)------->- V
If Pj = P(KC) denotes the indecomposable projective module with trivial head, there
clearly exists an epimorphism from P, ® F onto F. Hence, since P, ® F is projective,
there is also an epimorphism from P, ® F onto P(F). But P, = (KH)G by (6.16), and
so by comparing dimensions and applying (6.13), we conclude that
P(F)SP,® F = (K„)C® Fs(KH® F„)cs(FH)c. □
We are now in a position to prove the final result of this section; it will be needed
in Section 4 of Chapter III.
(7.18) Theorem (Cossey, Hawkes, and Willems [!]). Let H be a Hall p'-subgroup of
a p-soluble group G. Let К be a field of characteristic p > 0, let К denote its algebraic
closure, and let F be a simple KG-module. If the composition factors of F ®к К have
p'-dimension (in particular, if F has p'-dimension over K), then FH is simple.
Proof. If К = K, we can apply (7.17) to conclude that P(F) = (FH)C. Since F„ is
semisimple and P( F) is indecomposable, it follows from (6.2) that FH is simple.
8. Homogeneous modules
153
If К is not algebraically closed, let L denote the amalgam of the fields К and F
where q p is the smallest power of the prime p such that pz - 1 is divisible bv the
л СХТСП1 °f °’ ThCn L iS a Splitt,n« field for C and its subgroups
by (5.20), and so the composition factors of T®KL and V К have the same
(p'-)dimension over their respective fields. Since L is evidently a Galois extension of
K, by (5.14) we have
L®KL = I/,© • @|Z,
where k\, Vr are simple LG-modules, all of the same dimension. Moreover, by
(5.17) we have r = |K(z): K|, where / is the character of G afforded by И
Let U be a simple submodule of VH. Then (5.14) again yields
17 ®k L = If @ • • © L's,
where the Ц are all simple LW-submodulcs of the same dimension and s =
I^-(Zh) ' K|. Since (17H ®K L) s (U ®K L)H and since Ij is an absolutely irreducible
LG-module of p'-dimension, by the first paragraph each (Ц)(, is simple, and con-
sequently If = (fj )H (after suitable renumbering).
Let g e G. By the conjugacy of Hall p'-subgroups in G we can find an x e G. an
h e H, and an element k of p-power order such that g* = hk = kh. If R is the matrix
representation afforded by К then the eigenvalues of Rlk) are all 1, and so we can
write R(k) = I + N, for some nilpotent triangular matrix TV. Let R(/i) = A. Since A
and TV commute, AN is also nilpotent, and so Тгасе(ЛК) = 0. Thus /(y) = /(g*) =
Тгасе(Л + Л/V) = Тгасе(Л) = y(h), and consequently K(%) = K(XiA Hence r = s,
and we obtain Dim(T) = r Dim(fj) = s Dim(L',) = Dim(L'). Therefore V = VH.
□
8. Homogeneous modules
In this section we collect together facts about homogeneous modules which will be
needed later. The importance of homogeneous modules is evident from Clifford s
theorems; also from Lemma 3.5, which shows how the submodules of a semisimple
module are determined by their intersections with its homogeneous components.
Since a homogeneous module Lean be characterized by the combined properties that
(a) T is semisimple, and
(b) the composition factors of V are pairwise isomorphic,
the following observations are clear.
(8.1) Lemma. Submodules and quotient modules of homogeneous modules are
homogeneous.
If U is a simple KG-module and 0 e EndKC(U), then {(«, °'0'''qX"
to be a submodule of V © V isomorphic with U. It follows that if the field К is infinite.
154
В. Prerequisites—representation iheory
then the homogeneous module has infinitely many submodules. (In contrast, if Lj
and V2 are non-tsomorphic simple KG-modules, then fj @ U2 has only four sub-
modules.) On the other hand, if К is a finite field, the submodules of a homogeneous
KG-module can be counted according to their composition length, as follows.
(8.2) Proposition. Let К be a field, let U be a simple KG-module, and let E =
EndKC(U). If a > 1 is a natural number, denote by f'„ the direct sum of a copies of V.
(a) If E is a field and 1 < b < a, then there is a bijection between the set Sab of
submodules of Va isomorphic with Vb and the set of (b — Ifdimensional subspaces of
projective E-space of dimension a — I.
(b) If |E| = n < oo. then the cardinality of Sa b is
(tl°-1)(»°-»)(»“-Я*"1)
J"[a’ ’ (nb-T)(nb-n)...(nb-nb~lY
Proof, (a) Let Ot, 02,..., Oabe not-necessarily-distinct elements of E. Then
{(0,и, в2и,..., 0au): и e U}
is evidently a submodule of isomorphic with U, provided that at least one is
non-zero; furthermore, it is not difficult to see, by considering projections, that every
submodule has this form. Two а-tuples (0,,..., ва) and (0j,.... f)'a) give rise to the
same submodule of fj, if and only if there exists a non-zero endomorphism ф in E
such that f? = фО1 for i = 1,..., a. Thus we obtain a bijection between the set S„ , of
submodules of fa isomorphic with V and the set of points of a projective geometry
of dimension a — 1 over E (which is a field by hypothesis). Furthermore, it is clear
that submodules of Va isomorphic with fj, correspond to projective subspaces of
dimension b — 1.
(b) If E is finite, it is a field by A, 4.8 and Theorem 3.17, and in this case the number
of projective subspaces of dimension b — 1 has the stated form f„(a, b) because
there are (n“ — l)(n“ — n)...(n“ — n*’-1) linearly independent b-element subsets of
an u-dimensional E-space and these are partioned into sets each containing
(nb — l)(nb — n)...(nb — nb~l) E-element subsets which are bases of the same b-
dimensional subspace. (Here we are using the well-known correspondence between
subspaces of a projective space and subspaces of a vector space in one dimension
higher.) □
We now apply a special case of this result to the situation encountered in Clifford’s
Theorem 7.11.
(8.3) Proposition. Let К be a finite field of characteristic p, and let E be a simple
KG-module. Let N be a normal subgroup of G of p-power index, and assume that If
is homogeneous. Then If is simple.
Proof. Let Г2 denote the set of simple submodules of If. If g e G and V e Q, then
Ug efl by (7.1) and Un = V for all ne N. Therefore 12 admits the structure of a
8. Homogeneous modules
155
|'Ле-<E«dKNi(l»)l ” n a P°T °f the рт'те p- If has composition length
«.then |П| (ц l)/(9 !)-<?“ +•+<? +1 by (8.2). Hence |Q| = 1 (mod p),
and from A, 5.4 we conclude that G/N fixes some Uo e fl; in other words, Unq = V
for all geG, Hence Uo = V since by hypothesis V is simple. ' q
The proof of the next result also involves an application of (8.2).
(8.4) Lemma. Let У be a normal subgroup of prime index in a finite group X Let V
be a homogeneous X-module over a finite field К = F„ where g is some prime power.
If V is a simple submodule of Ц assume that Lly is also simple, and assume farther that
has a simple submodule which is not X-invariant. Then |X: У| divides the K-
dimension of U.
Proof Let V be a simple X-submodule of К By hypothesis Vt is simple, and so by
Schur’s Lemma and Wedderburn’s Theorem 3.17 the K-algebras EndKX(l7) and
EndK1 (I/) are finite extension fields of K. If в denotes the representation of X afforded
by U, then EndKr(U) is the centralizer in EndJU) of 0(У) and EndKX(l/) is the
centralizer of 0(X). In fact, в(Х) acts on EndK1(L7) by conjugation because 0( Y) <
в(Х), and via this action X induces a group of field automorphisms on EndK1(l7)
with fixed field EndKJ(L'). Let e and f be the K-dimensions of EndKr(l7) and
EndK V(L') respectively, let a denote the composition length of Ц and use (8.2) to count
the simple X- and У-submodules of V. Our hypothesis then yields
(< - 1)/(<?Z - 1) < (<?“ - l)/(<?e - 1),
and so it follows that f < e. Therefore the fixed field EndKj(L') s Fe/ of X acting on
EndKr(l7) S F4.. is a proper subfield, and so the image of the induced homomorphism
X -> Aut(Fe./(f,/) is non-trivial. Since У is a maximal subgroup of X and is contained
in the kernel of this homomorphism, it follows that У equals this kernel and hence
that X/Y is isomorphic with a subgroup of Aut(F,./F4/) = Ze//; in particular, |X: У|
divides e. Since VY is an EndKr( L/)-module. we conclude that DimK(l7) is a multiple
of DimK(Endir(l/)) = e and hence a multiple of \X : У|. □
The next result concerns a simple КG-module whose restriction to a normal subgroup
is homogeneous. Its arithmetical conclusions will be applied in Chapter XI, Section
1 and Chapter IX. Section 2.
(8.5) Lemma. Let N be a normal subgroup of a group G such that |G: N| is « power
of a prime r Let К = F,. and let V he a simple KG-module such that f„. = V ® • • • © U
a direct sum of s( > 1) copies of a simple KN-module V. Let e denote the K-dimension
of EndKN(l7). Then
(i) e divides Dimt (L'), and
(ii) r divides qels l} + qels~2' + + ’•
Furthermore, if ]G: N\ = r, additionally we have
(iii) s < r, and
(iv) s = r if and only if r divides q* — I.
156
В. Prerequisites representation theory
Finally, if s < r, then
(v) s = о(дг) (mod r), and
(vi) s divides o(q) (mod r).
[We recall that if n is a natural number, then o(n) (mod r) denotes the order of n as
an element of the multiplicative group of Fr.]
Proof. Since К is finite, the K-algebra E = EndKA(L') is a finite extension field of К
by Schur’s Lemma and Wedderburn’s Theorem 3.17. Since V is an E-module, we
have DimK(l7) = |E: K|DimE(G) = e DimE(LT). and Assertion (i) holds.
By (8.2) the set SF of simple TV-submodules of V has cardinality (q‘s — — 1).
Because V < V and V is KG-simple, the group G/N, which clearly acts as a permuta-
tion group on У, has no orbit of length 1. Therefore the length of each orbit is divisible
by a positive power of the prime r, and Statement (ii) now follows.
From now on suppose that |G: 2V| = r, and let {gj,..., g,} be a transversal to TV
in G. If V is a simple submodule of If, evidently Vgi + + Ugr is a non-zero
G-submodule of V and therefore coincides with F; hence, by comparing dimensions,
we have s <r, and (iii) holds.
To prove Statement (iv) first suppose that s = r. Then r divides (qer — l)/(qe — 1)
by (ii), and so, if qe f 1 (mod r), we have q" = 1 (mod r). However, by Fermat’s
theorem that nr s n (mod r) for any integer n, we deduce that qe = qer (mod r) and
hence that q‘ = 1 (mod r). Therefore, in any case, qr = 1 (mod r). Conversely, if q" =
1 (mod r), again from (ii) we ha ve 0 = qe<s~11 + + qe + 1 = s (mod r), whence s = r
by (iii).
Now assume that s <r. Then q* — 1 is invertible in Fr by (iv), and so q“ = l (mod r)
by (ii). Thus, if s' denotes the order of q‘ modulo r, we have s'|s. Suppose, for a
contradiction that s' < s. Then the set У of (proper) subspaces of F isomorphic with
U © • • © V (s' copies) has cardinality
f (. H =
’ ’ S (9я' - !)(«“' (9я' - 9e's'~u)
by (8.2). Since G/N acts fixed-point-freely as a permutation group on У, it follows
that /,.(s, s') = 0 (mod r). Now q” - q“ = 1 - q“ = q‘s’ f 0 (mod r) for i = 1, ...,
s' — 1, and so r divides
(qts - l)/(q“ - 1) = g»»-1' + • • + + 1,
which is congruent modulo r to к = s/s'. But this is impossible because s' < s < r by
assumption. Therefore s' = s and Assertion (v) is justified.
Finally, because by Langrange’s theorem the order of д' divides the order of q
modulo r, Assertion (vi) follows at once from (v). □
(8.6) Definition. Let M be a module for a group G over a field K. An element g e G
is said to have scalar action on M if there exists a fixed 2 e К such that mg — Am for
all m e M.
9. Representations of abelian and extraspecial groups
Finally, we give a useful criterion for a group element to have scalar action on a
module.
(8.7) Lemma. Let h be an element of a group H, and let V be an H-module over a field
K. Then h has scalar action on V if and only if Mh £ M for all maximal K-subspaces
M of К
Proof. If h has scalar action, evidently every subspace of V is an <)i>-submodule, and
so the condition is certainly necessary.
Conversely, assume that h does not have scalar action on К Then certainly
DimK(f) > 2. Suppose, for a contradiction, that uhe <u> for all и eV. Since the
action is not scalar, there exist distinct elements z and p in К and linearly independent
vectors a and w in V such that vh = zt> and wh = gw. But then (i> + w)h <f <(t> + w)li>.
This contradiction shows that V has an element и such that и and uh are linearly
independent. But then we can find a maximal subspace M of V with ueM and
uh ф M, and so the condition Mh £ M fails to hold. Hence the condition is also
sufficient. □
9. Representations of abelian and extraspecial groups
Abelian groups which possess a faithful irreducible representation are cyclic, what-
ever the field of the representation. This important fact, together with Clifford's
Theorem 7.3, implies that a faithful irreducible representation for a group which has
a non-cyclic abelian normal subgroup is always induced from a representation of a
proper subgroup. However, if all the abelian normal subgroups of a soluble group G
are cyclic, either G is metabelian or it possesses certain normal extraspecial subgroups
on which G induces (by conjugation) groups of automorphisms which incorporate
significant information about the structure of G/F(G). In view of Clifford s theorems,
this suggests that the simple modules for abelian and extraspecial groups play an
important part in the representation theory of soluble groups in general. This is
indeed the case, and it is our aim in this section to present the basic facts about these
modules.
The following elementary result is straightforward to verify.
(9.1) Lemma. Let G = (g) be a cyclic group of order n, and let К be* a field
which contains a primitive nth root of unity, £ say. Then the map G-> К define
by
W) = (0 < J < n — 1)
is an irreducible representation of G for i = 0,1... , n — 1, moreover {z0,2,,..., Я i ,
is a complete set of irreducible representations of G over К and the subset
{zm: (m, n) = 1} comprises the ф(п} faithful representations among them.
158 В. Prerequisites- representation theory
Terminology. Representations of degree 1, such as the above, are sometimes called
linear. This usage is ambiguous because “linear representation” also means a homo-
morphism (of a group or an algebra) into a general linear group.
(9.2) Lemma. Let G be an abelian group of order n, let К be a field, and let T be a
simple KG-module. If either
(i) the polynomial x" — 1 splits into a product of linear factors in К [x] (in particular,
if К contains a primitive nth root of unity), or
(ii) V is absolutely irreducible,
then DimK(F) = 1.
Proof. Let p denote the representation of G afforded by F, let g e G, and let r = o(g).
Since p(g)r = p(gr) = b the identity linear transformation, the minimum polynomial
in of a = p(g) divides x' — 1, which divides x” — 1 because o(g) divides |G| = n. Thus
m splits in К[x], and we can write m = (x — e)f for some i. e К and f e К [х]. Let v be
a vector in V such that the vector w = if (a) is non-zero; then w(a — a) = wn(a) = 0,
and >v is an eigenvector of a with eigenvalue a. Let
W — W(e, g) = {w e V: wg = ew},
clearly a non-zero subspace of V. For any x e G and >v e W, we have
(wx)g = (wg)x = t(wx)
because G is abelian; hence их e W, and W is a submodule of V. Because V is simple,
we conclude that V = W and hence that g, an arbitrary element of G, has scalar action
on V. It follows that every subspace of V is G-invariant; in particular, a subspace U
of dimension 1 is a submodule and therefore coincides with V by the hypothesis that
V is simple. Thus DimK(F) = 1 in case (i).
Now suppose that V is absolutely irreducible. If L denotes the algebraic closure of
K, by (5.6) the module U = V L is a simple LG-module. Since x" — 1 splits in
L[x], by the first part we have Dim,(L') = 1. But DimK( V) = Dimb(t/) and we again
have the desired conclusion. □
(9.3) Proposition. Let G be a group, К a field, and V a simple KG-module.
(a) I/DimA(F) = 1, then G' < Ker(G on F);
(b) If G' < Ker(G on V), then the group G/Ker(G on V) is cyclic and has p'-order
if Char К = p > 0.
Proof, (a) If DimK(F) = 1, the representation p afforded by V is a homomorphism
from G into the abelian group K'. Thus G/Ker(G on F) = p(G) is abelian, and the
conclusion of (a) follows.
(b) By regarding F as a G/Ker(G on F)-module by deflation, we may suppose
without loss of generality that Ker(G on F) = 1 and hence that G is abelian. If
Char(K) = p > 0, it follows from (3.12)(b) that OP(G) = 1 and hence that (p, |G|) = 1.
Let L be a splitting field for x|c| — 1 over K. Then L is a Galois extension of K, and
9. Representations of abelian and
extraspecial groups
159
by (5.14) we have
И ®K L = W, © • • © W„
a sum of Galois conjugate simple LG-modules W,, ..., W,. Since Ker(G on W ) =
Ker(G on I) by (5.2) and (5.26)(a), it follows that W, is a faithful LG-module'and
from (9.2) we conclude that Dimt( IT,) = 1. Thus the representation of G afforded by
is a monomorphism from G into L‘, and consequently G is isomorphic with a
finite subgroup of the multiplicative group of a field, which is well known to be cyclic.
□
(9.4) Corollary. Let У be a simple KG-module.
(a) If N < Z(G), then is homogeneous;
(b) If V is faithful for Z(G), then Z(G) is cyclic.
Proof, (a) Let W be a simple submodule of LK, and let geG. Then the map w -» wg
is an TV-isomorphism from W to Wg because (wg)z = (wz)g for all z e TV. By (7.1) the
restriction If is a sum of the submodules {Wg :geG}; it is therefore a direct sum of
a subset of them and is consequently homogeneous.
(b) If fz(C) = W © © W, where W is a simple Z(G)-module, then
Ker(Z(G)on W) = Ker(Z(G)on V) = 1, and so Z(G) = Z(G)/Ker(Z(G) on И7), which
is cyclic by (9.3) (b). □
Next we need some elementary arithmetical facts about the units in = Z/nZ
(9.5) Notation, (a) Let n be a fixed natural number and let m e I. If (m, n) = 1, then
m + nl is a unit in Z„. (We recall that the group of units in consists of the ф(п)
congruence classes {m + nl: 1 < ni < n — 1, (m, n) = 1}, where ф is the Euler func-
tion of number theory.) The order of m + nl in this group of units will be denoted
by o(m) (mod n) (or simply by o(tn) when the ring l„ is understood.) Thus o(m) (mod n)
is the smallest positive integer i such that m‘ — 1 is divisible by n.
(b) If r is a prime, we will write r' || d if t( > 0) is the highest power of r dividing an
integer d.
(9.6) Lemma. Let rbea prime, let a,beN, and let w e I with (r, w) = 1.
(a) If w = 1 (mod r"), then w'b = 1 (mod r°+b). Suppose further that r“||w - 1. If
r° > 2, then r‘+b || wrb - 1, and if r" = 2 and 2c+l || w + 1, then 2°+b+c || (w2 - 1).
(b) Let m — o(w) (mod r) and r“||wm — 1. If r° > 2, then
(m ifl<s<a.
If r° = 2, then
Г 1
o(w) (mod rs)= -( 2
[ 2S“C-1
if s = 1,
if I <s<c + 2,
if s> c + 2.
160
В. Prerequisites—representation theory
Proof, (a) We begin by proving the conclusions of Part (a) under the assumption that
b = 1. Let tv = 1 + nr‘ with neZ. Then
(9,a) w' = (1 + nr‘Y = 1 + rnr‘ + £
Since a > 1 and r > 1, it is clear that wr = 1 (mod r“+1). Moreover, if r“ > 2, it follows
from (9.a) that
(9./?) wr = 1 + nra+l (mod r“+2)
because the prime r divides the binomial coefficient when r > 3. If r° || w — 1, then
(r, n) = 1, and so r“+11| wr — 1 by (9./J). If r“ = 2, then w = 3 (mod 4), and so, if
2c+1 ||w + 1, we have c > 1. Certainly 2a+<:+11| w2 — 1, and we have proved the state-
ments in Part (a) when b = 1.
We will now prove the general case by induction on b. Suppose that b > 1 and
that we know that the conclusions of Part (a) are true for smaller values of b, in
particular that wrb 1 = 1 (mod r‘,+i’-1). Then from the case b = 1 with и1'1' in place
of w we obtain и/" = 1 (mod r°+b). If r° || w — 1, then by induction
r«+b-i||wr‘-> _ i ifr“>2, and
(.«+b+c-1||wr,’-< _ i ifr« = 2.
From the case b = 1 with wr<’1 in place of w it follows that
r“+b || wr” — 1 if r° > 2, and that
ro+b+c||wrb_ i ifr« = 2
in view of the fact that r‘>+i,+c-1 > 2 since b > 2 and c > 1.
(b) First suppose that r° > 2. If s > 1, then m = o(w) (mod r) divides o(w) (mod rs).
But r“ || wm — 1 and consequently o(w) = m (mod rs) for 1 < s < a. Next let s > a,
and let I = o(w) (mod rs). We aim to prove by induction on s — a that I = mr’ °,
which is certainly true when s — a = 0. Let s — a > 0, and note that by induction
o(w) (mod rs-1) = mrs a *. Since o(w) (mod rs-1) divides o(w) (mod rs), it follows that
(9.y) mrs“ l divides I.
But applying Part (a) with b = s — a and wm in place of w, we obtain rs || wmr" ° — 1,
and therefore
(9.6) I divides mrs~°.
Since r is a prime, it follows from (9.y) and (9.Й) that I equals mr’-”-1 or mrs “. But if
9. Representations of abelian and extraspecial groups igj
L=T'J71’W!^thaI^ividesw"r' “ ' - ^wbich contradicts the conclusion from
Part (a) that r || w™ - i. Hence I = mr'“, as desired.
The final statement of Part (b) follows by a similar argument from Part (a). □
(9.7) Construction. Let A = <n> be a cyclic group of order n, and let К be a field
containing a primitive nth root of unity e. (If q is a prime power and n divides q - 1,
then the field К = F, satisfies this hypothesis.) Let F be a subfield of K. When К is
viewed as an F-space, it becomes a faithful А-module over F if we define
(9 £) xa‘ = xe‘ (field multiplication)
for all x e К and 0 < i < n — 1
Proof. It is well known that the multiplicative group F,* of a finite field F, is cyclic.
Since |Fex | = q - 1 and n divides q - 1, it contains an element of order n.
It follows easily from the axioms of a field that the ЕЛ-action on К defined by (9.e)
fulfils the module axioms. □
The following theorem leads to a complete description of the simple modules for a
finite abelian group over an arbitrary field.
(9.8) Theorem. Let A be abelian group of order n, let Fbea field, and let Vbea simple
FA-module which is faithful for A. Then:
(a) A is cyclic, and (n, p) = 1 if Char(F) = p > 0;
(b) There exists an extension field К of F containing a primitive nth root of unity e
such that V is isomorphic with К regarded as the FA-module described in (9.7);
(с) К = F(e) = F«/l) and Dimr(P) = |K: F|;
(d) If F = F,, where q is a prime power, and if m is the smallest natural number such
that n divides qm — 1 (that is to say, m = o(q) (mod n)), then К = Fe„, and, in particular,
DimF(H = m. Furthermore, End^fF) = К in this case.
Proof. Assertion (a) has already been proved in (9.3) (b).
Let F = End„(F)S EndF(H> and let p: E4-»EndF(H denote the representation
of FA afforded by И thus
vb = vp(b)
for all v e V b e FA. Since A is abelian, FA is commutative, and therefore p(FA) S E;
in fact, p(FA) is contained in the centre, Z(E), of E because E is the centralizer of
p(FA) in Endf(F). Since by Schur’s Lemma E is a division algebra, of finite dimension
over F, it follows that Z(E) is an extension field of F with |Z(E): F| finite Let Я - <a>
and set p(a) = e e Z(E). Since FA = fa‘: z, e F}, its image p(FA) is the F-
subalgebra ofZ(E)generated by eand, sincef.is algebraic overF, thiscmncides
F(e), the smallest subfield of Z(E) containing F and e. Thus, setting К - F(e), we ha
К = p(FA). Regarding V as an EndF(П-module in the natural way, we may also
regard V as a К-module (vector space over K) by restriction. If V is a non-zero
162
В. Prerequisites—representation theory
K-subspace of И then U = UK = Up(FA) = U(FA); hence U is an /’/-submodule of
|z and it follows from the simplicity of V that Dirndl7) = 1. Thus if w is a fixed
non-zero vector in Ц the map
0: x -► H’K
is a К-module isomorphism from onto V since 0(K) is a non-zero K-subspace of
И Because V is faithful for A by hypothesis, the image p(A) is a subgroup of order n
of the multiplicative group К " of non-zero elements in K ; in particular, e = p(a) has
order n. Hence F(e) = £(^1). Regarding К as an / Л-modulc via the /-action defined
in (9.e), for к e К we obtain
(ка)в = (к/;) fl = w(K«) = (wx)e
= юкр(а) = (к)вр(а) = MOa,
and therefore в is the desired /’/-isomorphism from К onto V. We have thus justified
Assertions (b) and (c).
Now suppose that F = Fe, and let d = |K : F|. Since К ', which is cyclic of order
— 1, contains the element e of order n, we have qd = 1 (mod n), and therefore m
divides d; in particular, F4„. is a subfield of F4„. But by hypothesis n divides qm — 1,
and so the multiplicative group of contains the unique subgroup of order
n in F,l. Since F4„ = К = £(e) by (c), we therefore have К £ F,„; hence К = Fg„,
and consequently d = m. Since V is a Z(£)-module, its К-dimension is
|Z(£): K| DimZ(£,(l/). But we have already shown that Dirndl7) = 1, and therefore
К = Z(E], When F is a finite field, the algebra £ is a finite division ring, and therefore
Z(£) = £ by Wedderburn’s Theorem 3.17. Thus we have justified the claims in
Statement (d). □
As a simple application of this result and Wedderburn’s Theorem 4.4 we have the
following.
(9.9) Corollary. Let A be an abelian group, let К = F4 (q a prime power), and let
KA/J(KA) = &№,
be a decomposition of the right КА-module KA/J(KA) into a direct sum of simple
modules M,. Let m, denote the order of q modulo | A : Kcr(/ on M,)|. Then DimK(M;) =
m„ Lnd£1(M,) is a finite field with qm‘ elements, and M, A Mj for all 1 < i A J < t.
Remarks. 1. If n is a natural number, we know that
q*"1 = 1 (mod n)
for all q e IL with (n, q) = 1. Therefore, if m = o(q) (mod n), we have m|^(n). If q is a
power of a prime p and if(n, p) = 1, then it is easy to see from (9.8) that acyclic group
9. Representations of abelian and extraspecial groups 163
of order n has ^(n)/m distinct faithful simple modules over IF (“distinct” means
"pairwise non-isomorphic”). ’
2. Another consequence of (9.9) is that the smallest splitting field for Z over IF is
where m = o(q) (mod n). " 9
(9.10) Corollary. Let F = F,, let r be a prime, let m = o(q) (mod r), and let r° ||« - 1).
Assume that r“ > 2. Let A be a cyclic group of order r‘(s > a), and let V be a simple
FA-module which is faithful for A. If В denotes the subgroup of A of order r°, then VB
is a direct sum of rs~“ simple modules, all isomorphic to a given faithful В-module U
and V^UB.
Proof. By (9.4)(a) the restriction L„ is homogeneous and has faithful summands, and
by (9.8)(d) the F-dimension of U is m. Let t = rs~° = \A : B|, and let r,, ..., r, be a
transversal to В in A. By (9.6)(b) and (9.8) we have DimJL) = t DimJL), and
therefore the sum Ur1 + + Vr„ which by (7.1) coincides with V, is direct. It is then
straightforward to verify that the map
I t
52 r, ® fj -> y, up-;
i-1 i=l
is a KG-isomorphism from U° onto V. □
Remark. In the case r° = 2, an analoguous result can be deduced from (9.6) and (9.8).
We now have enough information about representations of abelian groups to
prove the following theorem of Huppert (cf. [H] VI, 8.1).
(9.11) Theorem. Let p be a prime, let G be a p-soluble group, and let К be a field of
characteristic p. Let V be a faithful, simple G-module of dimension n over K, and assume
that (n, IG|) = 1. Then
(a) G is cyclic, and
(b) if К = F,,, then |G| divides p" — 1.
Proof. In order to prove that G is cyclic, it will be sufficient to prove that G is abelian
by (9.3)(b), and this we do by induction on |G|. By (5.19) there is a Galois extension
field L of К which is a splitting field for G and its subgroups. By (5.14) we have
V ®K L = Wr ф • • ф Wr with DimjHJ dividing DimJL ®K L) = Dim JL) and
Ker(G on Wf = Ker(G on l')= 1. Hence we may replace the К-module V by the
L-module Wt without changing the hypotheses and can therefore assume without
loss of generality that К is a splitting field for G and its subgroups. But now the
hypotheses of 7.14(b) are satisfied and from that result we can deduce that DimJL)
divides |G|. It follows by hypothesis that DimJL) = 1 and hence that G is cyclic by
Proposition 9.3.
Assertion (b) follows at once from (a) and (9.8) (d).
Another application of Theorem 9.8 which is useful in studying representations of
soluble groups over finite fields is the following result, which we state without proof.
164 В. Prerequisites—representation theory
(9.12) Theorem ([H] II, 3.11). Let К = F,, let V = l(n, K), a vector space of dimen-
sion n over K, and assume that V is also a faithful module for a group G. Let A be an
abelian normal subgroup of G such that VA is homogeneous, let d denote the dimension
of the composition factors of TA, and set s = n/d, the composition length of TA. Then, if
L = there exists a homomorphism from G into Aut(L/K) sending an element ge G
to a map 2 -> 29(2 e L) such that V, viewed as a G-set, is isomorphic with a G-set
У = Pfs, L) on which G acts as a group of L-semilinear transformations satisfying
(v\ + v2)g = v\g + v2g, and (fvfg = 29(e'g)
for all v', v'i, v2 e У,де G, and Ле L. In this case Сс(Л) is the subgroup of G which
induces L-linear transformations on У.
This result is helpful when V is not induced from a module for a proper subgroup.
It implies that in this case G has a normal subgroup N (= Сс(Л)) such that
(i) G/N is isomorphic with a subgroup of the cyclic group Aut(L/K),
(ii) V may be viewed as an L/V-module of smaller L-dimension than K-dimension,
and
(iii) A has scalar action when V is so viewed.
The following more specialized result, related to the situation described in (9.12),
is also useful.
(9.13) Proposition. Let A = Сс(Л) < G, let q be a power of a prime not dividing |G|,
and let V be a simple ^G-module, faithful for G. If VA is homogeneous, then VA is
irreducible.
In order to prove Proposition 9.13 we shall need the following.
(9.14) Lemma. Let A be a self-centralizing normal subgroup of a group G, and assume
that each subgroup of G containing A is normal in G. Suppose that G has a faithful,
absolutely irreducible representation p over a field K. Then p has degree | G: A |.
Proof. Let V be a KG-module affording the representation p. Since p is absolutely
irreducible, by (5.6) we can assume without loss of generality that К is algebraically
closed. Let Dim^L) = n (the degree of p), and let
•л = к © Ф И
be the decomposition of VA into homogeneous components. Denote the stabilizer of
Ц by Sf; since Sf contains A, by hypothesis it is a normal subgroup of G, and since
S,, .... S, are G-conjugates by Clifford’s Theorem 7.3(b), they coincide, with S say.
Let At = СЛ(Ц). Since A is abelian and К is algebraically closed, and because Ц is a
homogeneous К Л-module, the cyclic group A/Л, acts faithfully on Ц by scalar
multiplication; hence A/At lies in the centre of S/A„ and we therefore have [Л, S] <
Ae Since V is faithful for G, the subgroups At, . ., A, have only 1 in common, and
so [Л, S] = 1; hence S < Сс(Л) = A and we conclude that S = A.
9. Representations of abelian and extraspecial groups
165
By Clifford’s Theorem 7.3(d), the KS-module Ц is simple, and since S ( = Л) is
abehan, we deduce from (9.2) (Case (ii)) that 1< has dimension 1; consequently n = t
But by Clifford’s Theorem 7.3(b) we have t = |G: S| = |G: Л|, and the result follows.
□
Proof of (9.13). Since VA is homogeneous and faithful, A is cyclic by (9.3)(b). Because
A<G, there is a homomorphism from G into Aut(4) with kernel Сс(Л) = A. Thus
G/A is isomorphic with a subgroup of Аш(Л), which is abelian by A, 2.21(a); in
particular, every subgroup of G containing A is normal.
Let W denote the group algebra FgG. Since (q, |G|) = 1, by (4.5) the algebra W is
semisimple, and therefore by Wedderburn’s Theorem 4.4
W = Wt ©© Ws,
a direct sum of complete rings Wt of n, x n; matrices over division rings K,. Since
each k, is finite-dimensional over Fg, by Wedderburn’s Theorem 3.17 it is a finite
field, of Fg-dimension say. Therefore И7, viewed as a right Fg G-module, is a direct
sum of n, isomorphic simple modules C',, each of Fg-dimension тр\. Since Kf is
commutative, we have End^t/J = Kf, and therefore l/„ regarded as a KjG-module,
is absolutely irreducible. Thus, if I/, is faithful for G (in particular, if If S E), then the
hypotheses of (9.14) are satisfied, and so by that result n, = DimK (GJ = |G : A\.
Let n = |G: A\, and let
VA = M
a direct sum of r copies of a simple Fg Л-module M. Since V г G, for some i e {1,..., s},
the module WA contains the submodule (G, ф • • ф UfA, which is isomorphic with rn
copies of M. However, the decomposition of G into |G : Л| left cosets of A yields a
decomposition of(FgG)4 = WA into the direct sum of |G: Л| copies of the right regular
module Fg A. But according to (9.9) a completely-reduced direct decomposition of Г,Л
contains only one copy of M. and therefore a comparison of the multiplicity of M in
composition series derived from these two expressions for WA yields the inequality
rn < |G : A\ = n. Hence r = 1 and VA = M. which is simple. □
When applying Clifford’s theorem to the restriction of a module to an abelian normal
subgroup, the information about the stabilizer provided by the following lemma is
often very useful.
(9.15) Lemma. Let A be an abelian normal subgroup of a group G, let К be a field,
and let W be a simple КА-module. Set В = Кег(Л on IE) and N = NG(B\ Then
(a) CN(A/B) < fJW7) < N;
(b) If DimjJM') = 1, then IG(W} = Cf(A/B}:
(c) If Dim.(B/) = 1 and, in addition. A has a complement, H say, in 1C(W), men
there is a one-to-one correspondence between the set of simple КН-modules and the set
of simple KG-modules V which have Was a summand of l'A. This correspondence may
be described as follows: Let U be a simple КН-module. Then W®K U is a simp
166
В. Prerequisites—representation theory
(Л/B) x H-module which should be viewed as a simple Ia(W)-module by inflation. Then
the induced module (IT® U)G is the simple KG-module which corresponds to V.
Proof. Recall that /G(HZ) = {9 e G : И79 = W}. Since the Definition 7.2 of We ob-
viously implies that Кег(Л on И/я) = дКег(Л on and since isomorphic
modules have equal kernels, it follows that fcffV) < WC(B). If geCN(A/B), then
wg~l ag = wa for all a e A and we W, and from the definition of We we conclude that
И79 s W. This proves Statement (a).
To see that (b) holds, we recall from (9.2) that A/В is a cyclic group, and from (9.1)
that if aB is a generator of A/В, then W affords a representation which sends aB to
a primitive | A/B\th root of unity t.. Let g e NG(B}. Then (Г'адВ = amB for some
1 < m < | Л/B). Since the representation afforded by И'9 sends gag'1 В to r, it sends
aB to cm; therefore W" = И' if and only if m = 1 and g~* agB = aB, and this holds if
and only if g e CN(aB) = CN(A/B). This proves Assertion (b).
To justify (c) we apply the Clifford correspondence. This asserts that induction is
a bijection from the set of simple B7G(H/)-modiJlcs Y with W a summand of YA to
the set of KG-modules V with W as a summand of VA. Since IC(W}/B 3 (Л/B) x H
and Wis 1-dimensional, it is straightforward to verify that each simple fG(H/)-modulc
У with WI Ya has the form Y = W ®к U, where (i) W is a viewed as an Л/B-module
by deflation and U is a simple KH-module, (ii) the action of (Л/В) x H on W ®K U
is that described in (1.12), and (iii) W ®к V is then regarded as an /(-(IT (-module by
means of the above isomorphism and inflation. □
We end this section with a description of the representations of extraspecial p-groups
over an arbitrary field К of characteristic # p. We recall from A, 20.5 that if E is an
extraspecial p-group, then Z(£) has order p and coincides with E' and Ф(Е); that
E/Z(E} is elementary abelian of order p2' for some natural number t; and that E is
the central product (with amalgamated centres) of t extraspecial groups of order p3.
If p is odd, there are two isomorphism types of extraspecial groups of order p3, one
of exponent p and one of exponent p2, and each type has a normal subgroup
isomorphic with Zp x Zp. Thus, viewing £ as a central product, we see that £ has an
elementary abelian subgroup, A say, of order p'+1; as indicated in A, 20.5, this
subgroup A is maximal among abelian subgroups of £ and is therefore, in particular,
self-centralizing. On the other hand, if p = 2, then either £ is a central product of t
copies of Dih(8), in which case £ has a (normal) maximal abelian subgroup which is
elementary (of order 2,+1) or £ is a central product of t — 1 copies of Dih(8) with a
copy of Quat(8), in which case all maximal abelian subgroups of £ are isomorphic
with Z4 x (Z2 x ••• x Z2).
We divide our discussion of the simple K£-modules into two separate cases. The
first is when £ has a normal abelian subgroup A which is elementary, self-centralizing,
and has order p,+*. Let m be the dimension of a non-trivial simple KZ^-modulc. Then
m = IK({/1): K| by (9.8), and from Wedderburn’s Theorem 4.4 we conclude, as in
(9.9), that there exist s = (p — l)/m non-trivial simple KZr-modulcs, Ut, ..., V, say.
Thus, for each maximal normal subgroup M of E, we can find s distinct K£-modules
W with Ker(E on W) = M, and since £ has (p21 — l)/(p — 1) maximal normal
subgroups, we obtain a total of s(p2' - 1 )/(p - 1) = (p2r — l)/m simple K£-modules
9. Representations of abelian and extraspecial groups
167
in this way. We assert that these, together with the trivial module KE are all the
simple KE-modulcs which are not faithful for E. Let W be a non-faithful simple
E-module over K. Since Z(£) is the unio,ue minimal normal subgroup of £ the
module W has Z(£) in its kernel and is therefore by deflation a module for the
elementary abelian group £/Z(£). Hence, if W is non-trivial, by (9.3)(b) we have
£/Ker(£ on W) S Zp; thus Ker(£ on W) is a maximal normal subgroup of £, and
W is one of the s simple modules with this kernel described above.
It remains to find the simple modules which are faithful for £. The maximal abelian
subgroup A obviously contains Z(£), and as A is elementary, we can find a subgroup
В such that
A = Z(£) x B.
Let W be a simple КЯ-module with В = Кег(Л on W); there are s distinct choices
for W and these are determined by the isomorphism class of Иад. Let g e N£(B}.
Then [A, g] = [Z(E)B, g] = [В, g] < B. But Г A </]<£' = Z(£), and so [Л, 9] <
Z(£)nB=l. Hence g e C£(A) = A, and therefore A = N£(B). It follows from
(9.15) (a) that A = /^(И7), and so by (7.4) the induced module IVе is simple. This
module WE depends ostensibly on the choice of complement В to Z(E) in A. Since
A has (p'+1 - 1 )/(p — 1) maximal subgroups, of which (p‘ — 1 )/(p — 1) contain Z(E),
there are
((p,+1 - 1) - (p' - D)/(P - 1) = P'
complements B, and since |£ : WE(B)| = |£: A\ = p', these complements are all con-
jugate in £. Therefore the simple module WE depends only on the isomorphism class
of the simple module И<,(Е) and not on the choice of B. Thus there are exactly s
distinct simple KE-modulcs obtained in this way, and each has K-dimension
| G : A | DimK( W) = mp'.
We claim that these are all the faithful modules. One way of seeing this would be
to show that EndKE(L) = K(</i) for any of these modules V = WE; to deduce that
V occurs with multiplicity p' as a summand in a direct decomposition of the regular
module; and then to conclude that the sum of the К-dimensions of the simple
KE-modulcs which we have already obtained (including their multiplicities) is
1 + m(p2‘ - l)/m + p'smp‘ = p2,+1 = |£|-
From this it would follow that we have found a full complement of simple KE-
modules. But instead, we take another approach.
Let V be a simple КE-modulc which is faithful for £. By (9.4)(a) the restricted
module VZi£) is homogeneous, say Vz>£l = V Ф • • • Ф G, where U is one о the s
isomorphism types of non-trivial simple KZ(£)-modulcs. Writing
VA = ЦФ---ФК.
with homogeneous components l<, we have Ксг(Л on fj) = Ксг(Л on Ifj), where
168
В. Prerequisites—representation theory
И7, is a simple summand of Vt, and Л/Кег(Л on И-j) s Zp. Since (И7,),,,, s t/ is
non-trivial, we deduce that A = Z(E) x Ксг(Л on Wt), and so И7, is one of the simple
Я-modules W described above. Since ^(H7,) = A as above, we conclude from (7.3) (c)
that V S (H-j)£, and therefore V is one of the s faithful E-modules already obtained.
Thus we have proved the following theorem.
(9.16) Theorem. Let p be a prime, let E be an extraspecial group of order p1,+i, and
assume that E has an elementary abelian subgroup A of order p'+l. (In particular, this
assumption is always satisfied when p # 2.) Let К be a field, let m(= |K(^/l): K\) be
the dimension of a simple non-trivial KZp-module, and set s = (p — l)/m. Then, in
addition to the trivial module, E has the following simple modules over K:
(i) (p21 — l)/m modules U of К-dimension m whose kernels contains Z(E) and have
index p in E',
(ii) s faithful modules V of К-dimension mp' which are induced from simple A-
modules and are uniquely determined by the isomorphism type of a composition factor
°f k'ztEi-
In particular, all the simple E-modules are absolutely irreducible when m = 1, namely,
when К contains a primitive pth root of unity.
The second case we have to consider is when the extraspecial p-group E has no
maximal abelian subgroup which is elementary; this occurs precisely when p = 2 and
E is isomorphic to the central product of t — 1 copies of Dih(8) and one copy of
Quat(8). Since E/Z(E) is an elementary abelian 2-group, E has |E/Z(E)| = 22' simple
modules of K-dimension 1. It turns out that there is one more simple module,
and this is the unique faithful simple module. To construct it, let A be a maximal
abelian subgroup of E. As before, we have A = С£(Л) < E, but in this case A s
Z4 x Z2 x ••• x Z2,andZ(E) = ?5,(Л). At this point, we distinguish two possibilities.
Case (a). К contains a primitive 4th. root of unity. In this case, A has a 1-dimensional
module U with Л/Ксг(Л on V) = Z4. It is straightforward to verify that IE(U) = A
and that, as before, Vе is the (unique) simple KE-module which is faithful for E.
(Although the group Л/Кег(Л on U) has two faithful simple modules over K, these
are conjugate in E because the normalizer in E of Ксг(Я on U) contains an element
which inverts a generator of Я/Кег(Л on U.)
Case (b). |K(,/ — 1): K| = 2. Let L = K(-J — 1). As in Case (a), let U be a simple
L/1-module with Я/Ксг(Я on U) 3 Z4, and let V = UE, the unique faithful simple
E-module over L. Let Vo denote V regarded as a KE-module (so that Dirndl7,,) =
2Dim,(l7)). Then evidently l'„ = (Uo)£, where Uo denotes U regarded as a KA-
module. Let X be a simple submodule of Vo. Since XA is semisimple, it follows from
Nakayama’s Reciprocity Theorem (6.8) that Uo is a submodule of XA. Let Y be the
homogeneous component of XA containing Uo, and let В = Кег(Я on U„). The
structure of E as a central product implies that В can be chosen to lie in the product
of the t - 1 copies of Dih(8), and that then A has index 2 in N£(B); furthermore. NE(B}
acts by inversion on A/В, and therefore sends (70 to a conjugate If. However, l/0 is
the unique faithful simple Л/B-modulc over К and therefore IE(U„) = NE(B). Since Y
9. Representations of abelian and extraspecial groups
169
is a simple /E(U0)-module by (7.3)(d), and since A/В is self-centralizing in NABU В it
foHows from (9 13) that YA is simple; in particular У s Uo. Hence X is isomorphic
with Uo induced from fE(l/0) up to E and has К-dimension 2' Since Dim (V ) = t+1
it follows from (5.13) that VB s X ® X. Finally, by arguments used in the eari.er case
it is easy to see that X is the unique simple KE-module which is faithful for E. Thus
we have proved the following result.
(9.17) Theorem. Let E be an extraspecial p-group all of whose maximal abelian
subgroups have exponent p2. Then E is isomorphic with a central product of t - 1 copies
of Dih(8) with one copy of Quat(8). Let К be a field whose characteristic is not 2. Then
E has 22‘ simple modules of dimension J and just one further simple module F; V is
faithful and has K-dimension 2'. If 1 e K, then V is induced from a 1-dimensional
module for a maximal abelian subgroup A Z4 x Z2 x • x Z2. If y/^-1 f K, then
V is induced from a 2-dimension simple KNE(B)-module U, where В = Кег(Л on V)
and | NF(B): A \ = 2. In particular, all simple KE-modules are absolutely irreducible.
We now state an important result about representations of certain cyclic extensions
of extraspecial groups. It may be seen as a non-modular version of the celebrated
“Theorem B” of Hall and Higman and, like that theorem, it finds many applications
in the study of finite groups, especially soluble groups. (A statement and proof of
Theorem В can be found in Huppert and Blackbum [1], Theorem IX, 2.9.)
(9.18) Theorem. Let E be an extraspecial p-group of order p1'*1, let H be a cyclic
p'-subgroup of Aut(E), and let G denote the semidirect product [E]H. Assume that H
acts regularly on E/Z(E) and trivially on ZlE). Let К be a field containing a primitive
pth roof of unity whose characteristic does not divide | G|, and let V be a KG-module
such that VE is simple and faithful for E. Then there exists 5 = + 1 such that | H | divides
p‘ — 6 and a 1-dimensional KH-module V such that
(i) if 6 — 1, then VH = m(KH) ф 17, and
(ii) if b = — 1, then F„ ® C! S m(KH),
where m = (p‘ - b)/\H\ and m(KH) denotes the direct sum of m copies of the regular
module KH.
Remarks, (a) Let IV be a faithful simple module for E over K. By Theorems 9.16 and
9.17 the isomorphism class of W is uniquely determined by the type of the homo-
geneous module and since by assumption G centralizes Z(E), it follows that W
is G-invariant. Moreover, in view of our assumptions about H and K, these theorems
also ensure that Dim^M7) (= p') and |H| are coprime. Therefore Theorem 7.12
applies and guarantees the existence of an extension of W to G. Therefore a module
with the stated properties of this theorem always exists.
(b) The non-faithful modules for G over К are easily described. Let V be such a
module, and set A = Ker(G on G). Then Z(£) < N, and either E £ N, m which
case G/N is a Frobenius group with Frobenius kernel EN/N,or , m w ic
case G/N e 0(H). If G/N is Frobenius, an easy application of Clifford s Theorem
7.3 in conjunction with Lemma 9.15(b) shows that U„ = KH (in particular, that
170
В. Prerequisites—representation theory
DimK(l7) = |H|), and if E < N. the module U may be viewed as a module for a
quotient of the cyclic group H and is determined by Theorem 9.8.
Proof of Theorem 9.18. We cite Satz V, 17.13 of [H], which gives a complete
description of the simple KG-modules when К is algebraically closed and includes
as Part (b) the stated structure of VH when V is faithful for E.
In the general case, denote the algebraic closure of К by L. Since К contains a
primitive pth root of unity, it follows from Theorems 9.16 and 9.17 that Vf (and hence
also И is absolutely irreducible of К-dimension p'. Hence the hypotheses of the
theorem are satisfied with I'* = V ®A L in place of V, and it follows from the cited
result that (P*)B (= VH ®k L) has the stated form. Thus, if & = 1, then m(LH)
(S m(KH)® L) is isomorphic with a direct summand of I'„ ® L, and by the Noether-
Deuring Theorem 5.24 we conclude that I'H s m(KH) © U for some К//-module U.
Since m(LH) has L-codimension 1 in I7*, it follows that m(KH) has K-codimension
1 in Г, and therefore DimK(L) = 1. On the other hand, if 6 = -1, a further applica-
tion of the Noether-Deuring theorem shows that If is a direct summand of m(KH).
and again the desired conclusion follows. □
(9.19) Proposition. Let К be a field, let H and L be groups, and let M and N be simple
modules for KH and KL respectively. Let V = M ®K N, regarded as a K(H x L)-
module according to (1.12). Set Ho = Ker (7/ on M) and Lo = Ker(L on N). Further,
set HJH„ = Z(H/H0} and Lx/L() = Z(L/L0), noting that both of these sections are
cyclic by (9.4).
(a) Let r = (\Hl/H0\, |/.|//.(l|), and let aH0 be a generator of the unique cyclic
subgroupof orderr in Ht/H0. Then Lf Lo contains an element bLoof order r such that
Ker(H x L on M ® N) = (afe'1, H0L0>.
(b) Now assume that Ho = Lo = 1, let T be a composition factor of V, and set
C = Ker(H x L on T). Then C < Z(H) x Z(L), and the subgroups (CLfoll and
(CH) n L are isomorphic subgroups of Z(H) and Z(L) respectively. In particular,
C= 1 if(IZ(H)l,IZ(L)l)= 1.
Proof, (a) Since Ho x Lo < Kcr(77 x L on k). we may assume without loss of gene-
rality that Ho = Lo = 1, and since kernels are unchanged by extending the field, we
may further assume that К is a splitting field for all subgroups of 77 x L. Let
R = Ker(/7 x L on V). Then R < H x L and Hr F = 1. Hence [77, R] = 1, simi-
larly [L, R] = 1, and therefore R < Z(H x L) = Z(H) x Z(L). Moreover, if de-
notes the projection of H x L onto H, then R n Ker(n„) < R n L = 1, and it follows
that nH (and likewise лг) is a monomorphism when restricted to R; in particular, |R|
divides |Z(77)| and |Z(L)| and therefore divides r = (|Z(77)|, |Z(L)|).
Since Z(H) and Z(L) are cyclic by (9.4)(b), they contain unique subgroups A = (a)
and В = respectively of order r. Having supposed that К is a splitting field for
A and B, we can deduce from (9.4) (a) and (9.1) that ma = cm and nb = tfn for all
m e M, for all n e N, and for suitable primitive rth roots of unity г, r' e K. Moreover,
since e' is a power of we can replace b by another generator of В and suppose
9. Representations of abelian and extraspecial groups
171
without loss of generality that i! = But then
(m®n)(u£ ') = (cm®®1,,) = ,n®„
for all m e M and n e TV, and consequently ab' e R. Since o(ab') = r, it now follows
that R = <ab 1 >, which is the desired conclusion.
(b) Since VH S ®M and L, s @N, it follows that TH and 7] are also direct sums
of copies of M and N respectively. Hence = CH(M) = H0=\, and similarly
CdTi) = 1- Consequently C is a normal subgroup of H x L which has trivial inter-
section with the direct components H and L. As in Part (a), it now follows that
C < Z(H} x Z(L), and the subgroups (C£)n/7 and (CH]r,L. which are respec-
tively the projections л„(С) and nL(C) of C onto H and L. are each isomorphic
with C. |-j
Remark. Let be extraspecial p-groups of order p3, and denote their central
product by E (every extraspecial group is isomorphic with such an £). Let Ц be a
faithful, simple £rmodule over a field К (of characteristic / p) for i = 1,..., t. Then
by repeated application of (9.19) we see that the module V = Lj ®K • • ®K regarded
asa K(£, x • • x £,(-module, has kernel C = <zizj’111 < i, j < t> for suitable choices
of generators z, of Z(£,). Since E = (E} x x £,)/C. the module V can be viewed
by deflation as a K£-module. In fact, the module V is homogeneous (even simple if
^/1 e K), and by taking a simple summand this procedure gives another way of
constructing the faithful simple modules for a general extraspecial group £.
(9.20) Corollary. Let V be a faithful, simple module for an extraspecial p-group E over
a field K, and let x be an element of order p in E\Z(E). Then Ркх> is a direct sum of
regular К (x^-modules.
Proof. Observe that by the Noether-Deuring Theorem 5.24 we may suppose without
loss of generality that the field К contains a primitive p2 root of unity. We first
consider the case |£| = p3. Let £ = (x, y>, and set A = O’. Z(Ef), a maximal abelian
normal subgroup of £. If A = Zp x Zr, the construction of V in Theorems 9.16
and 9.17 shows that there exists a 1-dimensional КЛ-module V such that 14,) 5
(Uc)<xy = (UA<xy)<,y, which by Mackey’s theorem in the special case (6.2l)(b) is
isomorphic with (lf(jj)<x> = L>n °ther hand, if A = Zri, then k< = If +
• • + U , where each 17, is a 1-dimensional КЛ-module which is faithful for A because
V is fafthful for Z(A); here we need the hypothesis that e K. By 9.15(b) the
stabilizer in G of If is С£(Л) = A, and so Lj, ..., Up are distinct Л-conjugates. By
Clifford’s Theorem 7.3(c) we have I® and, as before, L'<Jt> S K<x>.
In the general case, the analysis of A, 20.4 shows that £ can be written m the form
E = E, y-yE,
with xe E.\Z(Et)- (Here one needs the easy fact that the decomposition A, 2O.«
of a symplectic space into an orthogonal sum of hyperbolic planes can chosen so
that the component H, contains a prescribed vector.) It then follows from
172
В. Prerequisites—representation theory
preceding Remark that V s Ц ® • • • ® F„ where each k, can be viewed as a simple
£, X X E,-module, with all components Ej(j / i) acting trivially on Ц. We there-
fore obtain
= (V, )<*> ® (h)<x> ® • • ® (K)<*>>
and since (Ц)<х> S K<x> by the initial paragraph and <x> acts trivially on V* =
V2 ® • • ® V„ we conclude that F(x> S Dim( V*)K <\>. □
Remark. An alternative proof using character theory runs as follows: By the Noether-
Deuring theorem suppose that К is algebraically closed. By Satz V, 17.13 of [H]
the character of the faithful simple E-module V is zero on £\Z(£) and has the
value p' on 1. Thus F<x> and p''(K(x)) have the same characters and are therefore
isomorphic modules.
10. Faithful and simple modules
For any field К and finite group G, the regular module (KG)C affords the regular
representation of G, in which each non-identity element is represented by a non-
identity permutation matrix. Thus Ker(G on (KG)C) = 1, and the regular represen-
tation is faithful; in other words, every finite group is isomorphic to a linear group
over any field. If we want to find a KG-module which is not only faithful but also
simple, we may not be so fortunate; for we saw in (3.12)(b) that Op(G) is in the kernel
of every simple module over a field of characteristic p > 0. Moreover, it follows from
(9.3)(b) that a non-cyclic abelian group has no faithful irreducible representations
over any field. Our two main objectives in this section are: first to characterize in
different ways those finite groups which have a faithful simple module over a given
field; second to describe a procedure, due to Steinberg, for obtaining all simple
modules from a given faithful module by using tensor products.
Before embarking on these enterprises, we mention the following result, which will
be useful later.
(10.1) Lemma. Let К be a field of characteristic p > 0, and let V be a faithful module
for a group G over K. Then
(10.a) Op(G) = Q {Ker(G on U): U a composition factor of V}.
Proof. Let R denote the right-hand side of Equation 10.a. Since each composition
factor U of V is simple, (3.12)(b) implies that OP(G) < R. Since R < G, to complete
the proof it will suffice to show that R is a p-group.
Let H be a p'-subgroup of R, and let W be a composition factor of V„. By the
Jordan-Holder theorem W is isomorphic with a composition factor of VH for some
composition factor V of V, and therefore W is a trivial H-module by definition of R.
Since Ht, is semisimple by Maschke’s theorem, it follows from (3.13) that H <
Ker(H on FH), and therefore H = 1 because by hypothesis V is faithful for G. It now
follows from Sylow’s theorem that R is a p-group. □
10. Faithful and simple modules
173
As a preliminary to our first characterization of the class of groups which possess a
faithful irreducible representation, we show that it contains the direct products of
non-abelian simple groups.
(10.2) Lemma. Let S = Si x ••• x S„ be a direct product of non-abelian simple groups
5„ and let К be an arbitrary field. Then S has a faithful simple module over K.
Proof For notational convenience regard S as an internal direct product. Since
Op(St) = 1, it follows from (10.1) if Char(K) = p > 0 and from Maschke’s theorem if
Char(K) = 0, that the regular KS,-module has a non-trivial composition factor Ut,
and so L , is faithful since S, is simple. Let V be a composition factor of U =
Ut ®x ®k U„, viewed as an S-module according to (1.12). Since Us is a sum of
copies of L7„ so also is Fs., and therefore Ker(S on F) n S; = 1 for i = 1,..., n. By A,
4.13(b) a normal subgroup of S is the product of a subset of S,......S„. Therefore
Ker(S on f ) = 1. □
Terminology. The product of all the abelian minimal normal subgroups of a group
G is called the abelian component of the socle and is denoted by Soc5,(G).
We are now ready to state and prove our first characterization of groups which
have a faithful simple module over a given field.
(10.3) Theorem (Shoda [I], Nakayama [1]). Let G be a finite group and К an
arbitrary field. Then the following conditions are equivalent:
(a) G has a faithful simple module over К;
(b) SocVJ(G) has a subgroup N such that
(i) CoreG(/V) = 1. and
(ii) Soc9|(G)/A' is cyclic and is a p'-group if Char(K) = p > 0.
Proof. Set A = SoCai(G). We prove first that
(a) => (b): Let V be a simple KG-module which is faithful for G, let G be a simple
submodule of VA, and let N = Кег(Л on U). By (9.8)(a) the quotient A/N is cyclic and
is a p’-group when Char(K) = p > 0. It follows from Clifford s Theorem 7.3(b) and
from the proof of Lemma 7.6 that CoreG(N) < Ker(G on F), and therefore, since V
is faithful, Corec(N) = 1. It remains to show that
(b) ==> (a): Assume that A has a subgroup N satisfying Properties (i) and (ii) of
Condition (b). If A/N = Z„, it follows from (9.1) that A has a simple module V over
K(</1) with Кег(Л on k) = N. Therefore, in view of (5.13) and Equation 5.a, there
exists a simple К Л-module W with Кег(Л on W) = N. By definition of the socle we
can write Soc(G) = A x S, where S is the product of the non-abelian minimal normal
subgroups of G and is therefore a direct product of non-abelian simple groups by A,
4.13(a). Hence by (10.2) there exists a faithful simple module U for S over K. Let 1
be a composition factor of W®K U viewed as an (Л x S)-module in the way described
in (1.12). Since (W® L )s, and therefore Ts, is a sum of copies of U, we ave
Ker( AS on У) П S < Ker(S on У) = Ker(S on U) = 1 and therefore Кег(Л5 on У15=
C«s(S) = A. But (W® С).< is a sum of copies of W and so we conclude t
Ker(AS on У) < Кег(Л on W) = N.
174
В. Prerequisiies representation theory
Let T be a simple submodule of Yc. By Nakayama’s Theorem 6.5 we have
Hom( TAS. T) / 0, and since TAS is semisimple by (7.1), the simple ЛХ-module Y is a
direct summand of T<s. Furthermore, Clifford’s Theorem 7.3(c) implies that T is
isomorphic with Xе, where X is a simple module for fG( У) and Xds is a sum of copies
of Y. Now by (6.4) we have
AS n Ker(G on 7 ) = Q Ker(4S on X)9.
But Ker(4S on X) = Ker(4S on У) = N, and by hypothesis Core6(A') = 1. It follows
that Ker(G on T) has trivial intersection with AS = Soc(G) and consequently, being
normal, is itself trivial. Thus T is the sought-after faithful simple module for G. □
The following result is the crucial step in justifying our second characterization of
groups which have a faithful irreducible representation.
(10.4) Proposition. Let pbe a prime, G a group, and U a simple H-pG-module. Let V be
the direct sum of n (>1) copies of U, and set E = EndE|G(L ). Then the following
conditions are equivalent '.
(a) V has a subspace of Fp-codimension 1 which contains no non-zero submodule of V;
(b) n DimPj(£) < Dimf (Lj.
Proof. Throughout the proof “Dim” will denote the Fp-dimension of an Fp-space. Set
d = Dim(£j, and let
F=
(regarded as an internal direct sum), and for i = 1, ..., n let т(: U -» If be a
fixed KG-isomorphism; if u e 17, let u1 denote the image ur,. Since £ is a field by
Schur’s lemma and Wedderburn’s Theorem 3.17, we can regard If as a vector space
over £ by setting u‘a = (иа)т,- for all и e U and a e E. Then, as we pointed out in
the proof of (8.2), every minimal submodule of V has the form Ur for some т =
aiTi + ' + и„т„ e HomKC(G, F), where a,,..., a„ are suitable elements of £, not all
zero.
Let W be a subspace of V such that
(Wl)
Dim(F/W)=l, and
(W2)
4,^.
Set M = 1Л ф ф so that V = M © l/„; in this direct decomposition denote
the projection of V onto M by л and its restriction to Wby nw. Then Dim(Im(n„,)) +
Dim(Ker(TtB.)) = Dim (IF) = Dim(F) - 1, and since 1т(лн) is a subspace of M and
Кег(ли.) = Wn Ker(n) = W n If < If, it is clear that Dim(Ker(nH,)) = d — 1 and
DiniHmln^)) = (n — l)d = Dim(M); in particular, tiw is surjective. Let {u1;..., ud}
be a basis of G with the property that {uj,..., uj_]} is a basis of Кег(ли,). Since
Wn = M, we can find elements w] e W such that
ч — m
10. Faithful and simple modules jyj
(Ю/0 WjJl = и J
for 1 < i < n - 1 and 1 < J < d, and since IT/Ker(n„,) s Л/, it follows that
M> • • » “’J, • - w?, и-Г1, uf,..., uj.,}
is a basis for W. Because {u,!: 1 < i < n, I < j < d] is obviously a basis for Ц for
1 < i < n — 1 and 1 < j < d by (10./J) we can write
wi = uj + hyi/d + xj
with Xj e <uj,..., uj_, > = Ker(7t„,). Since the elements
{w] - x] : 1 < i < n - I, 1 < j < d],
together with uj, ..., uj „ obviously form a linearly independent set, by defining
h«i = ''' = Лта-ц = ° and h„d = -1, we obtain a spanning set for W of the form
(10.y) {u; + hijU;: 1 < i < n, 1 <j<d},
which becomes a basis when the element corresponding to i = n and j = d is omitted
(this element is zero). Let Hi denote the d x 1 matrix whose transpose is (hit ...hid).
Thus with each subspace W of V satisfying (Wl) and (W2) we have associated a set
{Hl,...,H„}ofn column vectors over Fp.
Conversely, given a basis {u,,..., ud} of V and a set of column vectors {H,.H„}
with = (0, 0,..., 0, — 1), we obtain a subspace W spanned by the elements in (lO.y)
which satisfies (Wl) and (W2).
Since U is an E-module, with respect to the basis {u,,..., ud} we obtain a matrix
representation a -» a — (a,,) e GL(d, Fp) of E given by uka = Ujajk for к = 1,...,
d. Since £ is a field, it is isomorphic with its image £ in GL(d, Fp), and we can allow
£ to act by right matrix multiplication on the vector space C(d. Fp) of all d x 1 column
vectors over Fp so that this space may also be viewed as a vector space over the field
£. Since the vectors Ht,..., H„ are in C(d, Fp), we can consider their linear dependence
over Ё. Let A,,..., A„ e Ё, set Л, = (a’k), and let a,,..., a„ be elements of £ such that
a, = Л, for i = 1,..., n. Let W be the subspace of V satisfying (Wl) and (W2), and let
{£/,} be the associated set of column vectors. Further, let a denote the projection of
I' = W © <uj> onto <uj>, and note that uja = - hi}u; since u] + hyuj e W by (10.;).
Then H, A, + • • + H„ Л„ = 0 if and only if for к = 1,..., d we have
- £ ( Z = °- which holds if and onIy if
- Z ( Z hva*J = 0 if and only if
176
В. Prerequisites—representation theory
(7 = 0
£ ( £ wJT,a;Jff = 0
if and only if
if and only if
£ ( £ иА‘к^)<7 = 0
if and only if
£ P = 0
for A: =
and this last equation clearly holds if and only if the elements iqr, udr are in
Кег(ст) (= IF), where r = cqt, + + a„r„ e HomKG(l/, F). Thus we have shown
that
(10.<5) {Hj,are linearly dependent over E if and only if Ut < W for some
non-zero r e HomKC(t/, F).
We can now complete the proof of the proposition. Suppose first that Condition
(a) holds, and let W be a subspace of V of codimension 1 that contains no non-zero
submodule of F. Then IF contains no minimal submodule of F, and since minimal
submodules necessarily have the form Ut for some non-zero t e HomKG(l7, F), it
follows from (10.<5) that the £-space C(d, Fp) has n linearly independent elements; but
these generate an Fp-subspace of Fp-dimension n Dim(E), which is therefore at most
equal to Dim(C(d, Fp)) = d = Dim(G). Hence Condition (b) is satisfied.
Conversely, if Condition (b) holds, the E-dimension of C(d, Fp) is at least n, and we
can find vectors H,....H„ in C(d, Fp) which are linearly independent over £ with
H„ = (0, 0,.... 0, — l)r. It follows from (10.5) that the associated space IF, which by
definition bas codimension 1, contains no subspace of the form Ut with 0 / re
HomKG(U, F). Consequently it contains no minimal submodule of F, and therefore
Condition (a) is satisfied. □
(10.5) Definition. Let p be a prime, and let U be a simple FpG-module. For each finite
group G define an integer ft,(G) (the frugality of U in G) as follows: Let d and e denote
respectively the Fp-dimensions of U and £ = Endr G(L) respectively. (Since £ is a
field containing Fp and since U is an £-vector space, we have d = e Dimf (L ), and so
d/e is a natural number.) If n denotes the composition length of the homogeneous
component of Soc(G) corresponding to U, we define
A/(G) = d/e - n.
(10.6) Theorem. A finite group G has a faithful simple module over a field К if and
only if
(a) /g(G) > 0 for all abelian minimal normal subgroups U of G, and
(b) Op(G) = 1 when Char(K) = p > 0.
10. Faithful and simple modules
177
Proof. First suppose that G has a faithful simple module over K. If Char(K) = p > о
then 0(G) = 1 by (3.12)(b). Furthermore, by (10.3) the subgroup Soc„(G) has a
normal subgroup N with Soc*(G)/N cyclic and Corec(/V) = 1, and so, in particular
N contains no non-tnvial submodule of Soc„(G) when this is regarded as a G-module’
If A denotes Socsl(G), the subgroup N is a product of the subgroups N n О (Л) as p
runs through the primes dividing | A |, and since Op(A) is elementary, it follows that
IOp(A) N ri Op(Л)| = p; moreover, if F is a homogeneous component of OP(A) corre-
sponding to a simple FpG-module 17, from the fact that N contains no non-trivial
submodule of V we conclude that W = I7 N is an Fp-subspace of codimension 1
in V satisfying Condition (a) of (10.4). We can therefore deduce from (10.4) that
n Dimj /hnd^JL')) < Dm,^'), in other words, that fv(G) > 0.
Conversely, suppose that Statements (a) and (b) are satisfied, and for q e P let
О,(Л) = Vt @ @ V„ where 14 is the homogeneous component with composition
factors isomorphic with 17„ say. Since /^(G) > 0, by (10.4) we can find a subgroup Щ
of index q in Vt which contains no minimal normal subgroup of G isomorphic with
Ц. Let be a maximal subgroup of O,(A) satisfying Ц r> TV, = W„ and let N denote
the product of the subgroups TV, as q | \A |. Then CoreG(/V) = 1 and G/N is cyclic of
order {q : q| |Soc2I(zl)|}, which is ap'-number when Char(K) = p > Oby Condition
(b). Therefore by (10.3) the group G has a faithful simple module over K. □
Since Dim(l7)/Dim(EndG(l7)) is at least 1, the integer fv(G) is non-negative if the
homogeneous components of Soc„(G) are simple. Thus we have
(10.7) Corollary. If the abelian minimal normal subgroups of a group G are pairwise
non-isomorphic as G-modules, then G has a faithful irreducible representation over any
field whose characteristic is either zero or does not divide |F(G)|.
Using the criterion of (10.6) and arguments with dual modules, one can prove the
following.
(10.8) Lemma (Akizuki—see Shoda [1]). An (FpG-module V contains a submodule of
codimension 1 which contains no non-trivial submodule if and only if V contains a
I-dimensional subspace which is contained in no proper submodule.
From this fact, which we shall not prove, one can deduce the following criterion,
due to Gaschiitz, which may be regarded as the dual to (10.3).
(10.9) Theorem (Gaschiitz [3]). A group G has a faithful irreducible representation
over a field К if and only if Soc9l(G) is generated by a single G-conjugacy class of
elements and Op(G) = I when Char(K) = p > 0.
The preceding results have been elegantly generalized by Zmud [1] to provide
criteria for a finite group to have a faithful semisimple module of composihon length
k; one criterion is that Soc9l(G) should be generated by к conjugacy classes of G,
another is that n < kd/e (in the notation of Definition 10.5) for each simple FpG-
module 17.
178
В. Prerequisites—representation theory
Before leaving the topic of the existence of faithful simple modules, we derive three
technical results. The first of these will be needed in Chapter VII, Section 2.
(10.10) Proposition. Let q be a prime, H a finite group, and let V be a simple
module which is faithful for H. Let rV denote the direct sum of r copies of V, and let
Gr denote the semidirect product [rh'JH. Let К be a field whose characteristic is not
q, and let m be the dimension of a faithful simple Z9-module over K. Then there exists
an r > 0 such that Gr has a faithful simple module over К of dimension mlHj.
Proof If Ii e H and if Mh = M for all maximal subgroups M of V. then h has scalar
action on V by (8.6) and, in particular, h e Z(H). If H is non-abelian, let H\Z(H) =
{Л[,..., hs}; then for each i e {1,..., s} we can find a maximal subgroup of V such
TH*' / Mj. Let A = sV = Ц ф • • • © V„ where f< is F,H-isomorphic with V. and let
Ц( < И) correspond to М,- under the isomorphism. Choose a maximal subgroup В
of A such that В r> f, = 17, for i = 1,..., s. If В were normalised by some ht, we should
have 17 = Br, I- = (Br, Ц)*' = Uf, contrary to the definition of Ц. Therefore, in this
case.
(10.C)
N„(B) < Z(W).
On the other hand, if H = Z(/7), set s = 1, A = К and let В be any maximal subgroup
of A. Then again (1 ().<:) is satisfied. Let G = [A]H S Gs.
Let L = and recall from (9.8) that |L : KI = m, the dimension of a non-
trivial simple KZe-module. Let IT be a simple I./l-module with Ker(A on W) = B;
since A/В a Ze and q Char(K), such a module exists and has L-dimension 1 and
К-dimension m. Because V is simple, is homogeneous, and because V is faithful
for H, each non-identity element of Z(H) acts fixed-point-freely on V and hence on
A. It therefore follows from (1 ().<:) that the centralizer in /VG(A) of A/В is A. (Because
V is simple, O,(H) = 1, and so the non-identity elements of Z(H} also act fixed-point-
freely on Z(H)-invariant sections of A by A, 12.1.) We can now apply (9.15) to this
situation and conclude that the induced module T = WG is a simple LG-module of
L-dimension |G : A \ = |H| and may therefore be regarded as a simple KG-module of
К-dimension m |W|.
If 1 / N < H, then [P, /V] = V because V is simple and faithful for H: hence
[Л, /V] = A. Let R = Ker(G on T). Since TA contains IT as a submodule, we have
A n R < A, and therefore [Л, RA] = [A, R] < A n R < A. But RA = NA for some
N s H, and by the above observation it follows that N = 1 and hence that R < A.
Consequently R s t V for some t < s, and so G/R S Gr with r = s — t. □
Our second technical result will be needed in Chapter IX, Section 2.
(10.11) Proposition. Let G = AH be a primitive group, where A denotes the socle of
G and is a q-group, and where H r\ A = 1. Let К be a perfect field whose characteristic
is distinct from q. If p is a prime such that OfH} / 1, then G has a faithful simple
module over К whose dimension is divisible by p.
10. Faithful and simple modules
179
Proof. Let L be a Galois extension of К which is a splitting field for G and its
subgroups. (Such a field exists by (5.11).) If V is a simple LG-module which is faithful
for G, by (5.14) there exists a simple KG-module V such that V is a summand of
U ®K L, and by (5.15) we know that Dim, (V) divides DimK( U). Since U is also faithful
for G, we may therefore suppose without loss of generality that К is a splitting field
for G and its subgroups.
Let P = Ор(Н)(/ 1). If [Л, P] < M for all maximal subgroups M of A, then
[A P] = I, contrary to the hypothesis that G is primitive, whence, in particular,
Сс(Л) = A. Therefore there exists some maximal subgroup В of A such that the
centralizer C of A/В in TVG(B) does not contain P. If W is a simple КЛ-module with
Кег(Л on W) = B, then W is 1-dimensional because К is a splitting field for A, and
it follows that
(io.<)
P £ fo(tv)
because Ic(kV) = C by (9.15)(b). Moreover, since A < C and G = AH, we have
/CW = ^(C n H), and as C/B = A/В x (Cn H), it is clear that IF extends to a
КС-module, W say, such that Ker(C on W) = B(C n H). (In the notation of (9.15)(c),
take U to be the trivial K(C r> H)-module and W=W®KU.) Then (9.15)(c)
implies that the induced module V = IVе is simple, and since DimK(lV) = 1, we
have DimK(F) = |G:IC(IT)| = |Ff : /и(И/)|, which is therefore divisible by |P:/Р(И/)|,
a non-trivial power of p by (10.f). Because W is a submodule of V'A and A f,
Кег(Л on IF), it follows that A -f Ker(G on I7) and hence that V is faithful for G. □
Our third technical result will be needed in Chapter VII. Section 2.
(10.12) Proposition. Let p, q( / p), and r be primes, and let n be an r’-number. Let К
be a finite field of characteristic p, let Q S Z,, and let U be a simple KQ-module,
faithful for Q. Let E = E(n/r), and let |Soc(£)| = rm. Further, let p be the primitive
permutation representation of E of degree rm, and set
W=[U]Q%E.
Then the following statements hold:
(a) The normal subgroup С of the wreath product W is a faithful simple module
over К for the complement QCE; its К-dimension is rm DimK(L).
(b) Assume that n = 1, and let Lbea field of characteristic q. If a is the dimension
of a non-trivial LZp-module, then W has a faithful simple module over L of dimension
aqb for some b > 0.
Proof, (a) Since U is a non-central minimal normal subgroup of UQ, and since p is
transitive, we can apply A, 18.5(a) to conclude that U" is simple as а И m u e an
hence as a (PP-module. Since p is a faithful representation of E, the subgroupQ is
self-centralizing in Q'E, and therefore Soc(Q<£) < But as U is faithful for Q, the
module U" is obviously faithful for Q“ and consequently also for Q t.
1^0 В. Prerequisites—representation theory
(b) If 11 = 1, then m = 1, and W s £(<7/p)rLrcg К, where R S Zr. Since (U’)„ is a
sum of regular (f^R-modules, we have [17’, R] < U", and so we can choose a maximal
subgroup M of U" such that R < C^fU^/M).
Let N = NW(M), and note that N < W because U •< W by Part (a). Since
U^R < N, it follows that we can write N = IPQqR with Qo = Ngt(N) < Q".
Let C = CK(UfM). Since N/C < Autfl/’/Af) = Z,_t, we can write C = Ul>QlR,
where |Q0: QJ = q or 1, according as q does or does not divide r — 1. Setting
C0 = Cr>Q0R, we therefore obtain N/MCos,Zp or E(q/p), and, in any case,
there exists a simple LN-module Y such that Ker (A on Y) = MC0; in particular,
Y,„ is a sum of non-trivial simple HU’/AfJ-modules, and so by hypothesis the
dimension of Y is a multiple of a.
Let A- be a simple submodule of Yv». Then /„-(A-) < N by (9.15)(a), and there-
fore YH is simple by (7.4)(b). Since 0 / [Y, G’] < [Y”', I/’], it follows that l/c
Ker(lV on Y"’). Therefore Yw is faithful because l/b is the unique minimal normal
subgroup of IV by Part (a). Finally we note that Dim, (Y^) = qb Dim, ( Y), where
qb = IIV: A| = |Q’: Qo| > 1; since Dim,(Y) is divisible by a, Assertion (b) is
justified. □
We now focus attention on our second objective in this section, which is to prove
the following important theorem of Steinberg; our proof is based on an elementary
approach due to Bryant and Kovacs [1].
(10.13) Theorem (Steinberg [1]). Let M be a faithful module for a group G over a field
K, and let N be a simple KG-module. Let M,r> denote the tensor power M ®K ®K M
of r copies of M regarded as a KG-module according to the diagonal G-action de-
scribed in (1.11). Then there exists an integer r e {1,..., |G| — 1} such that N appears
both as a factor module and a submodule of Af(r).
We will lead up to a proof of this theorem through a series of elementary lemmas.
(10.14) Lemma. An element geG has scalar action (see (8.6)) on a KG-module M if
and only iffor all m e M the subspace fnuf) generated by mg contains m.
Proof. Suppose that g has scalar action A on M. Since g induces a non-singular
linear transformation on M, the element A is non-zero, and so <mg> contains
/Г1 (mg) = A-1 (Am) = m.
Conversely, assume that m e (mg) for all m e M. Then certainly for each m e M
there exists a non-zero A = A(m) e К such that mg = J.m. If m and m' are linearly
independent and if Л' = A(m'), then m + m' e (Am + A'm'), and it follows that A = A'.
From this one easily deduces that A(m) = A(m') for all m, m' e M, in other words, that
g has scalar action on M. □
(10.15) Lemma. A KG-module M has a submodule isomorphic with the regular module
KG if and only if M contains an element m such that
тф <m(G\l)>.
10. Faithful and simple modules
181
(Notation. If X c G, then mX denotes the set {nix: x 6 X}.)
Proof. Suppose first that there exists a KG-monomorphism p: KG -»M, and let
/1(1) = m. Since the elements of G form a К-basis for the regular module KG we have
1 «f <G\1>; therefore m = д(1) p<G\l> = <m(G\l)>.
Conversely, if m j <jn(G\ 1)>, then mg $ <m(G\g)> for all g e G, and so the subset
mG of M is linearly mdependent over K. It follows that KG-homomorphism /i from
KG to M defined by setting /r(l) = ni is a monomorphism since its image has the
same dimension as its domain. Thus /i(KG) = <mG> is a submodule of M isomorphic
(10.16) Lemma. Let R and S be subsets of G, and let m and n be elements of the
KG-module M such that m ф <mR) and n ф <nS>. Then the element m®n in M®KN
satisfies
m®n$ (m® n(R иS)>.
Proof. Set m, = m, and let {m2,..., mr} be a basis of <mR>. Sincemt ф {m2,.... m,},
the set {ni,, m2,..., mr[ is linearly independent and can be extended to a basis
{m„ ..., ms} of M. Then the map a sending mt to 1 and m2,..., mr to 0 can be ex-
tended to a linear transformation from jW to K. Similarly there exists a linear trans-
formation fl: M -» К such that nfi = 1 and <nS>/? = 0. Then the К-linear map
a ® f}: M ® N -» К (see Theorem 1.8) satisfies (m ® n)(a ® fl) = ma ® nfi = 1, and if
g e R и S, then (m ® n)g(a ® P) = mga ® ngP = 0. Thus m ® n ф Ker(a ® P), which
contains (m ® n(R и S)>. □
(10.17) Theorem (Bryant and Kovacs [1]). Let К be a field, let {<?,, denote
the non-identity elements of a group G, and for i = 1,..., t let denote a KG-module
on which gt does not have scalar action. Then the regular KG-module is isomorphic with
a direct summand of the tensor product T = Мг ®K • • ®K M,.
Proof. If g; does not have scalar action on Mh by (10.14) there exists an such that
m, <»i,0,>. By repeated application of (10.16) it follows that if w = mt ® •• • ® m„
then w £ <w(G\ 1)>, and so T has a submodule isomorphic with KG by (10.15). Since
the regular module is injective by (2.3) and (2.10), we conclude from (2.9) that KG is a
direct summand of T. □
We are now ready to prove Steinberg’s Theorem 10.13 which states that every simple
G-module N appears both as a quotient module and a submodule of a suitable tensor
power of a faithful G-module.
Proof of Theorem 10.13. Let M be a faithful module for G over a field K, and let Kc
denote the trivial KG-module. Let L = KG @ M. If 1 ge G, by hypothesis t ere
exists an element me M such that mg / m; then (1 + m)g = 1 + mg is not a sea ar
multiple of 1 + m in L, and so g does not have scalar action on L. Therefore by (10. )
the regular KG-module is a direct summand of the tensor power
182
В. Prerequisites—representation theory
S = G\1 )S|
If P is the projective indecomposable module with F/Rad P 3 N (and hence
Soc P = N), then P is a direct summand of the regular KG-module by B, 4.6, and by
the Krull-Remak-Schmidt Theorem A, 4.9 is therefore isomorphic with a direct
summand of one of the summands of L,lcl-11. □
11. Modules with special properties
In this section we gather together an assortment of results about the existence,
construction, or properties of modules which satisfy various special conditions,
usually related to the groups acting on them. The first is a list of useful properties of
the regular module.
(11.1) Lemma. Let pbe a prime, let G be a finite group, and let В = F,,G, the regular
G-module over the field with p elements. Let H denote the semidirect product [B]G
(ns we mentioned in (3.161(a), there is an isomorphism from H to ZpQj G which sends
В to the base group Z‘,}. Let X and Y be subgroups of G. Then:
(a) |CB(X)| =
(b) X < Y if and only if CB(X) > CB(Y);
(C) |[B, X]| = p<IG|-|G = *l>;
(d) X < Y if and only if [В, X] < [B, YJ;
(e) If [B, A-] < CB( Y), then either X = I or Y=1 or X = Y and p = | Y| = 2.
Proof, (a) Let G = |J"=1 g,X be the decomposition of G into left X-cosets, set
« = E x-
xiX
and Uj = gtu for i= 1, ...,n. We claim that {wj,...,^} is an Fp-basis of CB(X),
whence, in particular, the F(,-dimension of CB(A ) is n = | G : X | and Statement (a)
follows. To justify this claim first note that их = и for all x e X and that therefore
each u, belongs to CB(X). Furthermore, if 1 < i # j < n, then u, and uf have disjoint
support, and so ut,..., u„ are linearly independent. Now let b = £9e g asg be a typical
element of CB(X) (with ag e (Fp). Since for x e X we have
E ae9x = E ae9.
geG geG
it follows from the linear independence of the elements of G in В that a№ = agx for all
x e X, in other words that the coefficients of b are constant on left cosets of X, and
therefore
b = E E = E a„,ui-
i=l \xeI / i=l
Thus [uj........u„} span CB(X) and our claim is justified.
11. Modules with special properties
183
(b) The inclusion CB(X) > Ce( Y) follows at once from X < Y. Conversely sunnose
« адгвд .«л Tta, »iSS
(c) With the notation of (a) we assert that the set
У — {</.( 1 + x): i = 1,..., n and 1 / x e X}
is an F(,-basis of [B, A-]. Certainly У is obviously a linearly independent set and is
contained in [В, X]. Let geG, x e X, and consider the element [g, x]. Writing
g = g,*' for some x' e X and i e {1,.n}, we have
[0, x] = -g + gx
= -gtx’ + s.x'x
= g,( — 1 + x'x) - g,( — 1 + x'),
which lies in the F(,-span of our assertion and Statement (c) now follows.
(d) If X < Y, then clearly [В, A-] < [В, YJ. Conversely, suppose that [В, X] <
[B, Y], and let G = i h, Y be the left coset decomposition of Y in G with ht = 1.
Let x e X. By (c) we have
- 1 + x = Y + у),
1=1
1#>еГ
and by comparing coefficients, we see first that aiy = 0 for i > 1 and then that at most
one of the coefficients al y (say ,.o) is non-zero. But this forces al n = 1, and
therefore x = y0 e Y
(e) Suppose that [В, X] < Ce(Y). By (a) and (c) we have |G| - |G: X| < |G: Y|,
whence |X|| Y| < |X| + |Y|, and the only possibilities are |X| = 1, or |Y| = 1, or
|X| = | Y| = 2. Suppose that |X| = | Y| = 2. Then by order considerations [В, X] =
C„(Y). If p / 2, by A, 12.5 we have
В = [В. У] x Ce(Y) = [В, Y] x [В, X],
and so, in particular, В = [В, GJ. Since this contradicts the fact that |[B, G]| = p1 1
by (c), it follows that p = 2. Let л be the involution generating X. Since p = 2, we
have b( — 1 + x)2 = 0, in other words [[/>, x], x] = 1 for all be B. Thus CB(Y) =
[В, X] < CB(X), and since | X | = | Y|, it follows from (a) that CB( Y) = CB(X). But then
from (b) we obtain X = Y, as desired. □
Remark. It follows easily from Statement (b) of (11.1) that
Q(CB(X)) = X
184
В. Prerequisites—representation theory
for all X < G. A G-module В with this property is called absolutely faithful, and such
modules have been investigated by Rose [2].
If N is a normal elementary abelian p-subgroup of a group G, we shall aim to show
how to extend a suitable G/A'-module over Fp by the trivial module to obtain a faithful
G-module First we consider the case where G splits over N.
(11.2) Lemma. Let G = NH, a semidirect product of a normal elementary abelian
p-subgroup by a complementary subgroup H. Then there exists an (rpG-module M with
the following properties:
(i) M has a submodule N* of codimension 1 which is isomorphic with N (when
regarded as an FpG-module in the usual way);
(ii) M/N* is a trivial FpG-module;
(iii) Ker(G on M) = Ker(H on N); in particular, if CH(N) = 1, then M is faithful
forG.
Proof Let m-^n be a G-isomorphism from N (written multiplicatively) onto an
additively written copy N; thus the bar map sends ne to ng. Let M = [(4n)J.ef(1
and n e A'J. Then M is an F(,-space with a subspace N* = {(0, n): ne Nj of codimen-
sion 1. We define an action of G on M as follows: If gt e G, write gt = htnt with
Ii, e H and n, e N (such an expression is unique), and set
(1 l.a) (2, n)fh = (Л Anj + nht).
Then with g, = h,n, (i = 1, 2), we obtain ((2, n)gf)g2 = (2, 2n2 + Xnlh2 + nhjh2).
Since gtg2 = hinlh2n2 = hth2(ni2n2), and since the bar map sends n,2n2 to nlh2 +
n2, it follows that ((>., n)gt)g2 = (2, n)(g1g2) for all (2, n) e M. Thus the G-action
defined by (1 l.a) makes M into an Fp-module. Moreover, it is immediate from (1 l.a)
that the subspace N* is a submodule isomorphic with N (and hence with N) and also
that the G-action on M/N* is trivial.
To justify (iii) let g e Ker(G on M) and write g = hn with hell and n e N. Then
(1,0) = (1,0)hn = (1, n), whence n = 0, n = 1, and geH; furthermore (0, n) =
(0, n)g = (0, nh), and therefore g e Ker(// on N). Conversely, if h e Ker(H on N), then
(2, ri)h = (2, nh) = (2, if), and so h e Ker(G on M). Hence Assertion (iii) holds. □
We now consider the general case.
(11.3) Proposition (The Magnus module, Magnus [1]) Let N(^ 1) be a normal ele-
mentary abelian p-subgroup of G,andlet W = N ^^(G/N). Denote the base group №
of W by B, and regard В as an HpG-module with Ker(G on B) = N. (Evidently В may
be viewed as a non-empty sum of regular Fp(G/Aj-modules, each faithful for G/N.)
Then there exists an FpG-module A such that
(i) A has a submodule B* of codimension 1 which is isomorphic with B,
(ii) A/В* is a trivial G-module, and
(iii) A is faithful for G.
II. Modules with special properties 185
Proof. Applying (11.2) to the semidirect product W = B(G/N), we obtain a faithful
fV-module M which has a submodule B* isomorphic with В and satisfying W/B* =
(Up)»” By A- 18 9 there exists a monomorphism p:G->W such that Bp(G) = W. We
identify G with its image p(G) and set A = Mc, the restriction of M to G ( = p(G)).
Since В is abelian and Bp(G) = W, it follows that B* is isomorphic with B. Further-
more, since M is faithful for IV, its restriction A is faithful for G, and it is clear that
Conditions (i), (ii), and (iii) of the Proposition are satisfied □
The following result may be seen as as generalization of (3.12)(b).
(11.4) Lemma. Let К be a field of characteristic p > 0, let Gbea p-soluble group, and
let V be a simple KG-module. Let N^G, and assume that N has a subgroup H of
p-power index such that CV(H) / 0. Then N < Ker(G on V).
Proof. Let R = <HG>, and let U be a simple submodule of with CV(H) A 0. Let
L = Ker(R on U), and observe from (3.12)(b) that OP(R/L) = 1. If L < R, from the
p-solubility of G we conclude that L < T, where T/L= Op(R/L). Moreover, T < HL,
since | R : f/| is a power of p. By A, 12.5 we have
U = [I/, T] © CV(T),
where both direct summands are KR-submodules and Ct,(T) > CV(H) > 0. Since U
is simple, it folllows that [I/. T] = 0. and hence T < Ker(R on U), a contradiction.
Hence L = R and C,,(R) > U > 0. Since R< G, the subspace C,,(R) is a G-submodule
of К whence Ct,(R) = V by the simplicity of V. Finally, since N/R is a p-group, we
conclude, again from (3.12)(b), that N < Ker(G on F). □
(11.5) Theorem (Dade). Let H be a p'-subgroup of a p-soluble group G, and let К be
a field of characteristic p. Then the following statements are equivalent:
(a) If V is a simple KG-module with C,,(H) A 0, then CR(H) = V',
(b) The subgroup H is a Hall p'-subgroup of some normal subgroup of G.
Proof, (a) => (b): We begin with a consequence of Condition (a):
(11.0) Let M denote the trivial КН-module, and let N be a simple submodule of MG.
Then H < Ker(G on N).
This follows from the fact that, by Nakayama’s Lemma 6.8 and Maschke’s theorem,
NH S M © M
for some //-submodule M, and so 0 / M < G(H). Therefore from (a) we conclude
that CK(H) = N. . , .
We will prove that (a) implies (b) by induction on | G|, the conclusion being obvious
when G = 1. Let L be a minimal normal subgroup of G, and observe tha /
p’-subgroup of G/Lsatisfying Condition (a) for K(G/L)-modules V. Then by induction
186
В. Prerequisites—representation theory
G/L has a normal subgroup T/L such that HL/L e Hallp(T/L). If L is a p-group, or
if L< H, then И e На11р.(7 ); consequently we can assume that L is a p'-group
with Li H. Let M = KH, the trivial module. By (6.4) we have Li Ker(G on Afc),
and so from the semisimplicity of (Mc)t we conclude that (MG)t has a simple
submodule, W say, which is not centralized by L. Since L < G, the sum S of all such
L-submodules IV is a G-submodule of MG, and so if N is a simple G-submodule
of S, by (1 l./J) we have H < CG(1V); moreover, [IV, L] # 0 by definition of S, and
therefore CL(N) = 0. Let D = Cc(N) n T, a normal subgroup of G containing H.
It then follows that
D n LH = (D n L)H < (CG(1V) n L)H = H,
and since LH e Hallp.(T), we conclude that H = D r. LH e Hallp.(D), as required,
(b) => (a): This implication follows at once from Lemma 11.4. □
Next we state and prove a technical result about the existence of a module with
some prescribed properties. In the stated generality, it is due to Forster; special cases
will be used at various points in the sequel.
(11.6) Proposition (Forster [2], Lemma 1.9). Let G be a group whose socle S is the
product of abelian minimal normal subgroups Nt,..., Nn which are pairwise non-
isomorphic as G-modules. Let H be a subgroup of G satisfying
Further, let К be a field whose characteristic, if non-zero, does not divide |S|, and let
U be a simple КН-module such that Nj i Ker(// on U) whenever Nt < H. Then
(a) there exists a simple KG-module V, faithful for G, whose restriction If has I) as
a quotient module. If, additionally, H r.S < Ker(// on U) (which can only happen
when H contains no NJ, then
(b) V can be so chosen that FHS has a submodule T such that H r S < Ker(//S on T).
Proof, (a) If Aj H for i = 1,..., t, then set r = 0; otherwise suppose that the min-
imal normal subgroups of G have been numbered so that
(i) H n Nj = Ni for i = 1,..., r,
(ii) HoNj< Nj and [G, (H m A'.)] / 0 for i = r + 1,..., s, and
(iii) H n < Aj and H n IV, < Ker(H on U) for i = s + l,...,t.
We begin with some preliminary observations. Let IV be a simple submodule of
Usr-u- Since H n Nf < H, the subspace CV(H n Nj) is an H-submodule of U, and the
simplicity of U implies that CV(H n A',) = 0 for i = 1,..., s and, in particular, that
CK.(H c- Nj) = 0. We assert that
(1 l.y) Ws has a simple submodule Z such that N, i Ker(S on Z) for all i e {1,..., t}.
To justify (1 l.y), let Cj be a complement to H n Nj in Aj for i = r + 1,..., t, and note
that C = Cr+1 x • - • x C, is a complement to H n S in S; in fact, S = (H n S) x C,
11. Modules with special properties
187
and, according to (6.21)(d), the module Ws can be identified with the tensor product
W® КС, where the direct product (HnS) x C acts componentwise (see (1 12)) Since
C is a direct product of elementary abelian subgroups C„ it is straightforwaid to
verify that C has a subgroup D such that
(1) C/D is cyclic (of square-free order), and
(2) D n C, has prime index in C, for i = r + 1,..., t.
By hypothesis Char(K) does not divide |C/D|, and so by (9.7) and (5.26)(b) there
exists a simple КС-module Y with Ker(C on Y) = D, whence [Y, C] = Y for i =
r + 1,..., t by (2). Now by (2.1)(b) and Maschke’s theorem Y may be regarded as a
submodule of the regular module К C, and then W ® Y can be viewed as a submodule
of Ws. Let Z be a simple submodule of the S-module W ® Y. Then Z„nX is a sum of
copies of W, and so [Z, H n Aj] / 0 for i = 1,..., s. Moreover, Zc is a sum of copies
of Y, and, in particular, for i = s + 1,..., t we have [Z, NJ > [Z, Q] = Z / 0. Thus
Z fulfils Condition (1 l.y).
By Mackey’s Theorem 6.20 the module (CG)S has a submodule isomorphic with
((W® l),,r„s)s (corresponding to the double (H, S)-coset representative 1) and hence
by (1 l.y) a simple submodule Z* isomorphic with Z. Let И be a simple submodule
of which is in turn a submodule of UG. By Nakayama’s Reciprocity
Theorem 6.8, we have Homt„(l/„, 17) = HomKG(l< 17е) 0, and so has a quotient
module isomorphic with U. Since the modules Z*g are conjugate to Z*, and since
Ker(S on Z*g) — g1 Ker(S on Z*)g, it follows that If acts non-trivially on each Z*g
for i = 1,..., t. But the composition factors of fj are conjugates of Z* (in fact, by
Clifford’s theorem, Ks is a sum of copies of the sum of a complete set of G-conjugates
of Z*), and therefore none of the minimal normal subgroups ,..., Nt acts trivially
on И Our hypothesis that Д' A'- for 1 < i / j < t implies that these are the only
G
minimal normal subgroups of G, and therefore V is faithful for G.
(b) If H n S < Ker(// on U), the hypotheses of the proposition imply that s = 0,
in other words, that cases (i) and (ii) described at the outset do not arise. Let Z* be
a submodule of Vs which is isomorphic with the module Z of (11 .y); since the subgroup
Hc.S acts trivially on U, it acts trivially on W ® КС and hence likewise on Z and
Z*. Set
T = £ Z*h<V.
heH
Since H normalizes S, it is clear that T is an SH-submodule of И Moreover
Ker(S on Z*h) = /Г’Кег(5 on Z*)ft > (HnS)” = HnS. Consequently HnS <
Ker(//S on T), as desired. □
The following special case of (11.6) is most frequently cited.
(11.7) Corollary. Let H be a subgroup of a group G such that CoreG(H) = 1, and let
К be a field whose characteristic does not divide |Soc(G)|. Assume that Soc(G) is tfte
product of pairwise non-isomorphic minimal normal subgroups of G. If U is a simp e
KH-module, there exists a simple KG-module Lfaithful for G, such that U isa quotient
module of L„; in particular, when U is the trivial module, the corresponding module V
satisfies [С, II] < И
Ijjj; В. Prerequisites—representation theory
Our final objective in this section will be to prove the following theorem.
(11.8) Theorem. Let pbe a prime, and let G be a group whose order is divisible by p.
Then there exists a group E with a normal, elementary abelian p-subgroup A A 1 such
that
(i) A < Ф(Е), and
(ii) E/A s G.
This fact will be applied in Chapter IV to prove that a local formation is saturated.
Using an idea due to Fotheringham [1], we shall deduce Theorem 11.8 from a
construction devised by Gaschiitz to show that any finite group can arise as the
quotient H/'¥(H) of a suitable group H by its Frattini dual 'f'(H). However, much
more is known about Frattini extensions E of an ffpG-module A by a group G
(namely extensions for which A < Ф(£)), and we present a more detailed account of
this in Appendix f!.
(11.9) Definition. Let G be a group. The Frattini dual T(G) of G is the subgroup
generated by the minimal subgroups of G. Thus we have
T(G) = <U: U < G, |U| e P>.
Evidently T(G) is a characteristic subgroup of G. It was introduced by Ito [1] as
the obvious dual to the Frattini subgroup Ф(С). But whereas ®(G) is always nilpotent,
there are no structural restrictions for G/T(G); Gaschiitz’s elegant proof of this fact
is our next goal.
(11.10) Lemma. Let R be a normal subgroup of a group G such that R = Ч'(й). Then
the following condition is necessary and sufficient for R = 'F(G):
(11-5) Whenever V/R is a minimal subgroup of G/R,thenRisnot complemented in V.
Proof. Necessity: Suppose that (11.5) fails to hold. If V/R is a minimal subgroup of
G/R and if V is a complement to R in V, then |U| e P. Consequently V < 'F(G), and
R'P(G) > V > R; in particular 'P(G) R.
Sufficiency: If lP(G) # R = 'I'(R), there exists a minimal subgroup U of G with
U f, R. Then U n R = 1, and on setting V = UR, we see that (11.5) is violated. □
Since properties of free groups will now be needed, for the rest of this section we
lift our blanket hypothesis that all groups under consideration are finite. In the proof
of the following result we need to cite Schreier’s theorem, and a full statement of this
can be found in Theorem f!.2 of Appendix [i.
(11.11) Proposition (Gaschiitz [9]). Let F be a finitely-generated free group, let p be
a prime, and let R be a normal subgroup of F of index p. Let S = Rw₽l, the smallest
11 Modules with special properties 189
normal subgroup of R whose quotient belongs to the class 9I(p) of elementary abelian
p-groups. Then R/S is not complemented in F/S.
Proof. Let .« be a set of free generators for F. By Schreier’s theorem, R is free on
p(r - 1) + 1 generators, where r = \ST\. Let M/S be a normal subgroup of F/S
maximal subject to the condition that M < R. Thus R/M is a chief factor in a chief
series of F above S, and so R/M < Z(F/M) since F/S is a p-group. Consequently
\F/M\ = p2 and F/M is abelian. Let T/S denote the intersection of all such normal
subgroups M/S, and note that F/T is abelian.
Let Г],..., r( be elements of R/S whose images in (R/S)/(T/S)(^R/T) form a
minimal generating set; thus t here denotes the rank of R/T. Then < T/S, ^,..., r > =
R/S, and so if N/S denotes the normal closure in F/S of the set (r,,..., r,}, it follows
that N/S = R/S; for if N/S were a proper subgroup of R/S, then (T/S)(N/S) would
also be proper by definition of T/S, and this is not the case. Since R/S is abelian, each
centralizer Сг/5(г() has index 1 or p, and therefore each rf has at most p conjugates in
F/S. It follows that R/S can be generated by ai most pt elements, and so pt >
p(r — 1) + 1, the rank of the elementary abelian group R/S. Thus t > r — 1 + 1/p,
and since t is an integer, we have t > r.
Now if R/S has a complement in F/S, certainly R/T has a complement in F/T, and
then F/T = R/T x Zp, since F/T is abelian. In this case, F/T is elementary abelian
of rank t + 1 > r + 1, and we have a contradiction because F, and hence all its
quotient groups, can be generated by \3.'\ = r elements. Therefore R/S is not com-
plemented in F/S. □
(11.12) Lemma. Let R be an abelian normal subgroup of a group V, let pbe a prime,
and assume that V/R is a p-group. Let R = P x Q with P e Sylp(R). Then R is
complemented in V if and only if R/Q is complemented in V/Q.
Proof. If U is a complement to R in V, then UQ/Q is obviously a complement to R/Q
in V/Q. Conversely, let U/Q be a complement to R/Q in V/Q, and let U e Sylp(C).
Since Q is a p'-group, U is a complement to Q in U. Therefore UR = UQP = UR = V,
also U n R = (U n U) n R = U rQ = 1, and hence R is complemented in И □
(11.13) Theorem (Gaschiitz [9]). For each finite group G there exists a finite group
H such that G = ///T(//) and 'P(H) is abelian with elementary Sylow subgroups.
Proof. Let F be a free group of rank |G| on the free generators {fg- g e G}, and let
в: F -» G be the uniquely determined epimorphism which satisfies 0(fe} = g. Let
R = Ker(fi), and for each prime divisor p of |G|, let S(p) = Kw₽l. Then S(p) is char-
acteristic in R and hence normal in F. Set
s= П s(p)
pea(G)
Then evidently R/S S ©pe p(C) R/S(p). and since each R/S(p) is an elementary abelian
p-group, we have 4>(R/S} = R/S.
190
В. Prerequisites—representation theory
Let V/R be a minimal subgroup of F/R, of order p say. By Schreier’s theorem, V
is a finitely-generated free group, and therefore by Proposition 11.11 there is no
complement to R/S(p} in V/S(p). Lemma 11.12 now implies that R/S is not comple-
mented in V/S; hence Lemma 11.10 applies, and we can deduce that R/S = 'f(F/S}.
Therefore with H = F/S, we obtain H/W(H} = (F/S)/(R/S) S F/R = F/Ker(0) s G.
Finally we can give the □
Proof of Theorem 11.8 (Fotheringham). As in the proof of (11.13), let F be a finitely-
generated free group with F/R = G, and let S(p) denote the '2I(p)-residual of R. Set
F = F/S(p} and R = R/S(p)(e'2I(p)), and let £ be a minimal supplement of R in F. If
A = £ n R, then E/A = F/R = F/R = G, and A < Ф(£) by A, 9.2(c). To complete the
proof of (11.8) we must show that A # 1. Since p divides the order of G S F/R, there
is a subgroup V/R of order p in F/R, and by (11.11) there is no complement to R
in V = V/S(p). It follows that R is not complemented in F, and therefore that
E n R > 1. □
12. Group constructions using modules
In this section we gather together various group constructions which will be used at
various points in subsequent chapters. Most of the constructions involve modules in
one way or another.
(12.1) Lemma. Let V be a group, let G and H be subgroups of Aut(lz), and assume that
(| VI, |GI) = 1. Then the semidirect products [l ]G and [VJH are isomorphic groups if
and only if G and H are conjugate in Aut(I ').
Proof. First suppose that there exists an isomorphism a: [F]G -»[VJH. Let it
denote the set of primes dividing |F|. Then {(r, l):t>e Г} is the unique Hall
л-subgroup of [F]G and is therefore mapped isomorphically by a onto the corre-
sponding subgroups of [VJH. Let
6 = {(1, g): g e G} < [F]G, and
H = {(\,h):heH} < [F]H.
Then G e Hall„.([F]G), H e Hall„.([£]//), and by the Schur-Zassenhaus Theorem
A, 11.3 the image of G under a is conjugate in [FJff to H. By composing a with a
suitable inner automorphism of [I ]H, we can therefore suppose without loss of
generality that Ga = H. It then follows easily that the map fl: G -» H defined by
is an isomorphism from G onto H. Similarly the map у: V —» V defined by
12. Group constructions using modules
(f'A 1) = (v, l)a
191
is an automorphism of И
lf ? f cAU,( V)’ T denote the image of V under 0 b> l«- Then, for all v
e G, the definition of a semidirect product (cf. A, 4.22) implies that
e V and
(»y,g 'P) = (yy,l)(l,g~lp)
= ((«. 0(1,0 '))“
l))a
= i)
= (»9~'y(9P), g-'P)
since (g 'P) 1 = gp. Consequently у = д~'у(дР), whence y-1gy = gp, and therefore
у *Gy = Gp = H.
Conversely, let p be an automorphism of V such that Ge’ = p 'Gp = H. We define
а тара: [F]G -> [K]H thus:
(t>, g)a = {ifl, g^).
Let t>,, t>2 e L and gt, g2 e G. Then, by the rule for multiplying the elements of a
semidirect product, we have
((»! Sl)(»2, 02»® ~ (f l(f 201 ‘ ), 9102)“
= Д). <?М) = ((1’1/Я((1’2/Я(/г'01-7Я),
= ((fiР)(Уг0(д?Г'19?9г) = (ViP, 9i)(v2P' 91) = (fi, 0i)a(e2.02>a-
The map a is therefore a homomorphism, and since it is clearly surjective, we conclude
by order considerations that it is an isomorphism. □
(12.2) Proposition. Let p be a prime, let G be a p'-group, and let V be a simple
IFpG-module. Then for each natural number e there exists a homocyclic abelian p-group
A o f exponent pr, which admits G as a group of operators in such a way that 4/Ф(4),
viewed as an lyG-module, is isomorphic with V.
Proof. If m e M, let denote the ring Z/mZ, and consider the group rings R = ^f-G
and R = ZpG = FpG. Then G is a (faithful) group of operators for their additive
groups R+ and R+, and clearly R+ is operator-isomorphic with R’/O(R’). Since
P11G I, by M aschke’s Theorem A, 11.5 we have R + = @?=1 Ц, where the G, are simple
F,,G-modules, and by (4.6) we know that V S Ц, for some к e {1,..., s}. By Theorem
A, 11.6 the G-operator group R + admits a decomposition
192
В. Prerequisites—representation theory
K+ = @«.
1=1
into G-admissible subgroups R, such that R,/®(R,) is a simple FpG-module; further-
more each R, is homocyclic (necessarily of exponent p‘). Evidently ®(RJ =
Rf n ®(R +), and therefore s = t, and with suitable numbering we have Ц =; R,/®(R,)
for i = 1,..., t; in particular, V s Rt/O(RJ. □
(12.3) Proposition, (a) Let A be a homocyclic group of exponent pe and rank n, and
use bar notation to denote the natural homomorphism from A to A = А/Ф(А). If
a 6 Aut(/1), let a: A -► A denote the map defined by setting
aS = (aa)
for all ae A.T hen a is a well-defined automorphism of A, and the map r: a -» a is an
epimorphism from Aut(/1) onto Aut(/I ) (sGL(n, p)), and its kernel is a p-group.
(b) If p and q are distinct primes, then the Sylow q-subgroups of Aut((Zp«)") are
isomorphic with those of GL(n, p).
Proof, (a) It is straightforward to verify that the map r: a -> a is well defined and is
a homomorphism. It is also clear from the definition that Ker(r) acts trivially on
A = Л/Ф(Л), and it then follows from A, 12.7 that Ker(r) is a p-group. It remains to
prove that r is surjective. Let {Oj, a2,..., a„} be a basis for A, so that
Л = <flj> x x <fl„>.
Let [i e Aut(A). Then there exist integers xy with 0 < x;j. < p — 1 such that
SjP = П afu
i=i
for j = 1,..., n. Since Det(xy), as an element of is non-zero, when Det(xfJ) is
computed in Z it is not divisible by p. Thus Det(x;j) computed in the ring R = Z/p'Z
is a unit, and (xu), viewed as a matrix over R, has an inverse. It follows that the map
a defined by
aja = П aiu
i=i
extends to an automorphism of A and satisfies a = /I. Hence r is an epimorphism.
(b) This follows at once from Part (a). □
(12.4) Theorem. Let e and n be natural numbers, and let p be a prime which does not
divide n. Then, to within isomorphism, there exists a unique group G with the following
properties:
(i) G has an abelian normal subgroup A, which is homocyclic of exponent pc:
12. Group constructions using modules
193
(ii) A has a complement C in G, and С г Z„;
(iii) C acts faithfully and indecomposably on A.
Proof Let C be a cyclic group of order n. Since 0„(C) = 1 by hypothesis, by (107)
there exists a faithful simple module V for C over F„, and by (9.8) the dimension of
V is the order of p modulo n. Then by (12.2) there exists a homocyclic abelian group
A of exponent pe such that C < Aut(A) and Л/Ф(Л) s V, as FpC-modules. By A, 11.7
the group C acts indecomposably (and obviously faithfully) on A, and therefore the
semidirect product G = [Л]С is a group with the three stated properties.
To prove the uniqueness, let i e {1, 2}, and let Gf = /1,6’,. be a group with Properties
(i), (ii), and (iii) of the statement of the theorem. Since C, acts faithfully on A„ by A,
12.7 it also acts faithfully on Af/O(A,); furthermore, by A, 11.7 it acts irreducibly on
4/Ф(Л)- Therefore |Л,-/Ф(Л()| = | V[ by (9.8)(d), and it follows that | At | = |Л2|. Thus
we can identify At and A2 with a fixed homocyclic group A of exponent p‘ and of
rank Dimfp(F) and transfer the action of C, on /1, to an action on A. It will the suffice
to show that [Л] C2 = [A]C2.
Let W = Л/Ф(Л), viewed as a faithful simple Cj-module over Fp for i = 1, 2. We
show first that [IPjC, = [W]C2. In fact, by Theorem 9.8 we can find an element £
of multiplicative order n in the field К with | W\ elements, and a generator c, of C„
such that
wc, = w£ (field multiplication)
for a suitable isomorphism w-> w from W to K+. It follows that [H'JQ s
[K+]<e> s [W]C2 for i = 1, 2, and we can then deduce from Lemma 12.1 that C2
and C2 are conjugate when viewed as subgroups of AutflT). If r denotes the homo-
morphism from Aut(/1) to Aut(lV) described in the statement of Proposition 12.3(a),
we conclude from that result that Ker(r)C! and Ker(r)C2 are conjugate in Aut(A); in
particular, there exists an element ft in Aut(/1) such that C2 < Ker(r)C,. Since Ker(r)
is a normal p-subgroup of Ker(r)Ct. by the Schur-Zassenhaus Theorem A, 11.3 the
Hall p'-subgroups C, and Cj are conjugate, and so Cf’ = Сл for some у e Ker(r). A
further application of (12.1) now yields [Л]C, S [Л]С2. □
(12.5) Notation. The uniquely-determined group G satisfying Properties (i), (ii), and
(iii) of (12.4) will be denoted by E(n/p‘).
Remarks, (a) Although a cyclic p'-group C of order n may have non-isomorphic
simple modules, Ц and V2 say, over Fp, Theorem 12.4 implies that the semidirect
products [Ц]С and [T2]C are isomorphic (each to the “unique” group E(n/p)).
(b) If E = E(n/p‘}andP = OP(E}, then clearly E/Qe-d(P} ~ E(n/pd)for 1 < d < e.
(12.6) Lemma. Let p and q be distinct primes, and let X = <x> be a cyclic group of
order qr which acts faithfully on a homocyclic abelian p-group A of exponent p
(a) There exist integers s, t,,.... ts and d, < • • • < d, such that
194
В. Prerequisites—representation theory
where, for a given i e {1,..., s} the Ait,..., Л,,. are indecomposable X-modules of rank
dj over 7Lp,..
(b) Let q‘l = Max{|X/Ker(X on Л;7)|:у = 1,t,-}.
There exists a monomorphism p from the semidirect product [A] X to the direct product
D=X W/P')F
i = l
such that Ap = 0p(D) and Im(p) < D. (Note that E(l/pe) = Zp« by convention.)
Proof The existence of the direct decomposition of A described in Statement (a)
follows from Theorem A, 11.6, which also shows that Лу/Ф(Лу) is a simple FpX-
module; furthermore, X has the same kernel on Л0-/Ф(Л(;) as on by A, 12.7.
(b) Let aif denote the automorphism induced by x on Ло-, and set q“o = o(ay). Then
by (12.4) there is an isomorphism from the semidirect product [Ло-] <afj- > to E(q°“lpf.
By definition of a, there exists an element j in {1,.... t,-} such that x induces on
(and hence on ЛЦ/Ф(Л;-)) an automorphism of order a,. From (9.8) (d) we can deduce
that q“‘ divides pd‘ — 1 and hence that E(q°“lpe) is isomorphic with a unique (normal)
subgroup of E(q°’/p‘). Hence, composing maps, we obtain a monomorphism
PtflAij\^ijy^E(q‘“lp‘}
such that АцРи = Op(E(q°,/pe)). For g e AX we can write g — ax" with a e A = ф Atj
and 0 < n < qr. We then define a map
p: AX ->D
by specifying the (/, j)-component of gp in the direct product D to be (а17а,")/г,7, where
av is the projection of a onto Evidently p is a homomorphism with Ap ~ Op(D).
Since x acts faithfully on A, it follows that a,- = r for some i and hence that p
is a monomorphism. Finally, we observe that D' <, OP(D) < (AX)p, and therefore
(AX)p < D. □
If A is a homocyclic group of rank n and exponent e, there is a natural isomorphism
between Aut(A) and GL(n, Zp«), the group of non-singular matrices with entries in
the ring Zp. = %-lp‘^-. The group of units of Zp« has order pe — p"~l = pfl(p — 1)
and has a unique subgroup К of order p — 1. We can therefore identify the multi-
plicative group F* with К by means of the map r + pZt->r* + p‘Z(l <, r s p ~ 1),
where r* + peZ is the unique element of Zp. which satisfies
(i) r* + pZ = r + pZ, and
(ii) the order of r*(mod pe) is prime to p.
(12.7) Lemma. Let x be a p'-element of GL(n, Zp.), and let x denote the n x n matrix
over Fp obtained by replacing each entry хц + prZ of x by the (well-defined) element
xtj + P^ °f — FP- Then Det(x) = Det(x), with the identification described above.
12. Group constructions using modules
195
Proof. Since x is invertible in Zp., its determinant is in the group U of units of Z
and since (Det(x))°'*> = Det(x°'*>) = 1, it follows that Det(x) has p'-order and so
belongs to the Hall p'-subgroup К of U. Thus Det(x) = r* + peZ, where the order
of r*(mod pc) divides p - 1. But, from the definition of x, evidently Det(x) =
Det(x) + pZ = r* + pZ, which is the element of F„ we have identified with Det(x).
□
We now describe a group construction, associated with a finite field, which provides
a valuable source of soluble groups of derived length 3; in fact, they are abelian-by-
metacyclic, being a 3-fold extension of the additive group by the multiplicative group
of the field by its Galois group.
(12.8) Definition. Let p be a prime, let n be a natural number, and let К denote the
field Fp„ with p" elements. A semi linear transformation of К is a map 0: К -> К of the
form
в = 0(a, b, y): x -» fix’ + a (for x e K),
where a, b e K, b # 0, and у 6 Aut(K).
(12.9) Proposition. The set of all semilinear transformations of К = Fp„ forms a sub-
group G of Sym(K). The group G is denoted by Г (pfand has the following subgroups:
(i) A normal subgroup A = {0(a, 1, i): a e K + } of order p", isomorphic with the
additive group K+ of K;
(ii) A cyclic subgroup В = {0(0, h, i): 0 b e К'} of order p" - 1, isomorphic with
the multiplicative group K* of K;
(iii) A cyclic subgroup C = {0(0, 1, у): у 6 Aut(K)} of order n, isomorphic with the
Galois group Aut(K) of K.
Furthermore, A rB = ABn C = 1 and C normalizes B: in particular, G = ABC and
so |Г(р")| = np”(p" — 1). Finally, В acts regularly on A, Cc(A) = A and Свс(В) = B.
Proof From the definition of 0(a, b, y) in (12.8) we obtain the following rule for
composition of two elements in Г(р"):
(12.a) 0(a, b, y)0(a', b', y') = 0(b'a1'' + a', bV, yy').
Thus we see that Г(р") is closed under multiplication (composition of maps). If 6
denotes у Л then the product 0(a, b, y}e(-b~V, b~6, 5) is the identity on K, and so
it follows that Г(р") is a subgroup of Sym(K).
If g = 0(a, b, y), then
(12./I) g~l6(a, 1, i)g = 0{ba', 1, i) e A,
and it is clear that A < G. In particular, Equation 12.
A corresponds to field multiplication (cf. Constructioi
to the action of Aut(R') on K. Therefore, if 1 / g e B.
[i shows that the action of В on
a 9.7) and that of C corresponds
then CA(g) = 1, in other words,
196
В. Prerequisites—representation theory
В acts regularly on A; moreover, C4(C) has order p because it corresponds to the
fixed field of Aut(K).
It is clear that A = K+, В =: K\ and С S Aut(K), and it is well known that these
groups have the stated structure. It is also obvious that AcB = ABnC = 1. From
(12.a) we obtain
в(0, I, y)W>, b, i)6(0. 1, y) = 0(0, b\ i)
and so C normalizes B, acting like Aut(K) on K'; in particular, Свс(В) = B. Finally,
let g = 6(a, b, у) 6 Cc(A). Then (12.(3) yields the equation a = bay for all a e A, and it
follows that b - 1 and у = i. Thus Cc(A) = A, and the proof is complete. □
Terminology. The group Г(р") is sometimes known as the extended affine group of
(12.10) Proposition. Let G = ABC denote the group Г(р") described in Proposi-
tion 12.9, where A is the normal, elementary abelian p-subgroup of G of order p".
Then A. viewed as an ^pBC-module, is absolutely irreducible. In particular, A is a
self-centralizing minimal normal subgroup of G, and G is a primitive group with
BC as a stabilizer. Furthermore, if I # Bo < В and ABo is simple, then Aut(AB0) S
Aut(G) s G.
Proof. Let К denote the algebraic closure of K, and let A = A ®K K. By Theorem
5.6, (c) => (a), it will suffice to show that A is simple. Let U be a simple submodule of
A, and let
UB = IF,®---® W„
where kkj,..., W, are the homogeneous components of UB. Since В acts regularly on
A by (12.9), it acts regularly on A, and hence also on U; therefore Ker(B on Wx) = 1.
By (9.1) the simple submodules of W, have dimension 1 (because К contains a
primitive (p" — l)st root of unity), and therefore by (9.15)(b) we have 1Вс(^) =
Свс(В) = В by (12.9). By Clifford’s Theorem 7.3(b) we have t = |BC: B| = n, and
so DimA(t7) > n. However, DimB(A) = DimKA = n, and we conclude that U = A,
which is therefore simple. In particular, A is a simple G-module and hence, by (12.9),
is a self-centralizing minimal normal subgroup of G. By A, 15.8(b) our G is a primitive
group with BC as a stabilizer.
Set Y = AB0, and note that by hypothesis У is a primitive group with Soc(F) = A.
In particular, Z(Y) = 1, and as usual we can identify Y with Inn(Y) < Aut(Y),
observing that with this identification the action of an element of Aut( Y) on Y is given
by conjugation. Moreover, since Y < G and Cc(Y) = 1, we can regard G as a sub-
group of Aut(Y) with Y = AB0 < AB <G< Aut(Y); in particular, |Aut(Y)| > |G|.
First we aim to show that A = CAul(n(A). Since A char Y = Inn(Y) < Aut(Y), we
have A < Aut( Y). Let At be a minimal normal subgroup of Aut( Y) distinct from A.
Then obviously N r Y = 1 and therefore N < CAut(n( Y) = 1; hence A = Soc(Aut( Y))
and consequently F(Aut(Y)) is a p-group. Let С = CAuIin(A) and F = F(C). Then F
12. Group constructions using modules
197
is a normal p-subgroup of Aut(T) with A = Z(F}, and if Ф(7 ) / ), it follows that
A < Ф(Е) because <f>(F) < Aut(T). But then we have [F, Bo] = [F, У] < Fn У =
A < O(F) and hence [F, Bo] = 1 by A, 12.7. However, this contradicts the fact that
[F, Bo] > [A. B0J = A; therefore O(F) = 1 and F = A x CP(B0) by A, 12.5. Since
CF(B0) — СГ(У) < G, it follows that CP(BO) = 1 and F = A, and we conclude from A,
10.6 (a) that A = Cc(A) = C. Thus we can regard Aut(T)/A as a subgroup of Aut(A)
(which is isomorphic with GL(n, p) of course).
Now let R/A = САи1(Г)(У/Л)(> У/Л). Since R/A induces (by conjugation) FPBO-
endomorphisms of A, we can regard R/A as a subgroup of End, Bo(A)', which is
isomorphic with (Fp„)* by (9.8)(d); in particular, |K: 4| < p" - f It follows that
R = AB and also that AB/A is a self-centralizing normal subgroup of Aut( Y)/A. Since
we have already shown that A is absolutely irreducible as an FpAB-module, and since
Aut(F)/K is isomorphic with a subgroup of Aut( Y/A) s Aut(Bo), which is abelian,
we can apply Lemma 9.14 and deduce that |Aut(T): AB| = n. However, with the
identifications described earlier, we have AB < G < Aut(Y) and |G: AB\ = |C| = n.
Therefore Aut( У) = G, and the last conclusion of the proposition is now clear. □
The last construction which we describe in this section is due to Brian Hartley. It
simultaneously generalizes the group of unitriangular matrices over a field and a
p-group construction used by Huppert ([H], Hilfssatz VI, 7.22) in the proof of the
Gaschiitz-Lubeseder theorem (see Theorem IV, 4.6 below). The raw material for the
construction is a set F = {Vt,Vn} of vector spaces over a field F. If V] is an
FG,-module fori = 1,..., n, the direct product G, x x G„ of the groups G, will be a
group of operators for the Hartley group
(12.11) Definition. Let У = {fj,..., V„} be a set of vector spaces over F, and, for
1 < i < j < n + 1, let V(i, j) denote the following tensor product
V(i,j)= K®F”®r Vj-t-
If i < j < k, make the natural identification between F(i, j)®P V(j, k) and F(i. k),
and write simply ® instead of ®P (since all tensor products will be over some fixed
but unspecified field F for the rest of this section). The Bartley group, which will
be denoted by H(1), is defined in the following way. The underlying set of Н(У )
consists of all (n + 1) x (n + 1) matrices h = (fi,;) whose entries fulfil the following
requirement:
(a) If 1 < i < j < n + 1, then /i,; is an element of F(f, j);
(b) If 1 < j < i < n + 1, then h,j = the Kronecker delta.
Thus the matrices have upper-triangular form. The binary operation for Н(У ) is given
by the usual rule for matrix multiplication. Thus, if h ~ (hti) and к — ( „) are two
elements of Н(У j, the product hk is the matrix m = (mJ whose entries are determine
by the equations
и+1
(12.y) mij='£hirkrj,
r=l
where
198
В. Prerequisites—representation theory
(a) for 1 < r < jwe define hirkrJ to be the tensor hir ® krj, regarded as an element
of F(i, j) by means of its identification with F(i, r) ® V(r, j) mentioned above, and
(b) we interpret multiplication by 0 and 1 in the usual way, so that (12.y) may in
fact be written as follows:
mt/ ~ + hi i+tki+l j + + hjj-tkj-jj + hjj.
(12.12) Proposition. Under the binary operation defined by (12.)) the set H(t ) forms
a group.
Proof. If i < j < к < I, the tensor products (P(i, j) ® V(j, k)) ® V(k, I) and P(i, j) ®
(F( j, k) ® V(ky 1)) are both identified naturally with P(I, /), and so it is clear that the
given binary operation is associative. Since the identity matrix Lt+1 obviously belongs
to ') and behaves as a multiplicative identity, in order to verify the group axioms
for H(i") it remains to prove that, for each h = (/i0) 6 H( U ), there exists a unique
к = (kJ e H(f ) such that hk = f„+1. Consider the system of equations obtained by
setting m,, = bis in (12.<5):
(12.e) kfj + + ’ + hjj-ikj-ij + htj =
for 1 < i < j < s. Given (htj) we must show that these equations have a unique
solution for kjj (1 s i < j < n + 1). We divide these equations into n + 1 subsystems
(12-e.s) according to the value ofj — i (which we denote by s) thus:
(12.E.S)
where s ranges over the values 0, 1,..., n. We then prove by induction on s that the
system of equations: (12.E.0) и (12.e.l) и • • и (12.s.s) has a unique solution for the ku's
that arise.
For s = 0, the equations (12.E.0) become
= 6u (1 < i < n + 1),
and these are clearly satisfied because hu = ku — 1. Fors = 1, we obtain «equations
kt.i+i + hi i+i =0 (1 < i < n),
and these yield the unique solution ki i+l = — liii+1 for the elements of the first
superdiagonal of k. Let s > 1, and suppose inductively that uniquely determined
solutions have already be found for Equations 12.E.0,12.E.1.and 12.s.(s - l).Then
consider the n + 1 — s Equations I2.e.s, which take the form
ki.i+s + h, i+iki+l l+s + ••• + ht i+s^iki+s_1 i+s + ht 1+, = 0
for 1 < i < n + 1 — s. Since ki+l i+s,..., ki+s_, i+s have already been determined by
12. Group constructions using modules
199
the earlier equations, it is clear that we obtain a unique value in F(i, i + s) for the
entry By induction we conclude that there exists a unique solution for the whole
system (12.c). Thus each element of ЩГ) has a right inverse, and it follows that ЩГ)
is a group. q
We will show next how to construct a group of operators for the Hartley group
ЩГ). Suppose that each F-space Ц 6 Г is an FG,-module for some group Gt, and
let G denote the direct product
G = G, x ••• x G„ = {(g,....g„): g, e GJ.
If i < j, regard the F-space F(i, j) as an FG-module by defining the action of an
element g = (gt,..., g„) on the pure tensors thus:
(12.£) (ц- ® • • • ® Vj,t)g = vigi ® ® and extending linearly
to the whole space in the usual way. This enables us to define an action h -► ha of G
on Н(Г) by setting
(12.4) h° = (htjg)
with the obvious conventions that 1g = 1 and Од = 0 for all g e G. Then certainly
he e ЩГ), and we assert that furthermore
(12.0) (hk)9 = for all g e G and h,ke H(t").
To justify this, observe that a typical element of F(i, r) has the form =
E„(u, ® ® a linear combination of pure tensors with uk e Vk. If we similarly
write krj = ® • • • ® with vk e Vk, by the definitions of the product hk and
of the G-action given by (12.Q and (1 2.ij) we have
E E (u< ®' ’' ® ® v' ®''' ® ₽'-i
t U,I’
E E (“•». ® ® “r-i0r-i ® Здг ® • •• <
r 4,v
e(E(«.® ®u-i))s(^^® "c
= X(hirg)(krjg) = haka,
and (12.0) is justified.
200
В Prerequisites—representation theory
Thus we have shown the following.
(12.13) Theorem. Let F be a field, let Gj,.... G„ be groups whose direct product is
denoted by G, and let У' = {Ц,..., Kb where ,s an FG,-module for i = 1,..., n.
Then the group Н(У) admits G as a group of operators through the action defined in
(12.0 and (12.1Д
If the field F is finite, say |F| = g, observe that Я(У') is finite; for if Dim(f<) = dh
evidently Dim(K(i,j)) = didi+l...dJ_i, and by summing over diagonal entries with
s = j — i fixed, we obtain:
(12.14) Lemma. If = then |Я(У')| = д<’, where e= £ ( £ dtdi+l ...di+s_i ).
S=1 \ « = 1 /
We now identify certain subgroups of Я(У ) and describe certain of their properties
which will be needed in the sequel; in particular, we show that Я(У') has at least n
distinct decompositions as a semidirect product.
(12.15) Definitions (Subgroups of the Hartley group}. For each integer m e {1,..., n}
let !/„ denote the set
= {(<,j):l <i<m<j<n + 1).
The matrix positions (i, j) which belong to lie above the diagonal and together
form a rectangular block whose bottom left-hand corner contains entries from the
space V(m, m + 1) S Vm. Viewed in another w ay, 5^, consists of precisely those (i, j)
for which Vm appears as a factor in the tensor product V(i, j}. We denote the set of
remaining above-diagonal matrix positions by .'Fm, which is therefore defined as
follows:
= {(>• j): 1 < i < j < n + 1 and either i > m or j < m}.
Next we define two subgroups A„ and Bm of Я(У ') thus:
= {(Ьц) e Я(У'): hy = 0 for all (i, J) e
Bm = {(hy) e Я(У'): hy = 0 for all (i, j) e </„}.
Since = 0, we see at once that An n Bm = {/„+1}. Other, less transparent,
properties of the subsets Am and Bm are listed in the following proposition.
(12.16) Proposition. Let Я = Я(У ) be the Hartley group on У' = {Ц,..., V„}, where
Ft is an FGrmodule for i — 1,..., n. Let 1 < m < n, and let A„ and Bm be the subsets
of H defined in (12.15). Regarding G = Gt x x Gm as an operator-group for H
according to (12.13), we have the following:
12. Group constructions using modules 201
(a) The subset Am is a G-invariant subgroup of H and is G-operator isomorphic with
the following direct sum of modules:
A» = @{HiJ):(iJ)6^,);
in particular, if each Ff is finite, then Am is an elementary abelian p-group for p =
Char(F).
(b) The subset is a G-invariant subgroup of H; it normalizes A„ and is centralized
by the subgroup G„ = {(1,..., 1, </m, 1,..., 1): gm e Gra) of G.
(с) H = A„Bm = AlA2...A„.
Proof. If a matrix h e H has a zero entry in a given position, then it is clear from
Equation 12.q that he also has a zero in that position; thus the subsets Am and are
certainly G-invariant.
To prove that Am is a subgroup, let h, ke Am and turn to Equation 12.<5, which
gives the formula for the (i, j)-entry mg of the product hk. It is clear from the definition
of Am that each term on the right-hand sum in (12.<5) is zero if i > m or j < m. Thus
mg = 0 whenever (i, j) e and therefore like Am. Now suppose that i < m < j, and
consider the (r + l)st term ht i+rki+r j of the sum on the right-hand side of (12.6) for
r — 1,. ,.,j — i — 1. If i + r < m, then hii+r = 0; on the other hand, if i + r > m, we
have ki+r j = 0. Hence, if h, к e Am and (i, j) e //,, then
(12.1)
= hg + kg,
and so the matrix — h = (— hg) is the inverse of h e Am. Therefore A~' A„ s A„ and
we conclude that Am is a subgroup of H. Moreover, the map в: Am -> Dm defined by
specifying that hg is the component of 0(h) in the summand f'(i, j) of Dn is obviously
a G-operator isomorphism, and Part (a) of the proposition is proved.
We now turn our attention to Part (b). Let h, ke Bm and let (i, j) e Then on
the right-hand side of Equation 12.5 we have ki+r 2 = 0 if i + r < m and A,,+, = 0 for
i + r > m by definition of B„. Hence mg = 0, and we conclude that hk e B„. If
H = Я(У ) is finite, which is always the case in subsequent applications, then it follows
at once that H is a group. One is easily led to the same conclusion in the general case
by scrutiny of the Formula 12.t: for calculating inverses.
Next we show that Bm normalizes A„, and to this end let h = (hg) e Am and
k = (kg)eBm. Put A1 = (AJ). We choose a pair (r, s)e^m and calculate the
(r, s)-entry of the matrix k~'hk. First let hk = (mg), and consider Equation 12.5. If
(i, j) e it follows that (i, i + 1) e for I = 1,..., j — i and hence that = 0.
Therefore mg = kg for all (i,j)e^m. Let k'hk = Kt): then (12.5) once more
yields
nrs = m,s + k^,+itnr+ls + ••• + k£s-ims-i.s +
If (r, s) 6 3-m. then (r + I, s) e 3Fm for I = 0, 1.s - r - 1, and we can replace each
mr+, , in this expression by In this case n„ is the (r, s)-entry in the product
A 1A = , and is therefore zero; consequently A 'hk e Am, as claimed. The asser i
702 В. Prerequisites—representation theory
that | B,„, G„] = 1 is a direct consequence of the fact that for (i, j) e ./„ the module
t„ doe"’no" appear as a factor of the tensor product f (f, y), and so each element
g" e Gm acts trivially on each entry of an element of Bm.
"For” e Н(У ), let x0 denote the matrix obtained from x by equating to zero the
//„-entries, and write x, = x - x0. Thus x0 e B„, and from Part (b) we conclude that
x0" 6 B„. Since all the entries of x,, apart from the //„-entries, are zero, it follows
that the ./„-entries of xtXo* are also zero, and therefore xx01 = (x0 + x1)x01 —
/n+i + x,Xo' e A„. Consequently x = (xx0')x0 e B„A„, which coincides with AmBm
by Part (b). Since x was an arbitrary element of H, we have shown that H = AmBm
and, in particular, that A„ < H. To complete the proof of Part (c), let x e H, and, for
1 < r < n, let х,и denote the matrix obtained from x by equating to zero all entries
above the diagonal which lie outside the rth row. A routine calculation then shows
that
x = x'",x,"-1,...x|1,eA„A„_1...A1 = AtA2...A„. □
In the special case when n = 2 and У' = {Lz2, L2}, it is straightforward to verify
that the derived group and centre of Н(У) coincide with the subgroup A, rU2, which
is G-isomorphic with И(1.2) = Vl®V2. Thus we obtain, as a special case of the
Hartley group, the construction of Huppert ([H] VI, 7.22) mentioned above.
1(12.17) Corollary. Let p be a prime, and, for i = 1, 2, let I/ be an IF^Gj-module. Then
the Hartley group H = Н(У') is a p-group of class two with H' = Z(H) = ® V2 and
H/H' = l’i ffi V2, where D denotes the direct product Gj x G2 acting on H as a group
of operators', H has exponent p when p is odd and exponent 4 when p — 2.
I We also record the following, more general, version of (12.17), omitting the proof,
which is straightforward.
(12.18) Theorem. Let H = H(i) denote the Hartley group on У' = {Ц,..., 1^},
where each V: is an FGj-module. Let 3ts denote the set {(i, j):j — i = s} of sth super-
diagonal matrix positions, and lef->7,= S#s. Then the (s + 1 )st term Ks+l(H)
of the descending central series of H consists of all matrices with zero entries in
the JU,-positions; in particular, H is nilpotent of class n. As F(Gt x • • • x G„)-module,
the quotient group K,{H)/K,+l(H) is isomorphic with @{P(i, y): (i, j) 6.<#s} for all
se{l,...,n}.
Remark. If G, — G2 = ••• = G in (12.17) and (12.18), we will regard G as a group of
operators for Н(У ) by identifying G with the diagonal subgroup of the direct product
Gj x G2 x • (see Remark following B, 1.12).
To conclude, we state, without proof, the following theorem of Bryant and Kovacs
which provides a valuable tool for the construction of soluble groups. A full account
of its proof, which involves Lie algebra methods, can be found in Chapter VIII,
Section 13 of Huppert and Blackburn [1].
12. Group constructions using modules
203
(12.19) Theorem (Bryant and Kovacs [1]). Let n>2, let V be a vector space of
dimension n over and let G a subgroup of GL(V). Then there exists a p-group P
such that Р/Ф(Р), regarded as an $f-space, has dimension n and can be identified with
V in such a way that Aut(P) induces on Р/Ф(Р] precisely the subgroup G; in particular,
Aut(P)/CAultf,,(P/0(P)) S G.
Chapter I
Introduction to soluble groups
1. Preparations for the p“^*-theorem of Burnside
Burnside’s celebrated theorem that groups whose orders contain at most two prime
divisors are soluble is the cornerstone of Philip Hall’s characterization of soluble
groups by the existence of Sylow p-complements. Burnside’s original proof, published
in 1904, is both short and elegant but requires certain facts about group characters;
its presentation in the classroom therefore calls for the development of some
representation theory. Our aim here is to give an elementary proof of this theorem,
which should be accessible to a student with some basic algebraic skills but with
little knowledge of group theory beyond the Sylow theorems, the Schur-Zassenhaus
theorem, and elementary properties of groups of prime power order. This purely
group-theoretical approach to Burnside’s theorem has evolved during the past two
decades from ideas of Feit and Thompson, Glauberman, Goldschmidt, Bender,
Matsuyama, among others.
This section contains the preliminary results needed for the proof, which is given
in full in Section 2. These results are cast in the weakest form required for the proof
and are proved by the simplest, if not always the shortest, methods. Of course, many
of them can be more generally formulated and more efficiently and elegantly proved
with more sophisticated tools; some of them even appear elsewhere in this book in
a more general form.
(1.1) Proposition. Let P be a p-group which has at most one subgroup of order p. If
p = 2, assume that P is abelian. Then P is cyclic.
Proof. We proceed by induction on |P|. If | P| < p2, then P is abelian and obviously
therefore cyclic. Next suppose that |P| = p3 and that P is not cyclic. By A, 9.7 the
group P/P' is not cyclic, and so we can find maximal subgroups X and Y (^X) of P
which have order p2 and are therefore cyclic. Writing X = <x>and Y = (y), we have
(x'’) = <y= X r> У, the unique subgroup of P order p and we may clearly suppose
that xpyp — 1 by replacing x with a suitable power. Since P/X r> Y is abelian, [y, x]
has order 1 or p and lies in the centre of P. Thus, appealing to A. 7.3(b) we have
(xy)1’ = ipy'[y, x]'"'’11'2.
Hence, if 2|(p — 1) (that is, if p is odd), or if P is abelian, we have (xy)’’ = 1. Since
xy £ X n У, this contradicts the hypothesis; therefore P is cyclic in this case. Now, to
1. Preparations for the p V theorem of Burnside 205
handle the general case, let N < P, recall from A, 8.3 that N is central of order p and
consider the possibility that P/N has 2 subgroups of order p, R/N and S/N say
Suppose without loss of generality that R/N < Z(P/N). Then RS is a subgroup of P
of order p3 with a unique subgroup of order p and is abelian if p = 2. But then, as
we have seen, RS is cyclic and so has a unique subgroup of order p2. Therefore R = S
a contradiction. Thus P/N satisfies the hypotheses of the proposition and is cyclic by
induction. Let P = (N, x>, and let X = <x>. If N r, X = 1, then P has two subgroups
of order p, namely Л' and Г1,(А'). Therefore N < X and P = NX = X. □
Remark. It is an immediate consequence of this result that a non-cyclic p-group P
of odd order contains a subgroup isomorphic with Zp x Zp; for Z(P) always contains
an element of order p.
(1.2) Lemma. Let P be a non-cyclic elementary abelian p-group which acts on an
elementary abelian q-group Q 1. Then there exists an element x in P, x / 1, such
that C0(x) > 1.
Proof. We suppose, without loss of generality, that |P| = p2. Then P = P(, the
union of its p + 1 subgroups of order p. We suppose that the conclusion is false and
aim to derive a contradiction. Write Q additively and regard it as a ZeP~module in
the usual way. Then if 1 g e P, the statement “x e Q and xg = x” implies that x = 0.
Let 0 у e Q. Then, for 1 he P, we have
( £ wV = E У(дМ = E У9
\geP / geP geP
and hence Eser У9 = °- A similar equation holds with Pf in place of P. Therefore
p+i / \
0 = E У9 = E E У9 I - РУ = ~ру-
geP »=1 \96 P. /
Since у has order it follows that p = q. But then [Q]P is nilpotent, and Q contains
an element of the centre by A, 8.3. This yields the desired contradiction. □
For the next result the Schur-Zassenhaus theorem is needed.
(1.3) Proposition. Let G be a group, and let A be a subgroup of Aut(G)t suchthat
(|A|, I G|) = 1. Assume that either A or G is soluble. Then there exists a Pe Sylp( )
such that P° = P for all a e A.
Remark. Here we regard A as a group of operators for G and according to Definition
A, 4.21 use exponential notation for the action of an automorphism.
Proof. Let H denote the semi-direct product [G]A, and let Po e Syl/G). By the
Frattini argument H = GI\’h(Pq)- Consequently
206
1. Introduction to soluble groups
A S H/G S NH(P0)G/G S Nh(P0)/Ng(P0)-
It follows from the hypotheses that the Schur-Zassenhaus Theorem A, 11.3 applies
to the group NH(P0) and hence that its normal subgroup A'(,(P0) has a complement,
call it L. Clearly L complements G in H, and therefore by A, 11.3 we have A = Le for
some je G. Since L normalizes Po. we conclude that A normalizes P = Pg e Syl„(G).
□
(1.4) Definition. Let Q be a group of operators on a group P. We say that Q stabilizes
the chain of subgroups
(l.«) P = PB>PI - >P„ = 1,
if [Л-i. Q] <^fori = l,...,n.
(1.5) Lemma. If Q stabilizes the chain (La) and if (|P|, |Q|) = 1, then [P, Q] = 1.
Proof. If n = 1, there is nothing to prove. Proceeding by induction on n, we may
suppose that n = 2 because [Pl; Q] = 1 by the induction hypothesis. Let a e P, b e Q,
and let o(b) = m. Then [a, b] e [P, Q] < P,, and so а1' = ac for some c e Pt. It then
follows that a = ak" = acm; hence cm = 1, and therefore c = 1 since (m, |PJ) = 1. We
conclude that [a, b] = 1, as desired. □
(1.6) Corollary, Let Q be a n-group of operators on a n-group P (n £ IP). If
[P, Q, Q] = I, then [P, Q] = 1.
Proof. If [P, Q, Q] = 1, then Q stabilizes the chain P > [P, QJ > 1. □
Another consequence of the Schur-Zassenhaus theorem is the following proposition.
(1.7) Proposition. Let Qbea n'-group of operators on a n-group P(n £ IP), and assume
that either P or Q is soluble. Then P = [P, 2]CP(Q).
Proof. Let G = [P]Q, the semidirect product of P with Q. Then [P, Q] < P and
[P, Q] < <P, Q> = G. Consequently, the subgroup К = [P, Q]Q is a normal sub-
group of G. Since Q is a complement to [P, Q] in К and all complements to [P, Q]
in К are conjugate in K, it follows from the Frattini argument that G = KblG(Q) =
[p,e]wc(e)Thus
p = pn [p, Q]/vc(e> = ел cjawc).
However, since P < G, we have [Nf(Q), Q] < P n Q = 1, and in consequence NP(Q) =
Cr(Q). □
(1.8) Lemma. Let p and q be distinct primes, and let P be a поп-cyclic elementary
abelian p-group. Let Q be an arbitrary q-group which admits P as a group of operators,
1. Preparations for the /'“(/-theorem of Burnside
207
and let t/(Q) denote the following subgroup of Q.
4(& = <.CQ(W): W < P, |P: fV| = p>.
Then Q = tjIQf, in particular, Q = <CQ(x): 1 7xe P>.
Proof. We proceed by induction on |P| + |Q|. First suppose that Q has a non-trivial
proper P-invariant normal subgroup Qo, so that Q/Qo and Qo both admit P as a
group of operators.
Let W < P and let R/Qo = CqiqJ-W). Then [R, И7] < Qo, and therefore R <
Q0Cq(W) by (1.7). Consequently we have ri(Q/Q0) < >1(Q)QO/QO. Since |Q/Q0| < |Q|,
it follows by induction that r;(Q/Q0) = Q/Qo, and therefore that Q = ti(Q)QB. Also
by induction we have Qo = t/(g0) because |g0| < |Q|, and since >/(Q0) < 4(Q). we
conclude that Q = g(Q) in this case.
Therefore suppose that Q is P-simple. Then, in particular, Q is elementary abelian,
and by (1.2) there exists a non-trivial element x e P such that Ce(x) > 1. Since P is
abelian, Ce(x) is a P-invariant subgroup of Q, and therefore by supposition CQ(x) =
Q. Hence we may regard P* = P/<x> as a group of operators for Q. If P* is cyclic,
then |P: <x>| = p, and Q = Ce((x>) < >/(Q). On the other hand, if P* is not cyclic,
by induction we have
Q < <Ce(lP*): IP* : ^*1 = P> < <.CQ(Wf. |P: IV| = p> = 4(Q). □
(1.9) Lemma. Let P be a p-subgroup of G and N a normal p'-subgroup of G. Then
NC(P)N/N = Ngin(PN/N).
Proof. It is clearly the case that NG(P)N/N < Ngin(PN/N). Let U/N — NGiri(PN/N).
Then PN < U and P e Syl„(PlV). By Sylow’s theorem and the Frattini argument we
have
U = NV(P)PN = NV(P}N.
But 1VG(P)W < U, and therefore U = NG(P)N.
□
(1.10) Proposition (Thompson). Let P and X be p-groups,Q a p'-group. and let P x Q
be a group of operators for X. If [Q, C\(P)] = 1. then [Q, X] = 1.
Proof. Suppose that (<?, X] I, and let No be a minimal element in the set of
P x Q-invariant subgroups A of X which satisfy [Q, IV] 1. Since No and P are
p-groups, we have [No, P] < by A, 8.3, and therefore [Л/о» P’ ffl “ ecause
|Л'О, P] is clearly P x ^-invariant. But [P, Q. 1VOJ = П- ^oJ = *’ a”d 1^-,°^
[Q, An P] = 1 by the Three Subgroups Lemma A, 7.6. Hence [A'fi, QJ -1.У- % J -
CX(P), and so by hypothesis [No, Q, Q] = L But then [IV0, Q] = 1 by (1.6), and this
contradicts the choice of No- Therefore our initial supposition is false.
208
I. Introduction to soluble groups
(1.11) Proposition. Let G be a soluble group and P a p-subgroup of G. Denote CG(P),
JVG(P) by C and N respectively. Then Op.(C) and OP(N) are contained in OP(G).
Proof. Since Op (C) char С < IV, we have OP (C) < OP (N). It will therefore suffice to
prove that OP (N) < Op.(G), and this we do by induction on |G|. Set К = Op(G), and
first suppose that К 1. Then induction yields Op(NGIK(PK/K)) < Op.(G/K) = 1.
Since Ng/k(PK/K) = NK/K by (1.9), we then obtain Op.(N)K/K < Op.(NK/K) = 1,
and hence OP (N) < K, as required.
We may therefore suppose that Op (G) = 1 and hence that Op(G) = F(G). Set
X = O,,(G) and Q = OP-(N), and observe that CX(P) < N < NG(g). It follows that
[g, GX(P)] <QoX=l, and since [P, g] < P r> Q = 1, we can apply (1.10) to
deduce that [g, A'J = 1. Consequently Q < CG(X) = Cc(F(G)) < F(G) by A, 10.6(a),
and since F(G) is a p-group, we conclude that Q = 1 (= Op.(G)\ □
(1.12) Lemma. Let p and q be distinct odd primes; let P be a p-subgroup and Q a
q-subgroup of the group G = GL(2, q). If Q normalizes P, then Q centralizes P.
Proof. Assume that Q < NG(P). If P = 1, there is nothing to prove. Therefore
suppose that P 1, and let Po = [P, g], a Pg-invariant subgroup of P. Clearly
Po < [G, G] < SL(2, q). Let V = F(2, q) be the natural module for GL(2, q), and first
suppose that there exist elements x in Po and t> in V such that vx = v 0. With respect
to a basis {c, w} of Ц this element x is represented by a matrix of the form
and since Det (A") = 1. we have b = 1 and therefore Xе = 1. Consequently x = 1
because Po is a (/'-group. From (1.2) we conclude that Po contains no non-cyclic
elementary abelian subgroup and hence from (1.1) that Po is cyclic. Since p divides
|GL(2, <?)| = q(q - l)2(q + 1) and q 2, it follows that p < q. Let |P0| = p“, a> 1;
then |Aut(P0)| = p“-1(P — 1) by A, 21.1(b), which is therefore not divisible by q. But
each element of Q induces by conjugation on Po an automorphism of (/-power order.
Hence we conclude that 1 = [Po, g] = [P, g, g] and then from (1.6) that [P, g] = 1.
□
(1.13) Lemma. Let q be an odd prime, V an elementary abelian q-group, and H a soluble
group of automorphisms of V. Assume that |H| is odd and that Oq(H) = 1. If h is a
q-element of H with |F: Cl,(h)| < q, then h = 1.
Proof. Without loss of generality we may suppose that о(й) = q and that | V: Cr(h)| =
q. Let H be a counterexample of minimal order, and set g = <h>. The Fitting
subgroup of H is a (/'-group because O,(H) = 1 and contains its centralizer by A, 10.6.
Therefore [ОДН), g] 1 for some prime p q. Let P = Op(H). Since g f PQ
and |g| = q, it follows that O,(Pg) = 1. Hence H = Pg, by the choice of H. Let
x e H\NH(Q). Since g and g* are distinct Sylow (/-subgroups of their join, we
have g f <g, gx>. As before, 0,(<g, g*>) = 1, and again the choice of H implies
that H = <g, g’>. Let W = F/Cr(H). Because | V: Cr(g*)| = | V: Cx(g)| = q
1. Preparations for the p^-theorem of Burnside
209
and Q.f/f) -G (0r, С,,(<2Л), we have | fF| < q1. From (1.5) it is clear that
P nKer(H on И') = 1, and therefore |Ker(71 on И')| divides |77:P| = q; hence
Ker(H on ll j < O,(H) = 1. Then H is faithfully represented as a group of
automorphisms of W and may evidently be viewed as a subgroup of GL(2 q)
Consequently from (1.12) we have [P, Q] = 1 and Q < PQ = H, a contradiction.
Therefore no counter example exists. q
(1.14) Theorem (Baer [2]). Let x be a p-element of a group G. Then the following
statements are equivalent:
(a) xeOp(G);
(b) <xh, xe > is a p-group for all h, g e G.
Proof (Alperin, Lyons [1]) (a) => (b): This is clear from the fact that <x‘„ x9> < 0 (G)
since O„(G) < G. "
(b) => (a): Let G be a counterexample and let C = {№}, the conjugacy class
containing x. Let P e Sylp(G). If C £ P, then <C> is a normal p-subgroup of G; it
follows that x e <C> < Op(G) and G is not a counterexample. Therefore C\P is
non-empty, and consequently (C n g)\(C r> P) is non-empty for some Q e Sylp(G). It
follows that the set
Л = {(P. 6): Лбе Sylp(G) and C n P * C n Q]
is also non-empty. Choose a pair (P, Q) e . К with |C r> P n as large as possible,
and let D = <C n P r> Q). The inner automorphism of G which maps P to Q sends
CnP to Cr-,ft and so |CnP| = |Cng|. Since CnP^Cnft it follows that
Cc.PfQ and hence that C r> P £ D By A, 8.3 we can find a series
D = Po <1 P, <□ — <1 P„ = P,
where | Л/Л-i | = p for i = 1,..., n. Since C n P„ D and C r> Po £ D, there exists a
smallest i for which C n P, $t D. Let и e (C n Pf)\O. Since the element и normalizes
C r> Pi_1 = C n D, it also normalizes <C n D> = D. Similarly there exists an element
v e (C n Q n Ng(D))\D. Because <u. t>> is a p-group by hypothesis, so also is <u, t >D;
therefore let R be a Sylow p-subgroup of G containing <u, t’>£>. Then CnPmR
Э CciPn <u, t>>£> 2(Cci£>)cj {u}. and so we have |CnPnR| > |Cm£>| >
\CcyPcyQ\. Hence (P, R) £ and so C n P = C n R. A similar argument shows
that C n Q = C n R. But then C n P = C n Q, contrary to the choice of (P, Q) e Ж
Therefore no counterexample exists. □
A different proof of (1.14) can be found in A, 14.11.
(1.15) Theorem (Baer-Suzuki [1]). Let t be an involution of a group G. If t ф O2(G),
there exists a 2'-element h / 1 of G such that h1 — h .
Proof. By (1.14) there is a g e G such that <t, t’> is not a 2-group. Since
r'ftt^t = tet = (tT1!-1 = («’Г1,
210
I. Introduction to soluble groups
the element t inverts by conjugation each element of <tt9>. Were <rr9> a 2-group, it
would follow that <t, r9> = <r, tt9> = <!><«"> were a 2-group. Therefore <H9>
contains a non-identity element h of odd order inverted by t. □
2. The proof of Burnside’s /^‘-theorem
Theorem (Burnside [1]). Let G be a group of order p“q'\ where p and q are primes.
Then G is soluble.
We are indebted to H. Bender who communicated to us the outline of the proof
that now follows:
Proof. Suppose that the theorem is false, and let G be a counterexample of minimal
order. The structure of G is carefully analysed, and eventually a contradiction is
reached. For ease of reading we break the argument into separately-stated steps.
(2.1) G is a non-abelian simple group all of whose proper subgroups are soluble; in
particular, p q, and the integers a and b are both positive.
If 1 N <3G, the choice of G forces the solubility of N and G/N; but then G itself
is soluble by A. 10.2(b). Hence G is simple, not abelian, and therefore, in particular,
not of prime power order.
Notational Remark. We shall assume henceforth that p < q. In order to present
without bias those results which are symmetrical in p and q, throughout the proof
{r, s) will denote the unordered pair {p, q}.
(2.2) If G has subgroups A and В such that G = AB and A / G, then В normalizes
no non-trivial subgroup of A.
If 1 * H < A and В < Ng(H), then 1 <HG> = <Нвл> = (HA> < A < G, and G
is not simple.
(2.3) If Re Sylr(G), then R normalizes no non-trivial s-subgroups of G.
To see this, simply apply (2.2) with В = R and A any Sylow s-subgroup of G.
(2.4) If S e Syls(G) and 1 Y < R e Sylr(G), then G = (S, Y>.
By A, 8.3 the group YnZ(R) contains an element z 1. By (2.1) we have R <
G;(z) < G. Let A — <S, z) and В = CG(z). Then G = SR <, AB. Since В normalizes
the non-trivial subgroup <z> of A, by (2.2) we have A = G. Hence <S, T> = G.
2. The proof of Burnside’s p V-theorem
211
(2.5) Let M and H be maximal subgroups of G. Assume that M has non-trivial normal
r- and s-subgroups R and S respectively such that R x S < H. Then
(a) R x S <; F(H) < M, and furthermore
(b) M = H.
Since G is simple and M is maximal, we have NG(R) = M = NC(S). Therefore S <
CH(R), and since H is soluble, we can apply (1.11) to conclude that S < 0,.(H) =
OS(H). Similarly R < and therefore 1 * Or(H) < CG(S) < M. Likewise 1 #
S S OfH] < M. Therefore (a) holds. We can now reverse the roles of M and H in the
above argument, replacing R and S by Or(H] and 0s(H) respectively, and conclude
that the counterpart of (a) holds, namely that F(H) = Or(H} x O,(H) < F(M) < H.
We can now apply the initial argument once more, with M and H in their original
roles but with Or(M} and OS(M) in place of R and S, and conclude this time that
F(M) = Or(M) x O,(M) < F(H}. Thus F(AT) — F(H), and consequently we have
M = Ng(F(M)) = Ng(F(H)) = H.
(2.6) If M <• G, then F(M) has prime power order.
We suppose that Or(M) 1 OS(M) and derive a contradiction. Let Rc = Z(Or(M}}
and Se = Z(OS(M)). Since Re and Se are non-trivial normal subgroups of M. Assertion
(2.5)(b) applies, and, in particular, CG(x) < M for any л e F(M), x 1.
We assert that RB is cyclic. If this is not the case, we can find an elementary abelian
subgroup T of RB of order r2 by (1.1). Let R e Sylr(M). Since R normalizes Se 1,
the step (2.3) implies that R ф Sylr(G), and so by A, 8.3(c) there exists an element g in
Ng(R)\M. Then we have
T < R = R’ < M9 * M,
whence T normalizes the normal subgroup S = Sg of M9. By (1.8) we have
S = <Cs(x): x e T> < <CG(x): x e T> < M.
Hence M contains R90 x S90 < M9, and therefore M < M9 by (2.5). This contradiction
proves that Re is cyclic. Similarly Sc is cyclic.
It follows that the non-identity group Po = Z(OP(M)) is cyclic. Let |Pe| = pc and
let Q e Syl,(M). Since the order of the «/-group of automorphisms induced by Q on
Po divides I Aut(P0)|, which equals pc'(p — D, it follows that Q centralizes Po because
p < q by assumption. Thus Po x Qo < NG[Q), where 1 # Qo = Z(O,(M)), and there-
fore Ng(Q) < M by (2.5). But then Q e Syl,(G), which is impossible by (2.3) because
Q normalizes Po. We conclude from this contradiction that either 0„(M) = 1 or
0,(M) = 1.
(2.7) Definition. Let t be a prime and E a finite group. A r-subgroup U of £ is called
locally central if U < Z(T) for some Те Syl,(£). (We make this definition for the
convenience of this section only; it is not in general use and is not used elsewhere in
this book).
212
I. Introduction to soluble groups
(2.8) (Matsuyama [1]). If a maximal subgroup M of G contains a non-identity locally
central r-subgroup Y of G, then F(M) is an r-group.
Suppose that, on the contrary, /(M) is an s-group, which by (2.6) is the only
alternative. Choose a Sylow s-subgroup of G containing F(M), and let Z denote its
centre. Since AG(F(M)) = M, we have Z < Cc(F(M)) < CM(F(Mf) < F(M) by A, 10.6
on account of the solubility of M. Let
L = <Z’ : у e У>.
Then L <, F(M). and so L is an s-group normalized by У. Let. tt denote the set of all
s-subgroups of G which are
(i) normalized by У, and
(ii) generated by G-conjugates of Z.
Then 1 Vs L e. //. Let К be a maximal element of . // containing L. and let S be a
Sylow s-subgroup of G containing K. By (2.4) we have G = <S, У), and by supposi-
tion Y normalizes К; therefore S does not normalize K, and we can find an element
x e NS(NS(K))\NS(K) by A, 8.3(c).
Then К K’< NS[K). Since Kx is generated by certain conjugates of Z, one of
them, Z9 say, is not contained in K. Since Y is locally central, we evidently have
G = Cc(Z)Cc(T); writing g = uv with u e Cc(Z) and v e Сс(У), we conclude that
Z9 = Z" K. Set L* = (Z"r: у e Y), and note that
L* = <Z9": у e У> = <fZy\ у e У>" = L".
Thus L* is an s-group normalized by У, and because У, Z" s /VG( K), it follows that
L* normalizes K. Therefore KL* is an s-group, and it is clearly normalized by У and
generated by conjugates of Z. But Z" f K, and so KL* is an element of.// strictly
containing K. This contradicts the choice of К and shows that F(M) is, like У, an
r-group.
(2.9) A locally central r-subgroup Y^lofG normalizes no non-trivial s-subgroup of G-
If S / 1 is an s-subgroup of G normalized by У, then a maximal subgroup M of G
containing /VC(S) contains У, as well as the centre Z of a Sylow s-subgroup of G
containing S. Now Z is certainly locally central, and therefore by (2.8) the Fitting
subgroup of M is at the same time an r-group and an s-group and is hence trivial.
This contradicts the solubility of M.
(2.10) G has odd order.
Certainly O2(G} = 1. If |G| is even, the centre of a Sylow 2-subgroup of G contains
an involution, which by (1.15) normalizes some non-trivial subgroup of odd order,
and this clearly contradicts (2.9).
(2.11) Let Rc be a non-trivial r-subgroup of G, and let CG(R0) < L <G. Further, let
< R] < R2 with Rr e Sylr(L) and R2 e Syl,(G). Then we have
2. The proof of Burnside’s pV-theorem
213
(a) F(L) = O,(L),
(b) fl(Z(R 2)) < fl(Z(O,(L))), and
(c) Cc(S2(Z(Or(L)))) is an r-group.
Since 1 J Z(R2) < Cc(Rc) < L and since Z(P2) is a locally central r-subgroup of G
by (2.9) we have OS(L) = 1. Hence F(L) = 0,(L). Since 0r(L} < R. < R, we have
Z(R2) < Cl(F(L)) < F(L) by A, 10.6 and (2.1).
Therefore Z(R2) < Z(Or(L)), and Conclusion (b) follows. Finally, let Se
Syls(Cc(Q(Z(Or(L))))). From (b) we deduce that [S, Q(Z(R2))] = 1. Since Q(Z(R2)) is
a locally-central r-group, we have S = 1 by (2.9), and the proof of (2.11) is complete.
(2.12) If Po is a non-trivial p-subgroup of G. then NG(Pe) has a cyclic Sylow q-subgroup.
(We recall that p < q.)
Suppose that, on the contrary, G has a p-subgroup Po # 1 whose normalizer has a
non-cyclic Sylow q-subgroup, and let V = fl(Z(P0)). Since the subgroup AC(F) con-
tains NqIPq), by (1.5) it contains an elementary abelian subgroup of order q2. Thus
the set .4' of ordered pairs (А, F) satisfying
(i) A is elementary abelian of order q2, and
(ii) К is a maximal А-invariant elementary abelian p-subgroup of G and F vM,
is non-empty, and from this we shall derive a contradiction. Let (A, F) be a pair in
J/' with |Cr(A)| as large as possible. First we assert that
(2.cr) F > CV(A).
Let Y = fi(Z(Op(Wc(F)))), and apply (2.11) with F. WG( F), and p in the roles of Ro, L,
and r respectively to conclude that Cc(T) is a p-group. However, Y is an A-invariant
elementary abelian p-subgroup of the centre of OpWG(F)), which obviously contains
F, and consequently VY is А-invariant and elementary abelian. Therefore Y < F by
the choice of F in the definition of .4', and it follows that Cc(F) is contained in Cc(F).
Thus Cg(F) is a p-group, and Assertion (2.a) is justified.
By (1.8) we have F = <Cr(x): 1 ± x e A>. and so there exists an element x e A\{1]
such that the subgroup U = Cr(x) properly contains C, (A). Since A centralizes x, the
subgroup U is А-invariant and is obviously not centralized by A. Therefore by (1.11)
the subgroup A induces on V /CV(A) a non-trivial q-group of automorphisms, and
because p < q and | Aut(Zp)| = p — 1. we conclude that
(2.0) |l//Cr(A)|>p2.
Let Z, = n(Z(Oe(Cc(xj))), and note that 1 x e Z,. By a further appeal to (2.11 )(c)
(this time with <x>, CG(x), and q in place of Rc, L, and r), we see that Q.fZJ is; a
q-group. Since 1 J U < CG(x), it follows that Z, is normalized but not centralized by
V; therefore the group of automorphisms induced by U on the section Z,/<x> is
non-trivial by (1.6). By (1.8) there is a subgroup W of index p in U which centralizes
a non-trivial subgroup Z2/<x> of Z,/<x>, and W centralizes Z2 by (1.6) once more.
Since Zj is elementary abelian, it follows that Z2 contains an e ementary a tan
214 I. Introduction to soluble groups
subgroup, A, say, of order q2, and because Аг centralizes W, we can find a maximal
Ai -invariant elementary abelian p-subgroup, F, say, containing W. Then evidently
(Л„ f'JeT, and IV <, Cj.fAJ. But by (2.J?) we have |1V| = |L'|/p > |Cr(A)|, which
contradicts the choice of the pair (А, V). Therefore (2.12) holds.
(2.13) Let M be a maximal subgroup of G, and assume that F(M) is an r-group. Then
M/F(M) has a cyclic Sylow r-subgroup.
Let R = F(M), and let SR/R = F(M/R) = OS(M/R) with S e Syl,(SR). By the Frattini
argument M = RS/VM(S) = RNM(S). If S = 1, then M = F(M) and (2.13) is certainly
true. Therefore suppose that S / 1. If s = p, by (2.12) the group /VG(S) has a cyclic
Sylow ^-subgroup; hence M/F(M) = RNM(S)/R S hlM(S}/NR(S) also has a cyclic
Sylow y-subgroup, the desired result since r = q. On the other hand, if s = q, then S
is cyclic; this is because S is contained in a Sylow q-subgroup of M and this is cyclic
by (2.12) since M = /VC(R). Now consider the soluble group M* = M/F(M). Its
Fitting subgroup is isomorphic with S and is self-centralizing by A, 10.6. Therefore
M*/F(M*} is isomorphic with a subgroup of Aut(S), which is cyclic by A, 21.1(b)
because q 2. Hence a Sylow r-subgroup of M*, which in this case is a Sylow
p-subgroup, is also cyclic. Thus (2.13) holds.
(2.14) Definition. Let R be an r-group, and define the subgroup J0(R) to be the
subgroup of R generated by all its elementary abelian subgroups of maximal order.
Clearly Je(R) is a characteristic subgroup of R, and if JC(R) < U < R, then JC(R) =
J0(U). (The Thompson subgroup J(R) is similarly defined by omitting the word
‘elementary’ from this definition. Subgroups of this kind play an important part in
the theory of insoluble groups.)
(2.15) Let M be a maximal subgroup of G, and assume that F(M) is an r-group. If
R e Sylr(M), then M = AC(JO(R)) and R e Sylr(G).
Let К = F(M). We suppose that JC(R) f, К and obtain a contradiction. Then among
the elementary abelian subgroups of R of maximal order, there exists one not
contained in K; call it A. By (2.13) the quotient AK/K is cyclic, and therefore
I A: A n К | = r. Let V — fl(Z(K')), and observe that (A n K)V is an elementary abelian
subgroup of R. Hence
|A| > |(/lr>K)F| = |Л r>K||F|/|.4 r> V\ = И||И|/г|Лг> F|.
It follows that
(2-y)
IV: A c F| 2 r.
Since 1 V char К <> M and M < G, we have M — NG(F). Therefore CG(F) < M,
and from (2.11 )(c) (taking V for Rc and M for L) we conclude that CG( F)isan r-group;
in fact, it is a normal r-subgroup of M and it therefore coincides with K. Let
H = M/K = M/CM(V).
2. The proof of Burnside’s pV-theorem 215
The group M acts by conjugation on V and induces a group of automorphisms
isomorphic with H. Moreover, since Or(M) = K, we have O(H)=I Let
a e A \(A n K) and h = aK. Then A r, V < Q(h), and so | F: Cr(h)| < r by (2 y)’ Now
the hypotheses of (1.13) are satisfied (with r instead of q), and therefore h = 1; in other
words, a e K. This contradiction shows that JG(R) < K.
Therefore JG(R) = JG(K) char К <3 M and it follows that M =- NG(J0(R)). If R <
R* e Sylr(G), we have J0(R) char R < N„.(R); but this cannot be the case because
NK-(R) < /VC(JC(R)) = M and R e Syl,(M). Therefore R e Sylr(G), as claimed.
(2.16) Let M be a maximal subgroup of G, and assume that F(M) is an r-group. If
g e G\M, then M r> M9 is an s-group.
Suppose this is not so, and choose age G\M so that a Sylow r-subgroup R of
has largest possible order. If R e Sylr(M), then R e Syl,(M9), and by (2.15)
we have M = /VC(JC(R)) — M9, a contradiction. Therefore 1 < R < R, for some R, e
SylJM), and hence R < NRt(R}. Let H be a maximal subgroup of G containing
NC(R). By (2.11 )(a) the group F(Я) is an r-group, and so it follows from (2.15) that
H = /VC(JO(R2)) for some R2 e Sylr(G). Also by (2.15) we have R, e Syl,(G), and it
follows that JC(R2) is conjugate to J0(R,) in G by Sylow’s theorem and the fact that
JC(X) char X; consequently H is conjugate to /Vg(Jc(Ri)) = M. Since R < NK](R) <
H r> M, the choice of R and M9 forces H = M. An identical argument applied to R
viewed as a subgroup of M9 yields H = M9, and again we have the contradiction
M — M9. Therefore (2.16) must hold.
(2.17) The conclusion of the proof of Burnside's theorem.
Let r be the prime for which a Sylow r-subgroup R of G has larger order than a Sylow
s-subgroup S, and let M be a maximal subgroup of G containing CG(Z(R)). From
(2.11 )(a) we see that F(M) is an r-group.
Ifge G\M, by (2.16) we have RnR9 = 1. Therefore RR9 is a subset of G containing
|R||R9| = |R|2 elements. However |R|2 > |R||S| = |G), and this final contradiction
completes the proof. О
Concluding Remarks. The group-theoretical proof just given yields less information
than Burnside’s original proof using character theory. There he proves the following.
(2.18) Theorem (Burnside). Let Gbea finite group containing a conjugacy class whose
cardinal is a prime power. Then G is not simple.
If |G| = p,'q1', the size of a conjugacy class containing a locally-central element is
clearly a prime power. Since G is not simple, it follows at once by induction on |G|
that G is soluble.
216
I. Introduction to soluble groups
3. Hall subgroups
(3.1) Definitions, (a) Let л be a set of primes, and recall that a л-number is an integer
whose prime divisors all belong to л. A subgroup H of a group G is called a Hall
n-subgroup if |H| is a л-number and |G : H| is a л'-number, where л' = Р\л. The
(possibly empty) set of Hall л-subgroups of G will be denoted by Hall„(G).
(b) A subgroup H of G is called a Hall subgroup if it is a Hall л-subgroup for some
л s P. Evidently H is a Hall subgroup of G if and only if (|G: H|, | H |) = 1.
Remarks. (Recall that cr(G) = {peP: p| | G|}.)
(1) Hall„(G) = НаЩ^сДС). Thus 1 is a Hall л-subgroup of G if and only if
л r> <r(G) = 0.
(2) A Hall л-subgroup is a maximal л-subgroup.
(3) If p e P, then HallIp((G) = Sylp(G).
(4) If p e P, P e Sylp(G) and H e Hall^fG), then G = HP and H n P = 1. For this
reason, Hall p'-subgroups are sometimes called Sylow p-complements.
As with Sylow subgroups, the following properties of Hall subgroups are straight-
forward to verify.
(3.2) Lemma. Let H e Hall„(G), and let M, G. Then
(a) He e Hall„(G) for all ge G,
(b) HN/N e Hall„(G/A),
(с) Hr.Ne Hall„(/V), and
(d) (H N}(H r,M) = Hr. MN e Hall,(MA).
Whereas by Sylow’s theorem a finite group has Sylow p-subgroups for each prime
p, the existence of Hall л-subgroups is not in general guaranteed; for example, it is
easy to see that the alternating group Alt(5) has no Hall {3, 5}-subgroup. Indeed, our
main aim in this section is to prove a fundamental theorem of Philip Hall which states
that Hall л-subgroups exist in G for each л S P if and only if G is soluble.
(3.3) Theorem (P. Hall, [1]). Let G be a soluble group and л a set of primes. Then
(a) Hall it-subgroups of G exist,
(b) they form a conjugacy class of G, and
(c) each л-subgroup ofG is contained in a Hall it-subgroup.
(A standard terminology has been widely adopted to denote the properties described
in Statements (a), (b) and (c) of this theorem. A finite group G is said to have Property
E„ if Hall„(G) is non-empty. Property C„ if Hall„(G) is a (non-empty) conjugacy class
of subgroups, and Property Dn if it has C„ and every л-subgroup is contained in some
Hall л-subgroup of G.)
Proof. We shall prove each of the three conclusions in turn by induction on |G|. If
G = 1, all conclusions clearly hold. Therefore suppose that G 1 and let N be a
3. Hall subgroup?
217
minimal normal subgroup of G; by A, 10.5 (a) the subgroup N is a p-group for some
prime p.
Conclusion (a): By induction G/N has a Hall л-subgroup H/N If pen then H is
already a Hall л-subgroup of G. If p £ л, then by the Schur-Zassenhaus theorem N
has a complement, U say, in H. Since |l/| = |H/N| and |G: l/| = |G:H||H: l/| =
|G : H| I TV I, it follows that U is a Hall л-subgroup of G.
Conclusion (b): Let Hl, H2 e Hall„(G). By (3.2) the group H.-N/N is a Hall л-subgroup
of G/N, i = 1, 2, and therefore by induction HlN = He2N for some geG. If p e л, we
have H, = HtN = H%N = , as desired. On the other hand, if p £ л, then H, and
H2 are complements to N in Ht N, and therefore, since N is abelian, Ht is conjugate
to He2 by the final statement of A, 11.3. This gives Conclusion (b).
Conclusion (c): Let U be a л-subgroup of G. It is a consequence of Conclusion (b)
and (3.2)(a) that every Hall л-subgroup of G/N has the form HN/N for some
H e Hall„(G). Therefore by induction UN/N < HN/N, and ifp e л, we conclude that
U < HN = H, as required. If p f л, then U nN = H nN = 1, and if we set V =
H r> UN, we have VN = HN nUN = UN. Thus U and V are Hall л-subgroups of
UN, and so U = Vх for some x e UN by (b) above. Therefore U S Hx e Hall„(G).
□
(3.4) Theorem (Wielandt, [5]). If a group G has three soluble subgroups Ht, H2 and
H3 whose indices \G: H^^G: H2\,\G: H3\ are pairwise coprime, then G is itself soluble.
Proof. By A, 1.6(b) we have G = H, H2 = H, H, = and we may certainly sup-
pose that # 1. Let N be a minimal normal subgroup of H,, and recall that N is
a p-group by A, 10.5. Because |G:H2| is coprime with |G:H3|, we can suppose
without loss of generality that pf|G:H2|. Let D = Hlr\H2. Since the
product ND is a subgroup of Ht, and the p-power |N: N n£>| = |ND:£>| divides
|Я, :D| = [HtH2 :H2\ = |G : H2|. Therefore |N : N nDi = 1 and N < D.
Let К = <№ > < G. Then К = (N1"^: Ь,еН;У = <Nh‘ :h2eH2>< H2, and con-
sequently K, as a subgroup of a soluble group, is soluble. The subgroups {H(K/K: i =
1, 2, 3} are soluble and have pairwise coprime indices in G/K. Since К # 1, we may
suppose by induction on the group order that G/K is soluble. Therefore G is soluble
by A, 10.2(b). 1=1
We can now prove that the existence of Hall л-subgroups (for all л S P) is a
sufficient condition for the solubility of a group. Hall’s original proof of this fact
appeared in 1937, some 9 years after he had published a proof of the necessity of the
condition.
(3.5) Theorem (P. Hall, [2]). Let G be agroup. and let | G| = JB=i p/\wl*Lep'\°'
are distinct primes. Assume that G possesses subgroups St. . , S, sue it lot | . , P, .
i = 1,..., r; in other words, that G has Sylow p-complements for all peP. 1 hen U
soluble.
218
I. Introduction to soluble groups
Proof. We prove this statement by induction on r = |<r(G)|. If r < 2, G is soluble by
Burnside’s p^-theorem (see Section 2 of this chapter). If r > 3, the subgroups S,, S2
and S, have pairwise coprime indices. Let 1 < i j < r, and let Tu = Sf r> Sy. By
A, 1.6(b) we have G = StSj, and therefore pf = |G : S?| = |S(: TJJ. Thus Tiy is a Sylow
p-complement of S„ and so S, fulfills the hypotheses of the theorem. Since |<r(S()| =
r — 1, by induction S, is soluble. In particular, .S\, S2 and S3 are soluble, and hence
G is soluble by (3.4). □
The following fundamental theorem follows directly from (3.3)(a) and (3.5).
(3.6) Theorem. A finite group is soluble if and only if it possesses Hall n-subgroups
for all sets n of primes.
(3.7) Concluding Remarks, (a) Theorem 3.5 is clearly a generalization of Burnside’s
theorem of Section 2. If |G| = p‘‘qb, then G is the product of two permutable nilpotent
groups, namely G = PQ with P e Sylp(G) and Q e Syl,(G). Burnside’s theorem is also
capable of generalization from this viewpoint, as the following theorem of Wielandt
and Kegel shows'.
If G is a product of pairwise-permutable nilpotent subgroups (in other words, if
G — Gt ...G„ with G; nilpotent and G^G, = G2Gj for all 1 < i, j <, n), then G is soluble.
We shall not use this result in the sequel and we offer no proof. The original
references are Wielandt [2] and Kegel [1]. and a complete proof may be found in
Huppert [5], Chapter VI, Section 4 (see Hauptsatz 4.3).
(b) The following special case of the Wielandt-Kegel theorem is due to Ito [2] and
is even correct without any finiteness assumptions:
If G = AB, the product of permutable abelian subgroups A and B, then G" = 1.
This suggests the following
Open question. Let G = MN, a permutable product of nilpotent groups M and N,
and let m and n denote the nilpotency class of M and N respectively. Let d be the
derived length of G. Does there exist a function f : N x N -» N such that
d < f(m, и)?
In particular, is it true that d < m + nl
Gross [1] and Pennington [1] have shown independently that it is sufficient to
find a bounding function / in the case where G is a p-group. Explicitly, it has been
shown that
(i) if n = o(M)r\o(N), then G<m+'” < <ti(G)r.O,(G) (in particular, G,m+"' is a nil-
potent л-subgroup, and
(ii) the Fitting subgroup F(G) is the permutable product (M r> F(G))(A r> F(G)).
(c) In the context of Theorem 3.5 we should also mention the following theorem
of Arad and Ward [1]:
3. Hall subgroups
219
. G ar°Uf’ Td aSSUme that G haS HaU «b^oups for all pairS
{p,q} £ IP- Then G is soluble.
Their proof of this theorem makes use of the classification of finite simple groups.
(d) The existence of Hall л-subgroups does not ensure their conjugacy; indeed a
group may have non-isomorphic Hall л-subgroups for the same set л (see Exercise
6 below). In his fundamental work on properties of Hall subgroups in arbitrary finite
groups, Hall [6] suggests that the Property E„ might imply the Property D„ when л
is a set of odd primes, but in [3] Gross shows that this is very far from being'true: for
every pair of odd primes {r, s) he constructs a general linear group GL(»i, q) over a
finite field of a suitably chosen characteristic p g {r, s) which, on the one hand, has
a Hall {r, s}-subgroup H and, on the other hand, has an {r, s)-subgroup that is not
isomorphic with any subgroup of H. A concrete counterexample is obtained by
observing with Gross [5] that SL(3, 34) has a Hall {3, 5}-subgroup H of order 312 • 52
which has a normal Sylow 3-subgroup; however, the normalizer of a Sylow 5-
subgroup contains a Frobenius group of order 3 52, which certainly has no normal
Sylow 3-subgroup and cannot therefore be isomorphic with any subgroup of H.
Nevertheless, Hall’s intuition that better behaviour could be expected for sets of odd
primes is indicated by another theorem of Gross:
Let it be a set of odd primes, and let G be a finite group. If the set Hall„(G) is
non-empty, then it is a conjugacy class of subgroups of G.
In other words, £„ implies C„ when 2 л. Gross’s proof of this theorem, begun [5]
and completed in [6], is made to depend on an interesting property of finite simple
groups, which can be deduced from their classification, namely the property that
every odd order Hall subgroup H of a finite simple group G has a Sylow tower with
respect to an ordering of the primes in <r(G) which depends only on G and not on H.
We bring this brief discussion to a close by stating an interesting sufficient condi-
tion for a group to satisfy D„. This is the following theorem of Wielandt [3]:
Let G be a finite group with a nilpotent Hall it-subgroup H (where л is an arbitrary
set of primes). If U is a it-subgroup of G, then U < Hs * for some ge G.
Thus, if G has Property E„ and if Hall„(G) £ 91, then G has Property D„.
Exercises
1. Verify that Alt(5) has Sylow p-complements if and only if p # 2.
2. Verify that GL(3, 2) has Sylow p-complements if and only if p / 3.
3. Find an insoluble group G whose maximal subgroups all have prime power index
and which contains a maximal subgroup of p-power index for each prime p
4. (Hall [6]' Theorem A 4) Let p and q be primes and n a natural number such that
p < q < n. Then Sym(n) has a Hall {p, (/[-subgroup if and only if p - 2, q - 3 an
n e {3, 4, 5, 7, 8). Deduce that these are the only Hall subgroups of symme nc
220
I. Introduction to soluble groups
groups which are soluble and do not have prime power order. (Thompson [1] has
proved that if H is an insoluble Hall subgroup of Sym(n), then either n > 5 and
H = Sym(n) or n is a prime >7 and H = Sym(n - 1).)
5. Show that for all ne N there exist soluble groups G, and G2 such that
(i) k(Gj)| = |<r(G2)| = n,
(ii) for all it £ P with |it| < n - 1 and for Hi e Hall„(G,), it follows that Ht s
H2, and
(iii) Gj 7= G2.
6. (a) Show that the Hall {2, 3}-subgroups ofGL(3, 2) are all isomorphic with Sym(4)
and that they fall into two conjugacy classes of subgroups.
(b) Show that PSL(2, 11) has Hall {2, 3}-subgroups H, and H2 such that Ht iH2.
4. Hall systems of a finite soluble group
The results of Section 3 show that the existence of Sylow p-complements for each
prime p is a characteristic property of finite soluble groups, and it is therefore not
surprising that in the study of such groups the Hall subgroups have an important
part to play. The central concept is that of a Hall system, originally called a Sylow
system when Hall introduced it in 1937 in his fundamental paper (P. Hall, [3]), where
many of the results of this section are to be found.
(4.1) Definition. Let G be a finite soluble group and let <r(G) denote as usual the set
of prime divisors of | G|. A Hall system of G is a set X of Hall subgroups of G satisfying
the following two properties:
HS1: For each rt £ P, E contains exactly one Hall it-subgroup, G„ say;
HS2: If H, Ke E, then НК = KH (i.e. H 1 K).
Condition HS1 means that G„ = Gt if and only if it n <r(G) = r n <r(G). Also we note
that E always contains 1 and G, these being Hall л-subgroups corresponding to
it = 0 and it = <r(G) respectively.
(4.2) Lemma. Let L be a Hall system of G and, for each it £ P, let Gr be the unique
element of Yd Hall,(G). Then the map
rt->G„ (for it £ <r(G))
from the power set of <r(G) to E is bijective and preserves the partial ordering of
inclusion.
Proof. That E n Hall„(G) contains a unique element and that the given map is
bijective is a direct consequence of Condition HS1. Suppose that p £ r £ <r(G). By
HS2 the product GfGr is a subgroup of G, and by A, 1.5 its order is |GP| IGt|/| G(, л GJ.
Therefore GPG, is a r-subgroup of G, and since G, is a maximal r-subgroup, we have
G„ < Gt. □
4. Hall systems of a finite soluble group
221
(43) Lemma. Let G be a finite group. Fori = 1,2 let be a set of primes, and let H,
be a Hall tij-subgroup of G. Assume that (|G : HJJG: H2|) = 1. ThenG = H H and
II t r ht is a Hall n, n n2-subgroup of G. 1 2’
Proof By A, 1.6(b) we haveG = H1H2 and |G: H, r,H2| = |G: H,||G-H,| Con-
sequently the prime divisors of |G: H, n H2| belong to ^n'2 = (тц n л2)'. Since
HtnH2 is obviously а я, n it2-group, it follows that it is a Hall nl r-. n2-subgroup
ofG. c
(4.4) Proposition. Let G be a finite soluble group, and let <r(G) = {₽!,..., p,}. For each
ie {!,...,r} let S’, be a Sylow prcomplement of G, and let К = {G, sj..., S,}. If
л £ <r(G), let n* = a(G)\n, and let G„ = Q {Sy pf 6 л*}. Then
(a) E := {G„: n £ <r(G)} is a Hall system of G, and
(b) E is the unique Hall system of G containing K.
Proof. The existence of the set К of Sylow p-complements is guaranteed by (3.3)(a).
Since Sf is a Hall p/-subgroup of G, repeated application of (4.3) shows that G„
is a Hall subgroup of G for the set of primes Q {p‘: p e л*} = (л*)'; because
(л*)' n cr(G) = л, it follows that G„ e Hall„(G), and therefore that E satisfies Condition
HS1; note that it is clearly only necessary to check this requirement for the subsets
л of <r(G).
Next let л, t £ <r(G). Then evidently G„ n Gr = Q {S;; p; e и* и т*} = G„,, since
n* и т* = (л n t)*. Hence |G„Gt| = |GJ |Gt|/|G„rt| = |Gnut|. However.it is clear from
their definition that G„ and G, are subgroups of G„,t. Therefore G„G, = G„ut and we
have G„ ± Gt. This proves that HS2 is satisfied and that X is a Hall system of G. as
asserted in (a).
To prove Statement (b), let E’ be a Hall system of G containing K, let n £ o(G),
and let H e E' n Hall„(G). If pf e it*, we have л £ p-. Since S£ is the Hall p,'-subgroup
in E', by (4.2) we have H < S, and can therefore conclude that H < Q {S;: p, e л*} =
G„, which is a Hall л-subgroup of G by Part (a). Hence H = G„, and it follows that
E' = E. □
(4.5) Definition. A set К comprising the group G together with exactly one Sylow
p-complement of G for each p e <r(G), is called a complement basis of G. (The inclusion
of the group G is a device to ensure that complement bases are preserved under
epimorphisms, etc.) We shall say that a Hall system E is generated by a complement
basis К if it is constructed from К in the way described in the statement of (4.4). We
have therefore shown the following
(4.6) Corollary. Each complement basis К is a subset of a unique Hall system E = EK
of G, and К generates EK. Each Hall system E of G contains a unique complement
basis К = KL, and E is generated by KL. The maps К — EK and E — KL are mutually-
inverse bijections between the set of complement bases and the set of Hall systems of G.
In (4.4) a Hall system of a finite soluble group G is represented as the set of
intersections of the subsets of a ‘basis’ of Sylow p-complements, (G itse eing
by convention the intersection of the empty subset). In a dual ashion it may
222
I. Introduction to soluble groups
represented as the set of joins of the subsets of a suitably defined ‘basis’ of Sylow
p-subgroups.
(4.7) Definition. A set В consisting of pairwise permutable Sylow p-subgroups of G,
exactly one for each p e <r(G), together with the identity subgroup, is called a Sylow
basis of G.
Let В = {1, Pt,..., Pr) be a Sylow basis of G. Thus, if <r(G) = {p,,..., pr}, we
have Pt e SylPl(G) and PjPj = PjPt for 1 < i, j < r. For n £ <r(G), make the following
definition:
= l<F': р‘еп^ ,{п* 0’
’ll ifn = 0.
If {ij,..., is} £ {1,.... r}, the permutability of each pair Pj, Pj ensures that
<Р;,,...,Р(1> =
Because |G„| is therefore the product of the orders of the Sylow p-subgroups
of G taken over pen, it follows that G„ e Hall„(G) and hence that the set
ZB = {G„: n £ <r(G)} is a Hall system of G. We say that the Sylow basis В generates
ZB in this case. In the other direction, if a Hall system L is given, the set BL of Sylow
subgroups in £ clearly fulfills the requirements of a Sylow basis, and it follows easily
from (4.2) that BL generates L. Thus we have justified the following analogue of (4.6).
(4.8) Lemma. Each Sylow basis В of a finite soluble group G is contained in a unique
Hall system of G, namely the Hall system EB which it generates. Each Hall system L
of G contains a unique Sylow basis BL and is generated by it.
Remark. A complete set of Sylow p-complements, the result of selecting at random
just one p-complement of G for each p e <r(G), is always a complement basis. The
same procedure applied to Sylow subgroups, however, does not always lead to a
Sylow basis. This is because the condition of pairwise permutability must also be
respected, although for groups G with | <r(G)| 2 this is no restriction since each
Sylow subgroup of such a group is simultaneously a Sylow complement. The class
Ш1 of groups for which every complete set of Sylow subgroups yields a Sylow basis
has been studied by Huppert [3]. His analysis shows that each ЯИ-group (that is, each
group in which two Sylow subgroups corresponding to different primes always
permute) is an epimorphic image of a direct product
Gt x G2 x • •• x G„
of groups G, satisfying |сг(С£)| <, 2 for i = 1, ..., n. (This result is also proved in
Huppert’s book [5], Chapter VI, Satz 3.1.)
Our next goal is to prove that a group G, acting by conjugation, is transitive on
its set of Hall systems. If a e Aut(G) and H e Hall„(G), it is obvious that H" e Hall„(G);
4. Hall systems of a finite soluble group 223
thus, if E is a Hall system of G, so also is E“ = {Я": H e E}. The set of Hall systems
of G is therefore characteristic, in the sense of being invariant under the action of
Aut(G). What we have to show is that Inn(G) already acts transitively on this set.
(4.9) Theorem. The number of Hall systems of G is
П |G:JVC(S)|,
Se К
where К is a complement basis of G.
Proof. By (4.6) there is a bijective map E -»KL from the set of Hall systems of G
onto its set of complement bases. By definition a complement basis is uniquely deter-
mined by the independent choice of one Sylow p-complement for each p e <r(G) =
{pi, , Pr}. If П, denotes the number of Sylow p,-complements of G, the number
of complement bases is therefore nin2...nr. But by (3.3)(b) the set Hallp.(G) is a
conjugacy class, and so its cardinal n, is |G: NC(S,)|, where Sf is any representative of
the conjugacy class. □
(4.10) Theorem. IfK' and К are complement bases of G.thenK' — K9 for some g e G.
Proof. The set В of all complement bases of G is certainly a G-set when G acts by
conjugation. Let Q be the orbit containing K. By A, 5.2 we have |O| = |G: T|, where
T = {g e G: К9 = K}, the stabilizer of K. Thus T = {g e G: S9 = S, for all S e K} =
Pls e к blG(S). Since К is a complement basis and NG(S) > S for each S e K, the groups
in {Nc(S):SeK} have pairwise coprime indices. Hence repeated application of
A, 1.6(b) yields
|G: Q AC(S)|= П IG:NC(S)|.
SeK SeK
Therefore by (4.9) we have |Q| = |В|, and consequently Q = B. In particular, K' is in
the G-orbit containing К. О
If a complement basis К generates a Hall system X, it is obvious that К9 generates
X9. If X and X' are arbitrary Hall systems of G. by (4.6) they are generated by the
unique complement bases, К and K' say, contained in them. Since К — К9 by (4.9),
it follows that X' = X9 for some g e G. Thus:
(4.11) Theorem. A group G acts transitively by conjugation on the set of Hall systems
of G.
Using (4.8), we can at once deduce the following result.
(4.12) Corollary. If В and B' are Sylow bases of G, then В' = B9 for some geG.
We shall now study the connection between Hall systems of a group and those of
its subgroups and quotient groups.
224
I. Introduction to soluble groups
Notation, (a) If S denotes a set of subgroups of a group G and if N < G, we shall
denote by S7V the following set of subgroups of G:
EN = (XN: X e =},
and by EN/N the following set of subgroups of G/N:
EN/N = {XN/N: X e =}.
(b) Weshall use H(G), K(G), and B(G) to denote respectively the set of Hall systems,
complement bases and Sylow bases of a finite soluble group G.
(4.13) Proposition. Let N be a normal subgroup of a group G.
(a) If e is an epimorphism from G to c(G), then c(H(Gj) = H(e(G)); thus H(G/N) =
{LN/N: E 6 H(G)}.
(b) Let n £ P, and assume that for each pen
(i) S(p) is a Hall p'-subgroup of its normal closure <S(p)G>, and
(ii) g(p) is an element of G such that Sipf^N = S(p)N.
Then there exists an element n e N, independent of p, such that
(4.a) S(p)™ = S(pf
for all pen.
(c) If Ее H(G) and L? N = EN for some geG, then E9 = E" for some n e N.
(4.14) Corollary. Analogous statements hold in Proposition 4.13 when H( ) is replaced
by each of K( ) and B( ) in turn.
Proofs, (a) By the first isomorphism theorem it will suffice to prove that H(G/7V) =
{EN/N: E e H(G)}. If E e H(G), we see from (3.2)(a) that E/V/7V is a Hall system of
G/N, although of course there may be repetitions in the list of HN/N as H runs
through E. By (4.11) the Hall systems of G/N form a G/N-orbit. Thus
H(G/N) = {(EN/Nf": gN e G/N}
= {E«N//V: geG}
= {EN/N: E e H(G)},
again by (4.10).
(b) If N = 1, we can choose n = 1. Therefore assume that N > 1 and choose a
minimal normal subgroup M of G with M < N. Let q be the prime dividing |Л/|.
Since the assumptions carry over to G/M with S(p) replaced by S(p)M/M and N
replaced by N/M, by induction on the order of G we obtain an element n in N such
that
225
4 Hall systems of a finite soluble group
Sfp^'M = S(p)"M
for all pen. Therefore we can assume that
= S(p)M
for a" p e n. Since S(p)and S(p)^ are Hall p'-subgroups of they are also
Hall p -subgroups of <S(p), S(p)»<₽'> < S(p)M. Hence for each p e n there exists an
m(p) e M such that
Sipy™ = S(pY'*n.
We aim to show that (4.a) holds with n = ni(q).
Let pen, and first suppose that M r> <S(p)°> = 1. Then [IM, S(p)] = 1, and so
S(p)”'” = S(p)"“” = S(p) = S(p)".
Now suppose that M n <S(p)c> * 1. If p q, then M < S(p) n S(p)91'’', and hence
S(p)’“” = S(p)^>M = S(p)M = S(p) = S(p)"
in this case. On the other hand, if p = q, then S(p)M = S(p)" by definition of n = m(q),
and therefore (4.a) holds for all pen.
(c) It follows from Part (b) that if К is a complement basis of G with K9W = KN for
some g e G, then there exists an n e N such that К 9 = K". Therefore Part (c) and the
analogous statements of Parts (a) and (c) for K( ) and B( ) follow directly from (4.6)
and (4.8). □
(4.15) Definitions. If E is a set of subgroups of a group G and if L < G, we shall
denote by E n L the following set of subgroups of G
E n L = {X n L: X e E}.
Let X be a Hall system of G, and let L < G. It is easy to see that, in general, E n L is
not a Hall system of L. However, when it so happens that E n L is a Hall system of
L, we say that E reduces into L and write
E\L
In this case we call E an extension of E n L. (The concepts of the reducibility and
extendibility of a Hall system are due to R.W. Carter [1].) Naturally we shall apply
the same terminology to complement bases and Sylow bases in the corresponding
situation.
Let E, be a Hall system of a subgroup L of a finite soluble group G, and let K,. be
the complement bases of E,, For each ре P, choose a Sylow p-complement G„. of G
containing the p-complemcnt L„. in K,; this is always possible by (3.3)(c). Let
226
I. Introduction to soluble groups
К = {Gp-: p e P}, a complement basis of G, and let E be the Hall system of G
generated by K. Since Lp. is contained in the p'-subgroup Gp. n L of L, we have
Lp. = Gp. ci L. Thus, if G„ denotes the Hall л-subgroup in E, it follows that
LriG„ = LrYQ Gp.) = Q (LnGp.)= L„. = L„.
\р€Я’ J Pen‘ Pen‘
Therefore E n L = EL, and E is an extension of E,. Moreover, if E' is any Hall system
of G, by (4.11) we have E' = E9, and therefore E' reduces into Le. Thus we have proved
the following:
(4.16) Proposition. Let L be a subgroup of a finite soluble group G. Each Hall system
of L extends to a Hall system of G. Each Hall system of G reduces into some conjugate
of L.
(4.17) Remark. Let L be a Hall system of G, let L < G, and let N < G. Then
(a) if E reduces into L, then 1.N/N reduces into LN/N, and
(b) if LN/N reduces into LN/N, then E reduces into LN.
Proof Let H e E n Hall„(G). If E \ L, then H n Le HallJL). The «-subgroup
HN/N r-,LN/N of LN/N contains the subgroup (Hr~.L)N/N, which belongs to
Hall„(L/V/N) because Hall subgroups are preserved under epimorphisms. Therefore
HN/N ci LN/N e LN/N n Hall„(L/V/W) and (a) is proved.
To prove Part (b) put Ho = H n LN. Then H0N = HN n LN, and by supposition
H0N/N e Hall„(L/V//V). By (3.2)(b) the index |N: N n Ho| is a «'-number; therefore
ILN: Ho| = I LN : H0N| |H0N : Ho| = \LN: H0N\\N: N n Hc| is a л'-number. Con-
sequently Ho e Hallp(LN). □
The next result shows that the reducibility of a Hall system into a subgroup is
determined by the reducibility of either of its bases.
(4.18) Proposition. Let The a Hall system of a group G, let К be its complement basis
and В its Sylow basis. Let L be a subgroup of G. Then the following statements are
pairwise equivalent:
(a) E reduces into L;
(b) К reduces into L;
(с) В reduces into L.
Proof. It is obvious that (a) implies (b) and (c). Assume that (b) holds, and let
К = {G, S„..., Sr}, where S( is a Sylow p,-complement of G. By assumption StcyL
is a Sylow p(-complement of L and, in particular, coincides with L when p, f a(L).
The set {S; n L: i = 1,...,г} и {L} therefore contains a complement basis, Kc say,
of L, and Ko generates some Hall system Ec of L. If G„ e E n Hall„(G), by (4.4)
G„ = Q {S;: Pi 6 <r(G)\«}. Therefore G„ c, L = Q {S; л L : p, e <r(G)\«} = Q {Sf л L :
Pi e c(L)\«}, which is the Hall л-subgroup of L in Eo by (4.4) again. Thus (a) follows
from (b).
4. Hall systems of a finite soluble group 227
Finally, assume that (c) is the case. The statement that В reduces into L means that
Br I. is a Sylow basts of L and includes the assumption that the elements of В rs L
are pairwise permutable. In fact, this latter condition is not needed in order to infer
that (a) holds. In a separately stated result we shall prove that (a) is a consequence
of the following weaker hypothesis: 4
(с') P r L is a Sylow subgroup of Lfor each P e B.
(4.19) Lemma. Let L<G and let В be the Sylow basis of a Hall system Z of G. If В
satisfies Hypothesis (c') above, then Z reduces into L.
Proof Let л £ <r(G) ={ p,,..., p,}, with the primes so numbered that л = {p,,..., p,}
say. For i = 1,. , t, let e В n SylPi(G); set Ln = fj'=1 (Pt n L) and G„ = f[:=1 A-
Clearly we have
(4./?)
£ Gn ry L.
Since В is a Sylow basis, G„ e HallJG) and so G„ n L is a л-subgroup of L. From
repeated application of A, 1.5 we conclude that the cardinal of the subset Ц, is
П?=11^1 n L|, which is clearly the order of a Hall л-subgroup of L because by
assumption n L e Sylp.(L). Therefore by Lagrange’s theorem |G„ n L| < |L„|. To-
gether with (4./J) this means that L„ = G„ n L and that L„ e Hall„(L). In particular,
the Sylow subgroups {Pf n L: i = 1,.... t} are pairwise permutable, and it follows
directly that Y \ L. This completes the proof of (4.19) and with it the proof of (4.18)
also. □
(4.20) Lemma. Let L be a subgroup of G, and let Z be a Hall system of G.
(a) Assume that | G: L| is a power of a prime p. Then, for each prime q distinct from
p, each Sylow q-complement of G reduces into L. In particular, Z reduces into L if and
only if L contains the Sylow p-complement of Z.
(b) Assume that Z reduces into L. Then there exists a maximal chain of subgroups
L = L,< <
< Li <L0 = G
such that Z x L,- for i = 0, 1,..., r.
Proof, (a) Using A, 1.6(b), we have |G: L\\L: LnH\ = |G: L nH| = |G :£||G.H|
for Я e Hall„.(G). Thus LnH has q-power index in L, which means that
Ln He Hall,.(L). Therefore, if К is the complement basis of a Hall system Z of
G, then К reduces into L if and only if the p-complement S of K s®t,sl^s
S n L e Hallp.(L), which is the same as saying S < L because Hall„ (L) £ Hallp.(G).
The implication: (b) => (a) of (4.18) now yields the desired conclusion.
(b) We first prove, using induction on |G|, that the assumption <L, Gp > or
all primes p and Gp. e Z leads to the conclusion that L = G. Let A be а пита
normal subgroup of G. Since ZN/N reduces into LN/N, and since (.LN/N,
(L, G -)N/N = G/N for all Gr-N/N e^N/N, by induction we conclude that LN
I. Introduction to soluble groups
If N is a p-group, it then follows from Part (a) that L contains the p-complement Gp.
of E. and consequently G = <L, Gp ) = L.
In order to prove Assertion (b) we can clearly suppose that L < G and hence by
the previous paragraph that <L, Gp ) < G for some Gp. e E. Let L, be a maximal
subgroup of G containing <L. Gp >. Then E \ Lj by Part (a), and since EnL,
reduces into L, it follows at once by induction on | G : L| that there is a maximal chain
of subgroups from L to L, such that E n Lt (and hence E) reduces into each subgroup
of the chain. □
(4.21) Proposition (Kegel [2]). Let Lbea subgroup of a finite soluble group G. Then
the following statements are equivalent:
(a) L is subnormal in G;
(b) Each Hall system of G reduces into L.
Proof (a) =>(b): Let L = Lr < о L, < Lo = G, and let К £ E e H(G) with К e
K(G). By (3.2)(b) we have К \ L1; and so by induction on the length r of the
subnormal chain we have К \ Lr. Then by (4.18) it follows that E x Lr, as required.
(b) => (a): We prove this by induction on | G |. Let 1 / N < G. From (4.13) it follows
that every Hall system of G/N reduces into LN/N. Hence by induction LN sn G. If
LN < G, then there exists a proper normal subgroup К of G such that LN < K. If
Eo e H(K), by (4.16) we have Eo = К n E for some E e H(G), and since E \ L, we
have Eo x L. Hence by induction L sn K, and therefore L sn G. Thus we may suppose
that LN = G.
On taking N = Corec(L), if this is non-trivial, we can conclude that L = G sn G.
Therefore suppose that L is core-free, in which case by taking N <G we may also
suppose that L is a maximal subgroup of G complementing N. In particular, it follows
that |G: L| = |N|, which is a power of some prime p. By (4.20) Assertion (b) then
implies that L contains R = <Hallp.(G)>, which is a normal subgroup of G by (3.3)(b).
Since L is supposed to be core-free, we conclude that R = 1 and hence that G is a
p-group. Then by A, 8.3 we have L sn G. □
Remark. For a soluble group G condition (b) of Proposition 4.21 is easily seen to be
equivalent to the following:
(b') For all primes p
P e Sylp(G) => P n L e Sylp(L).
Condition (b') is evidently a necessary condition for a subgroup L to be subnormal
in an arbitrary finite group G. The long-standing open question whether it is also a
sufficient condition has now been settled positively by Kleidman [1], using the
classification of finite simple groups.
(4.22) Theorem. Let E be a Hall system of a group G, and let U and V be subgroups
into which E reduces. Then
(a) (Shamash [1], Proposition 9) E reduces into U г, К and
(b) (Lockett [1]) if, in addition, UV = VU, then E reduces into UV.
4. Hall systems of a finite soluble group 229
Proof. Let В = {1, Pi,..., P,} be the Sylow basis in E, where Pf g Syl (G).
(a) Since G = P1P^...P an elementary counting argument shows that each q e G
has a unique expression of the form
9 = 9,92 9„
with each g, e P,. Since В x U, we have U = U, U2... U„ where Ц = P, n U; similarly
V ~ viv2. - v,y where V, = P( n V. Hence, if g e U n V, the uniqueness of the expres-
sion (4.y) implies that g,, e Ц n IZ = P- n (U г, V). Therefore we have
UnV = n^niW),
1=1
whence | U r, F| = |P; n U r> I/|. Consequently Pt n U г, V e Syl (U n V), and
by (4.18) we have I \ (/nF. '
(b) Recall that for any group X we denote by |Ajp the highest power of a prime p
dividing |X|. Let P be a Sylow p-subgroup in B. Then we have
(4.6) |Pn 1/Ц < |UP|p = |I7|p|I/|p/|C7ri I/|„
< |Pn L7| |Pn I/|/|Pn U n I/| = |(Pn U)(Pn F)|.
Evidently (P n U)(P n V) £ Pn UV, and therefore (Pn U)(Pr, V) = Pr. UV. It
follows that equality holds throughout the expression (4.6) and, in particular, that
Pn UV e Sylp(C7I/). We conclude once more from (4.18) that E x CL □
Remark. The proof shows that Theorem 4.22(b) holds equally well with a single Hall
л-subgroup Я in place of a system E; in other words, if
f(i) UV = 1/(7,and
(4.£) <
((ii) H r U and Я n T are Hall л-subgroups of U and V respectively,
then Я n UV e Hall „(СТ). In contrast, the proof which we have given for Part (a)
depends essentially on the fact that the full Hall system reduces into U and V. It is
apparently unknown (cf. Shamash [1], p. 299), even for soluble groups, whether the
assumption (4.e)(ii) alone is sufficient to ensure that Яп U n Ve Hall„(l/ri V),
although this certainly holds with the additional assumption (4.e)(i) (cf. Exercise 5
below).
If E is a Hall system of G, let £?(E) denote the set of subgroups of G into which E
reduces. Thus
Й?(Е) = {Я < G : E x Я}.
The preceding result may be summarized by saying that .4?(E) is closed under
intersections and permutable joins of subgroups. We now describe an examp e w ic
shows that the permutability hypothesis cannot be dispensed with here
230
I Introduction to soluble groups
(4.23) Example. Let S = Sym(4), and let IV = Z3'liS, where the wreath product is
taken with respect to the natural permutation representation of S of degree 4. Let
V = «12» and И = «134».
Then one can easily verify by direct calculation that S = <G, E>. Let В denote the
base group of W, and let {x,,..., л.,} be its natural basis. Thus, for neS and
1 < i < 4, the equation
defines the action of S on В viewed as an F3S-module. Since the normal subgroup
E = {1, (12)(34), (13)(24), (14)(23)] of S permutes the elements of this basis transi-
tively, we have CB(E) = <x1x2x3x4>. Let
E = {1, BV, EU, fP},
which is evidently a Hall system of W, and put b = Xi-X2 6 B. Clearly [12, b] = 1, and
therefore Eb = {1, BV, EbV, W] is a Hall system of W which reduces into U and
V. If Eb \ S, then EbU e Syl2(S); consequently E = BE nS = BEb nS = Eb, and it
follows that beBr> NW(E) = CB(E) = <x1x2x3x4>. Since this is clearly not the case,
we have found a group W with subgroups U and V, and a Hall system Eb of W which
reduces into V and into V but not into S = <G, F>.
Therefore ./'(E) is not in general a sublattice of the subgroup lattice '/'(G) of G.
However, .9?(E) does contain several interesting sublattices of ./(G). For example, by
(4.21) it contains the well-known sublattice .//(G) of all subnormal subgroups of G
(cf.A, 14.2 and A, 14.4). Another example is the set of normally embedded subgroups
into which X reduces, and this will be discussed at length in Section 7 of this chapter.
We now investigate a third example.
(4.24) Definitions. Let E be a set of subgroups of G. A subgroup 17 of G is said to be
3-permutable if
(4.e) VX = XU for all X e E.
We shall write GTE in this case, and shall denote the set of all E-permutable
subgroups of G by -'/(E).
A subgroup of G is called system permutable if it is E-permutable for some E e
H(G). Maximal subgroups and permutable subgroups of G, and in particular normal
subgroups, are examples of system permutable subgroups of G.
if G ± H e Hall„(G), we have |G: H n G| = \UH: H\, which is a тг'-number, and
therefore Hr.L e Hall„(G). This justifies the first of the following remarks; the
second is obvious, and the third follows directly from the Dedekind law.
(4.25) Remarks. Let E be a Hall system of G, and let U be a E-permutable subgroup
of G. Then:
4. Hall systems of a finite soluble group 231
(a) £ \ I', and consequently ^(E) e .^(E);
For all K<G, the quotient group UK/K is a XK/K-permutable subgroup of
(c) If U < H < G and E \ H, then U is a(Xr, Hfpermutable subgroup of H.
(4.26) Proposition. Let 2. be a Hall system of G with complement and Sylow bases К
and В respectively. Then each of the following conditions:
(a) U L К and (b) U 1 В
is both necessary and sufficient for a subgroup U of G to be 1,-pennutable.
Proof. It is clear that both conditions are necessary. Since the elements of E are the
permutable products of certain subgroups in B, it follows at once from A, 1.6(a) that
Condition (b) is also sufficient.
It remains to show that Condition (a) is sufficient. Let S’,,.... S, be a set of Sylow
complements in K, and put H, = QJ=1 Sj. We prove by induction on t that U L H„
and as H, is a typical element of E by (4.4), it will follow that L! _L E. By assumption
we have U LS, and so U 1H,. If t > 1, by induction C permutes with H,., and
therefore by A, 1.6(c) it also permutes with H,_t n S, = H, because and S, have
coprime indices. □
(4.27) Corollary. Let T. be a Hall system of G. let p e P. and let P and S denote
respectively the Sylow p-subgroup and the Sylow p-complement in E. Then a p-subgroup
U of G is 2,-pertnutable if and only if the following two conditions hold:
(i) U <P, and (ii) US = SU.
Proof. Since U has p-power order, we have U _L P if and only if U < P. The necessity
of (i) and (ii) is therefore clear.
Now suppose that Conditions (i) and (ii) hold. Let К be the complement basis of
E, and let T 6 K. If T = S, certainly U L T by (ii). On the other hand, if T / S, by
(i) we have U < P < T, and again LIT Therefore U 1K, and so U 1E by (4.26).
□
(4.28) Lemma. Let Xbea Hall system of G. Then a subgroup U of G is 1,-permutable
if and only if the following two conditions hold:
(i) E reduces into U, and
(ii) P LE for all Sylow subgroups PofXrsU.
Proof. Suppose that E \ U. Let p e P, let P e Sylp(C). and let S e E n Hall (G) Since
Sn I/ is a p-complement of U. we have U = P(Sn 17) = (Sn U)P. Therefore US
PS and SU = SP, and it follows that U 1S if and only if P 1S.
If U is E-permutable, by (4.25) we have ZyU. Therefore the subgroup E n Syi (Ы
contains a Sylow p-subgroup P of U, and P IS as just remarked. Hence PL2. by
(4.27). Conversely, if (i) and (ii) hold, then U L S for all S in the complement basis of
E, and so I' 12 by (4.26).
232
I. Introduction to soluble groups
As promised, we now prove that ^(E) is a sublattice of //(G).
(4.29) Proposition. Let E be a Hall system of G. If U and V are L-permutable
subgroups of G, then so also are U n E and <12, E).
Proof. Let U, V e £?(Е). It follows directly from A, 1.6(a) that <12, E) e .^(l). In order
to prove that U n V e we show that Conditions (i) and (ii) of (4.28) are fulfilled
by U n V. Certainly L reduces into V n V by (4.22)(a). Let P e Syip( L2 n E), and
let S e L n Hall„.(G). If T e 1 n Syl„(G), then T n (12 n E) e Sy 1„(U n V), and there-
fore P = (Tn(Uri V))x for some x e U n И Let Q = (Tn I7)x e SylpfL) and R =
(Tn V)x e Syl^E). Then P = Q n R and QR £ Tx. Since Tx is a transversal to S in
G, it follows that = |S||QR|. Moreover, from (4.28) and the fact that L per-
mutes with U and V by assumption, we have SQ = QS and SR = RS, and therefore
(SQ)(SR) = SQR. It then follows that
|SCnSR| = |SCI|SR|/|(S(2)(SR)|
= |S|2ICI|R|/|S||CR|
= |S|ienR| = |SP|.
Since we clearly have SP £ SQ n SR, we conclude that SP = SQ n SR and, in par-
ticular, that SP is a subgroup of G. It follows that P permutes with the complement
basis of E and hence with L itself. Therefore by (4.28) the subgroup U n V is
E-permutable. □
Evidently the Hall subgroups in £P(E) are precisely those in E, and therefore
гЖ) = if and only if 1 = E*. Kegel [2] has shown that Q {.^(E): E e H(G)}
is contained in .‘/2(G), the lattice of subnormal subgroups of G.
We end this section with two results that are needed in Chapter VII, Section 2.
They are derived from Venske [1] and [2].
(4.30) Lemma. Let Lbea Hall system of G, let К < G, and let Z be a cyclic p-subgroup
of G such that ZK/K is LK/K-permutable. Then there exists a L-permutable cyclic
p-subgroup U of G such that UK = ZK.
Proof. Let P and S denote respectively the Sylow p-subgroup and p-complement in
E. By (4.27) it is enough to show that P has a cyclic subgroup U such that UK = ZK
and US — SU. We prove this assertion by induction on |G|, assuming without loss
of generality that К is a minimal normal subgroup of G.
Case 1: К is a p'-group. Since ZK/K is EK/K-permutable by hypothesis, wc have
ZK < PK, and so in this case P has a subgroup U = P n ZK such that ZK = UK;
moreover, Z is conjugate to U, which is therefore cyclic. To conclude the argument,
we observe that US = UKS = ZKS = SZK = SUK = SKU = SU, as desired.
4. Hall systems of a finite soluble group
233
2' * ,₽’gr°“J’’ Then Z - PK = P’ the complex SKZ is a subgroup of G,
and 1 reduces into SKZ. If SKZ < G, then the conclusion of the lemma follows at
once by induction. Therefore suppose that SKZ - G if К < <J>(G), we have SZ =
G = ZS by A, 9.2 and can take U = Z. Hence we can suppose that К is complemented
in G and can find a complement M containing S. Let U = ZK n M, and note that
— ZK n MK = ZK; since U ~ UK/K = ZK/K e q(Z), we conclude that U is a
cyclic subgroup of P (= ZK}. Finally, we observe that US = SU, as in Case 1. □
(4.31) Lemma. Let M be a maximal subgroup of G.
(a) If M has prime index p in G, then there exists a Hall system LcfG reducing into
M and a cyclic p-subgroup U supplementing M in G such that U is both 1,-permutable
and L n M-permutable.
(b) If M has a cyclic supplement U in G and if U permutes with a Hall system A
of M, then M has prime index in G.
Proof, (a) By A, 10.5(b) the maximal subgroup M complements a chief factor H/K
of G with | H/K | = | G: M | — p. Let Z be a minimal supplement to К in H. Then
Zr K < <S>(Z), and Z/(Z n K) = H/K is a cyclic p-group. Hence Z is a cyclic p-
group by A, 9.8. Let 1 be a Hall system of G reducing into M. Since ZKIK = H/K <
G/K, the quotient ZK/K is certainly SK/K-permutable. By (4.30) there exists a
E-permutable cyclic p-subgroup U of G such that UK = ZK; moreover, MU —
MKU = MZK — MH — G. Since U is a p-group and M has p-power index in G, by
(4.27) we deduce from the X-permutability of U the fact that it is also Ir.Af-
permutable.
(b) If p is the prime dividing |G : Ml, then by (4.28) we may clearly suppose that
U = is a p-group. We will prove the claim that |G : M\ = p by induction on |G|,
and evidently can therefore suppose without loss of generality the Coree(M) = 1.
Since 17 is abelian, we have <(17 n Af)M> = <(17 n Af)t,M> < G, and therefore
UnM = 1.
Let P e An Syl„(M), and set W = <uF>. Since 17 permutes with P. clearly UPe
Sylp(G) and P < UP. Let T be a maximal subgroup of UP containing P. Then
P(Tn U}= TnUP = Tandp = |17P: T| = |H : Tn U |, and consequently Tn U =
W; thus W permutes with P. Let Q 6 A n Hal lp( AT); then the group QU has a
non-trivial cyclic Sylow p-subgroup and therefore possesses a complemented chief
factor of order p by A, 11.8(c). It follows that QL has a maximal subgroup L of index
P and that L can be chosen to contain Q. As before, we then conclude that L n U = W
and hence that W permutes with Q. It therefore follows that W permutes with
QP = M, and, since U n M = I, that MW < G. Consequently W < U n M = 1, and
p = |l/| = |G:M|. °
Some concluding remarks on terminology
(a) The concept to which we have given the name “Hall system was ongina у
called a “Sylow system” by P. Hall [3]. Although there has been considerable
inconsistency over the nomenclature of the two types of basis, t e term у о
system", used in Hall’s sense, has been more or less universally adopted over the
years by writers in the English language, and the major break with tradition whic
234
I. Introduction to soluble groups
we are now suggesting needs some justification. Our reasons for the change are as
follows:
(i) It seems appropriate that Hall’s name should be attached to the central concept,
which he originated.
(ii) It is helpful to have the first word of each concept relating to the constituent
objects: thus a Hall system comprises Hall subgroups, and a complement basis [Sylow
basis') consists essentially of Sylow complements (Sylow subgroups).
(iii) The distinction between the complete set of 2r Hall subgroups in a system and
its two distinguished generating subsets of cardinality r + 1 (the bases) is emphasised
by our proposed terminology.
By way of final apology we point out that the terms we have chosen do not seem
to conflict too seriously with accepted usage. For what we call a complement basis
(Sylow basis) Hall [3] uses the phrase “a complete set of Sylow complements (per-
mutable Sylow subgroups)”. In his book [5] Huppert uses the words “Komplement-
system” and “Sylowsystem” for these two bases, but has no terminology to describe
a Hall system.
(b) A subgroup which permutes with every subgroup of a group was called quasi-
normal by Ore when he first introduced the concept in (Ore [1]). More recently the
substitute term permutable has gained currency (cf. Gross [2] and Lennox and
Stonehewer [1] for example), and so we have adopted it for this and related concepts.
Thus we have chosen X-permutable to describe a subgroup which permutes with every
member of a Hall system 1 in preference to the alternative X-quasinormal used by
Venzke [1], for example; for at this level of generalization the concept bears scant
resemblance to normality.
Exercises
1. Let /V <J G, and let X* be a Hall system of G/N. Then there exists a Hall system
E of G such that E/V/W = £*,and ifY.0N/N = E*,thenE0 = X" for some n 6 N.
2. Let 1 be a Hall system of G. Prove that 1 reduces into every subgroup of G if
and only if G is nilpotent.
3. Let N < G, U < G, and let X be a Hall system of G such that XN/N reduces into
UN/N. Then:
(a) There exists a Hall system l0 of G such that Y.0N/N ~ "LN/N and Xo\ I';
(b) If A < G, then 1ч U.
4. Let V, V < G and H e Hall„(G). If H n U and H n P are Hall л-subgroups of U
and V respectively, does it follow that <H n U, H n F) is a Hall л-subgroup of
<U, P>?
5. Let H 6 HallJG), and let V and V be subgroups of G such that Equations (4.t)
on p. 229 are satisfied. Prove that H n V n V is a Hall л-subgroup of V n F.
6. Let G be a soluble group, and let H be a subgroup which permutes with every
Hall system of G. Prove that H sn G.
7. Let G = Sym(4). How many Hall systems does G possess? Show that:
(a) G has subnormal subgroups which are not system permutable.
(b) If H is a system permutable subgroup of G and if 1 is a Hall system of G
reducing into H, it does not follow that H ± 1.
5. System normalizers
235
(c) If H is a system permutable subgroup of G and if H < L < G it cannot he
inferred that L is system permutable in G.
8. Find a group G and a system permutable subgroup U such that
(i) <GC> = G, (ii) /VC(G) = V, and (iii) Corec(G) = 1.
9. Show that the class consisting of the finite soluble groups all of whose subgroups
are system permutable lies strictly between the class of nilpotent groups and the
class of supersoluble groups.
10. Show that a cover-avoidance subgroup need not be system permutable.
11. (Carter [3]) Let V be a subgroup of a finite soluble group G.
(a) Let z0(U) denote the product in a L'-composition series of G of the orders of
the L’-central factors avoided by V. Show that z0(U) is independent of the series
chosen.
(b) Let cu(G) denote the number of Hall systems of G and o(Lj the number of
those that reduce into V. Show that
<t(G) = m(G)z0(U)/|G : U|.
12. (A. Mann) A subgroup V of a soluble group G is system permutable if and only
if it can be written V = QЦ, where L!, is a subgroup of p,-power index in G and
/ Pj for i / j.
13. (H. Wielandt [4]) Let L be a Hall system of G. If X reduces into subgroups U
and V of G and if V, V sn <L!, V>, then X reduces into <L!. F>.
5. System normalizers
Soluble groups are defined in terms of their commutator structure, by requiring that
the derived series should reach the identity subgroup in a finite number of steps or,
equivalently, that all the composition factors should be abelian. In Section 3 they are
characterized by the property that each Sylow subgroup has a complement. This
suggests an intimate connection between, on the one hand, the commutator or normal
structure of a group and, on the other hand, the Sylow structure, by which we shall
mean the manner in which a group contains its Sylow subgroups. Nowhere is this
connection made more explicit than in the class of nilpotent groups, which, as we
saw in A, 8.1 and 8.3. is characterized by either of the following properties:
(1) For some natural number c, the equation Kc+1(G) = 1 holds;
(2) The Sylow subgroups of G arc all normal in G.
A connecting link for these two aspects of structure in an arbitrary finite soluble
group is provided by the concept of a system normalizer, which Philip Hall
introduced and fully investigated in P. Hall. [4]. It was proved in the previous
section that a finite soluble group has a transitive permutation representation when
it acts by conjugation on the set of its Hall systems; the system normalizers are the
‘stabilizers of a point’ in this representation. Thus the index of a system normalizer
in G is the number of Hall systems of G. Its order turns out to be the product of the
orders of the central chief factors in a chief senes of G. It is a nilpotent subgroup о
236
I Introduction to soluble groups
G whose relative size may be viewed as a measure of how close a group comes to
being nilpotent.
In this section we confine ourselves to just the most important basic facts about
system normalizers, all of which can be found in Hall's original paper cited above.
This is because in Chapter V we shall develop at length the more general concept
of an 5-normalizer within the framework of formation theory and shall then be able
to read off properties of system normalizers by specializing to the case where g is the
class of nilpotent groups.
(5.1) Definition. If 1 is a Hall system of a group G, the subgroup
NG(Z} = {g 6 G: H = H9 for all H 6 X}
is called the normalizer of X. A system normalizer of G is a subgroup of the form NG(Z)
for some X e H(G).
If К and В denote respectively the complement basis and the Sylow basis of L, it
is clear from (4.4) and the discussion after Definition 4.7 that
NB(Z) = AC(K):= Q{A6(S): SeK)
= AC(B) := П {NB(Py. P e B}.
Hence by A, 8.3 it is immediate from the definition of a system normalizer that G is
nilpotent if and only if G is a system normalizer of G. Moreover, if a 6 Aut(G), it is
obvious that Ac(2?) = Af,(X)’. Since Aut(G) permutes the Hall systems of G and
because by (4.11) the subgroup Inn(G) is already transitive, the following theorem is
evidently true.
(5.2) Theorem. The system normalizers of a finite soluble group form a characteristic
conjugacy class of subgroups.
With a further appeal to (4.11) the next result follows from the Orbit-Stabilizer
Theorem (see A. 5.2).
(5.3) Theorem. The number of Hall systems of a finite soluble group G is the index of
a system normalizer in G.
We shall now prove the earlier assertion that system normalizers are nilpotent.
(5.4) Theorem. Let Z be a Hall system of G, let p e P, and let P and S denote
respectively the Sylow p-subgroup and p-complement in Z. Then
(a) P n A0(S) is a Sylow p-subgroup of both NG(S) and Ao(£),
(b) A(;(l) is Z-permutable (in particular, L reduces into NG(Z)), and
(c) A6(L) is nilpotent.
Proof, (a) Let A = AC(S). Since G = PS = PA, we have (P n A)S = PS n A = A.
Therefore |P n A| = | A : S| = |А|р, and so P n A e Sylp(A). Let К be the com-
5. System normalizers
237
plemenf basis of К If S* e K\ {S}, we have P < S* < Wc(8‘). Therefore P n N <
/VG(K) = /vc(L) < N, and it follows that P n N e Syl (AC(E)).
(b) In view of Part (a), we have 1 ч NC(X) by (4.19); each Sylow subgroup of Nr(E)
is X-permutable by Sylow s theorem and (4.27); and therefore AL(L) is L-permutablc
by (4.28).
(c) Let D = WO(X). By Part (b) we have S n D e Hallp.(D). Because D normalizes
Pe£ and SnD <S < IV, certainly S n D normalizes PnjV. Therefore the Sylow
p-subgroup P n N of D is normal in (P n N)(S n D) = D, and consequently D is
nilpotent by A, 8.3.
One of several properties of finite soluble groups used implicitly in the formulation
and proof of the next result is the fact that a characteristically simple normal subgroup
is an elementary abelian p-group.
(5.5) Lemma. Let A be a minimal normal subgroup of a group G. Assume that A is
contained in a soluble normal subgroup M of G, let p be the prime dividing |A|, and let
S 6 Hallp.(M). If A n NC(S) 1, then A < Z(M}.
Proof. Let С = CM(A) = M n Cc(A) < G. By A, 13.6 we have OfM/C} = 1. Let
K/C = Op(M/C), a characteristic subgroup of М/С. Then К < G, and so C(K n S) —
К by Parts (b) and (c) of (3.2). By viewing A as an Fp(K/C)-module, we can apply
Maschke’s theorem and conclude that
A = [A, K] © G(K).
Since СЛ(К) = СЛ(К nS) > An NC(S) 1, we have [A, K] < A. Because К < G,
we have [A, K] < G, and hence [A, K] = 1 since A •< G. Therefore К = C and
O,(M/C) = 1 for all r 6 P. Since М/С is soluble, it follows that M = C, in other words
that A < Z(M). □
(5.6) Theorem. A system normalizer of a finite soluble group G covers the central chief
factors and avoids the eccentric chief factors of G.
Proof. Let I e H(G), and put D = NC(I). Let H/K be a p-chief factor of G, and let
8 e X n Hall„(G) and P e Io Syl„(G).
First suppose that H/K is central. Then clearly we have SH/K = (SK/K) x (H/K),
and, in particular. SK is a normal subgroup of SH of index p = \H/K\. Applying the
Frattini argument to the p-complement S of HS. we have HS - KSIVHS(S) < NC(S)K.
Therefore the normal p-subgroup H/K of G/K is contained in every Sylow p-
subgroup of NK/K, where A denotes NC(S). Let Po = Pn/V. Then Po < D an
Po e Syl,(N) by (5.4). Hence PoK/K 6 Syl„(KK/K). and so we have H/K < P0K/K <
DK/K'. in other words, D covers H/K.
Now suppose that H/K is eccentric. Take the special case of (5.5) where G - M,
and apply it to the group G/K with H/K in place of A and SK/K in p асе о . e
it follows that (H/K)n Kc/k(SK/K) = 1. Since H/K is a p-group,(H/K)r\(DK/K)£
contained in the Sylow p-subgroup P0K/K of DK/K. But Р0К/К _ c/*(
238
I. Introduction to soluble groups
because Po normalizes S, and consequently (H/K) n(DK/K) = I.Thus Hr, DK = K;
in other words, D avoids H/K. □
Combining the preceding result with A, 1.7, we obtain the following.
(5.7) Theorem. The order of a system normalizer of G is the product of the orders of
the central chief factors in a chief series of G.
Using these facts, we can now prove the following important property of system
normalizers.
(5.8) Theorem. Let I he a Hall system of G, and let К < G. Then N(fL]K/K =
NgikILK/K). In other words, system normalizers are preserved by epimorphisms.
Proof Put D = /VG(X), and recall from (4.14) that T.K/K = {HK/K: H e 1} is a Hall
system of G/K. It is obvious that
(5.a) DK/K < NOIKfLK/Kf
By (5.6) and A. 1.7 the index |D: D n K| is the product of the orders of the central chief
factors above К in a chief series of G passing through К and therefore coincides with
the product of the orders of the central chief factors in a chief series of G/K. By (5.7)
the index |D: Dr-. K\ is therefore the order of a system normalizer of G/K. Since
\DK/K\ = |D: Dn K|, it follows that equality holds in (5.a). □
(5.9) Theorem. Let D be a system normalizer of a soluble group G. Then
(a) <DC> = G, and
(b) QeeGDe = Z.,(G), the hypercentre of G.
Proof. If (a) fails to hold, then D is contained in a proper normal subgroup and hence
in some maximal normal subgroup N of G. Since G is soluble, G/N is abelian and is
therefore a central chief factor of G. Then by (5.6) we have DN = G, which is
impossible because DN = N < G. Therefore (a) is true.
For the proof of Assertion (b) put К = QeeGIX = CoreG(D). From the definition
of the hypercentre we know that the chief factors of G below Z,r(G) are central, and
therefore Z,r(G) < D by (5.6). Consequently Z.z(G) < K. Let
1 = Ko < K, < • < Kr = К
be part of a G-chief series passing through K. Since D covers KJK^^, i = 1....r,
we have [K;, G] g К;_1; again by (5.6). Therefore [K, G, .Г., G] = 1, and we con-
clude that К < Z,r(G). Hence К = Z„(G). □
(5.10) Remark. In conjunction with (5.8). Theorem 5.9(b) implies that the hyper-
centre of an epimorphic image of a soluble group is the core of the image of a system
normalizer. In Chapter V we shall see that other important characteristic subgroups
5. System normalizers
239
(e.g. the Frattini subgroup) can be obtained as the cores of certain canonical
jugacy classes of subgroups in this epimorphism-invariant fashion.
con-
(5 11) Example. (Shamash [1]). The immediate purpose of the example that we are
about to describe is to give a negative answer to the following question raised bv
Carter [3J: ’
If a subgroup H of a soluble group G contains a system normalizer D of G does D
normalize a Hall system of HI
Let S = Sym(4), and let
G = Z/IjS,
where the wreath product is taken with respect to the natural permutation representa-
tion of S of degree 4. Let В = <bt,b4> be the base group of G, where, for aeS,
we have
n 1bin =
Let E = <(12)(34),(13)(24))c S, let A = <(123)), and let T = <(12)). The subgroup
L = A T is a complement to E in S and is isomorphic with Sym(3).
Step 1: Let E — {1, BET, A, G}, and let
D = T x <6,6263) x <b4).
We shall show that E is a Hall system of G and that NG(E) = D. Since BET e Syl2(G)
and A 6 Sylj(G), it is clear that E e H(G), and armed with the obvious fact that
CB(A) = (blb2b3) x <b4), one easily checks that D normalizes E. Let Z—(bib2bibi')
and У = <6]62, b2b3, h3h4). Then it is easy to verify that
1 < Z <y <B < BE < BEA < G
is a chief series of G. It contains three central chief factors, namely Z/l, B/Y, and
G/BEA, each of order 2. Therefore from (5.7) we conclude that NG(E) = D.
Step 2: Let H = BL; this is certainly a subgroup of G and is, in fact, maximal.
Moreover, E\.H, because ErtH = {1. ВТ, A, H] eH(W). Let N = <Ь,Ь2, b2b3),
U = </il/>2/i1), and V = <h4). Since by inspection L( = <(12), (123))) conjugates the
three non-identity elements of N transitively, it is clear that N, U, and Fare minimal
normal subgroups of H, with N eccentric and U and V central. Thus a chief senes ol
H through В has two central chief factors below В and, because H/B s Sym(3 , also
one above, and each has order 2. Hence by (5.7) the subgroup D of H, which clearly
normalizes H and E, coincides with NH(E n H), a system normalizer о
240
I. Introduction to soluble groups
Step 3: Let D* denote an arbitrary system normalizer of H. By (5.2) we have £>* = [)''
for some h e H, and therefore
(5,/i) D* г. В = Dh r\В — (D c. = (IJ l']>'— IJ V.
Let g = (12)(34), and note that [T, g] = 1. Then Ds = T x <b,b2b3>’ x <b4>® =
T x (,b1b2bi} x <b3> < H. Since £>9nB = </>!Ь2Ь4> x <b3> UV, it follows from
(5./i) that D" normalizes no system of H. Since De is a system normalizer of G
contained in H. we have found the promised answer to Carter’s question. (Alperin
[1] was the first to answer it, but his example is not quite so easy to present as the
one due to Shamash which we have just described.)
Exercises
1. Find system normalizers in each of the following groups: Sym(4), Alt(4), GL(2, 3),
SL(2, 3), Zplireg G (where G is a group whose system normalizers are known and
p e P). Generalise the last example.
2. Provide a counterexample to the following statement: “If D is a system normalizer
and £ a Hall system of a group G, then £ X D if and only if D = 7VG(£).”
3. Let M be a maximal subgroup of a soluble group G.
(a) If M is not normal, a system normalizer of M contains a system normalizer
of G.
(b) If M is normal and if D is a system normalizer of G, then D n M is contained
in a system normalizer of M.
Show that tn both (a) and (b) the stated inclusion can be proper
4. (P. Hall [4]) Let G be a soluble group with abelian Sylow subgroups. Prove that:
(a) Z„(G)nG' = 1;
(b) A system normalizer of G complements G' in G.
(c) A system normalizer of G is self-normalizing if G" — 1, and may or may not
be self-normalizing when G has derived length 3.
5. (Carter [3]) Let D be a system normalizer of G, let D < H < G, and let D* be a
system normalizer of H. Show that:
(a) |D| divides |f)*|;
(b) If D = NC(L>), then D is a system normalizer of H;
(c) If D = Ng(D) and geG, then g e <£>, De).
6. Show that the following statement is false: If D is a system normalizer of G and if
К < G, then Nc/k(DK/K) = NC(D)K/K. (Hint: Use Example 5.11.)
7. The following question was posed by Alperin [1] and resolved negatively by
Shamash [1]: Let D be a system normalizer of G, let D < H < G, and assume that
D normalizes a Hall system of H. Does D normalize a Hall system of G which
reduces into H?
8. (Alperin [1]) Let D be a system normalizer of G, and let n(D) — {Pi, p,}- For
i = 1,..., r, let Pj be a Sylow p,-subgroup of some system normalizer of G. If
D* = PtP2... Pr is a subgroup of G, then D* is a system normalizer of G.
6. Pronormal subgroups
241
6. Pronormal subgroups
The last two sections of this introductory chapter on soluble groups are devoted to
various embedding properties of subgroups. The concept of pronormality is one of
the most important of these and was first introduced by P. Hall in his lectures in
Cambridge.
(6.1) Definition. Let G be a group and U < G. Then U is said to be pronormal in G
U рГ G) if’for each 9 eC’the subgroups U and L™ are conjugate in their join
One reason for the importance of pronormality, as we shall see in the sequel, is
that one is frequently concerned with a situation where each group G in a certain
universe possesses a specified set .Г(С) of subgroups fulfilling the following two
requirements:
(i) ^(G) is a conjugacy class of G, and
(ii) If T e .:T(G) and T < S < G, then T e T(S).
(The set T(G) = Sylp(G) is obviously an example of such a set for the universe of all
finite groups.) The members of FIG) are then pronormal subgroups of G. For, if
U e 5"(G) and geG, then Ue e r(G) by (i), and so if J denotes (C, l/9>, it follows
from (ii) that U and Us belong to T(J). But then, by (i) once more, U and Vя are
conjugate in J.
(6.2) Illustrations, (a) A normal subgroup of a group is obviously pronormal.
(b) A Hall subgroup of a soluble normal subgroup N of a group G is a pronormal
subgroup of G. Let H e HallJA) and geG. Then clearly Ня e Hall„(A), whence H
and Ha are both Hall л-subgroups of their join J = (W, H’>. Since J is soluble, by
(3.3)(b) the subgroups H and He are conjugate in J.
(c) A maximal subgroup is a pronormal subgroup. For, if M <• G and M G, then
(M, M") = G when g i M, and so g always belongs to <M, Msf If pronormality
were a transitive embedding property, if, in other words, one could infer from U pr
V pr W that U pr W, it would follow that every subgroup of a group is a pronormal
subgroup. But this is not the case, witness the subgroup <((12)(34)> of Alt(4).
For much of this section we restrict the universe to finite soluble groups and leave
the reader to decide which of the results are true more generally in the category of
all finite groups.
(6.3) Lemma. Let U be a pronormal subgroup of a group G.
(a) If U < L< G, then U pr L;
(b) If U < К G, then G = NG(U)K;in other words, the Frattini argument applies
to pronormal subgroups; „nnv-
(c) IfK< G,then UК prG; furthermore, UK/Kpr G/K,and NG(UK} = NG(L)K,
(d) If U sn G, then U < G;
(e) Ng( U) is both pronormal and self-normalizing in G.
Proof, (a) This is obvious. . -i
(b) Let g e G. Then V = Ux for some x e <U, U’> < К and consequently gx e
Ng(U) with x e K. Therefore g e NG(U}K, and the result follows.
242
I. Introduction to soluble groups
(c) If g e G, then U9 = Ux with x e J = (U, U9>. Thus (UK)3 = U9K = UXK =
(UK)X, and xeJK = CUKfUKff Consequently UK pt G, and it follows directly
that UK/K pr G/K. Finally, let N = NG(UK). Clearly we have NG(U)K < N. Since
N = Nn(U)K by Part (b), it follows that N = NG(U)K.
(d) By hypothesis there exists a series of the form
U = Uo < 17, < < U„ = G.
We prove by induction on n that U < G, noting that this is certainly true when n = 1.
Since U pr 17„_,, by induction we have U„_t < NG(U). An application of Part (b) with
U„_t as К then yields the equation G = Nc(U)U„_t = NC(U), as required.
(e) Let g e G and J = <17, l/®>. Then U9 = Ux with x eJ, and therefore NC(U)9 —
NC(U9) = NG(UX) = NG(U)X. Since x e J < <NO(U), Nc(Uf), it follows that NG(U) pr
G. Because U sn NG(NG(U)), Parts (a) and (b) imply that U is normal in NG(NG(U)),
which therefore coincides with blG(U). □
The next result gives a test for pronormality which is particularly useful when an
inductive argument involving quotient groups is applied.
(6.4) Proposition (Gaschiitz—unpublished). If U < G and К <G, the following two
statements are equivalent'.
(a) U pr G;
(b) U pr NC(UK) and UK pr G.
Proof. From (6.3), (a) and (c), it is clear that Statement (a) implies (b). Therefore
suppose that Statement (b) holds. Let geG, and set J = <1/, U9). Since
<(7K,((7K)9) = KJ, and since by supposition UK pr G, we have (UK)9 = (UK)kx for
some ke К and x e J. Therefore gx~' e blG(UK). Because U pr NG(UK), it follows
that U9X ' = Uy for some у e <C, U9X '>. But since x e J, we have U9X ' < J and
therefore у e J. Thus U9 = U,x and yx e J. Consequently U pr G. □
(6.5) Lemma. Let M be a maximal subgroup of a soluble group G. Let p be the
prime dividing |G: M), and let S e Hallp (Af). Then the two following statements are
equivalent:
(a) M is not normal in G;
(b) Ng(S) < M.
Proof. Let N = NC(S). If M <! G, by the Frattini argument we have G = NM and
therefore N £M. This proves that Statement (b) implies (a). Now suppose that
Statement (a) holds, and let К = CoreG(M). Proceeding by induction on |G|, we note
that M/K is a non-normal maximal subgroup of G/K and that SK/K e Hallp.(M/K);
therefore, if К > I, we have
Ng(S)K/K < NG(SK/K) < M/K,
which yields the desired conclusion. Therefore we may suppose that К = I, hence
6. Pronormal subgroups
243
is primitive and- in particular, that the subgroup A = F(G) coincides with
Op(G) and >s complemented by M in G. Let R = Op.(M). Since O„(M) = 1 we Ьа‘е
R -A 1 and hence M - NG(R). Obviously we also have R < S. Therefore R = S n RA
and because RA < G, we conclude that NC(S) < NC(S r RA]-- NG(R) = m. □
Criterion (b) of the next result shows that there is a natural way of associating with
a given Hall system Z of G a unique member of each conjugacy class of pronormal
subgroups of G.
(6.6) Theorem (Mann [3]). Let U be a subgroup of a group G. Then the following
statements are equivalent in pairs:
(a) U pr G:
(b) Each Hall system Z of G reduces into exactly one conjugate of U;
(c) If geG and if E is a Hall system of G such that Z and T? reduce into U then
geNG(U).
(Remark. From the results of Section 4 it is clear that a similar theorem holds when
“Hall system” is replaced in Statements (b) and (c) by either “complement basis” or
“Sylow basis”.)
Proof (a) => (b): We shall proceed by induction on |G|. By (4.16) a given Hall system
Z of G certainly reduces into some conjugate of U and this we may take to be U itself.
Suppose that Z also reduces into I/9. Let N < G, and denote images under the natural
homomorphism G -» G/N = G with bars. By (4.17) and (6.3)(c) we havcZ \ U pr G
and therefore U = Ue by induction; in other words U’N = UN. Let L = UN. Since
<C'9, Uj < L, we have Ue = U‘ for some I e L. By (4.21) Z\N. Therefore by
(4.22) (b) the set Z n L is a Hall system of L and clearly reduces into U and U'. If
L < G, by induction we have U = U‘ = Ue, as required. Therefore we can suppose
that UN = G for all N < G. Hence either U = G, or CoreG(l/) = 1 and G is a
primitive group with stabilizer U. If U = G or U = 1, the desired conclusion holds.
Therefore suppose that U is a non-normal maximal subgroup of G, of p-power index
say, and let S e Z n Hallp.(G). By (4.20) we have S < U n I/9, whence S and S’ ' are
Sylow p-complements of U. It follows that Ss — S“ for some u e U and hence that
ug e Na(S) < U by (6.5). Consequently g e U and U = U", as required.
(b) => (c): If Z and Z9 reduce into U, then Z9 reduces into U and U". Hence by
assumption we have U = I/9, and Assertion (c) follows.
(c) => (a): Let g e G, and let Zo be a Hall system of U. Extend Zo to a Hall system
Zt of J = <1/, l/9>, and extend Z, in turn to a Hall system Z of G. By (4.16) there
exists an x e J such that ZJ \ I/9. Then Z* \ I/9, and so Z*9 ‘ \ U. Since by definition
Z \ U, we conclude that xg e NG(U) from our supposition that (c) is true. Therefore
U» = Ux with x e <1/, C9> and U is pronormal in G. □
(6.7) Corollary. Let U be a system permutable, pronormal subgroup of G. If a Hall
system L reduces into U. then U permutes with E.
244
I. Introduction to soluble groups
Proof. By hypothesis there exists a Hall system Xo of G such that U L Xo. and then
by (4.28) we have Xo \ U. It then follows from (4.11) and (6.6) that X = Xg for some
g e NG(U). Since U = I/® and Vе 1 Xg, we have ULI. □
(6.8) Proposition (Lockett [1]). Let X be a Hall system of a group G, and let H be a
pronormal subgroup into which X reduces. Then X reduces into NG(H}; furthermore,
NC(X) < NC(H).
Proof. By (4.16) there exists an element geG such that X reduces into NC(H)®. Since
He < NG(H®) = NG(H)®, by (4.21) the system X reduces into He. From (6.6) we then
conclude that g e NG(H), whence X reduces into NG(H)® = NG(H).
Finally, if d e WG(X), then X = X1* reduces into Hd, and therefore d e blG(H) by (6.6)
again. □
If H and L are pronormal subgroups of a group G, it does not in general follow
that H n Lis pronormal in G, even if H and L permute; nor is it the case that <H, Lj
is necessarily pronormal in G. (See Exercises 1 and 2 below.) However, <H, L) is
indeed pronormal in G if there exists a Hall system X of G reducing into both H and
K. This is a consequence of the following important result.
(6.9) Theorem (Fischer—unpublished). Let X be a Hall system of a group G, and let
[H/. 2 e A} be a set of pronormal subgroups of G such that X \ for all 2 e A. Let
if denote the set of all subgroups of the form
Then the minimal elements of LT, partially ordered by inclusion, form a conjugacy class
of pronormal subgroups of G', furthermore, the join {H,'. 2 e A> is the unique member
of this class into which X reduces and, in particular, is pronormal.
Proof Since G is finite, we may suppose without loss of generality that A =
{l,2,...,n} for some neN. Let ./ = <//,, H2,..., H„>. Let LeSf say L =
<H®',..., H®">. By (4.16) there is an element geG such that X® \ L. We assert that
(6.a)
ifX®\L, then J” < L.
Let i e {1, 2,.... n}. Again by (4.16) there exists an /, e L such that the Hall system
X® n L of L reduces into He‘‘‘, and in consequence X® reduces into both H? and Hf',‘.
Since W, is pronormal, by (6.6) we have Hf = Щ'1' < L, and therefore
J® = <H?,..., H»> = <Нр'-,..., < L,
as desired. Thus we have shown that each member of the set У contains some
conjugate of J, and since the conjugates of J themselves belong to У, it follows that
these are precisely the minimal elements of
6. Pronormal subgroups
245
If Z \ J", then L9 \ J, and it follows from (6.a) that J" '< J. Therefore J — Je
rnd T0 cxaclly one conjugate of J, namely J itself. It therefore follows
from (6.6) that J pr G. ™
Another sufficient condition for the pronormality of a join of two pronormal
subgroups U and V is that U permutes with V.
(6.10) Theorem. Let U and V be pronormal subgroups of a soluble group G such that
UV = VU. Then UV is pronormal in G.
Proof. By (4.16) the group UV has a Hall system Z reducing into U. By the same
result wc know that E* \ V for some x e UV. Let x = uv with ueU and г e V. Then
Z" \ F" = V and £“ \ U“ = U. If E* is an extension of E“ to G, we have E* \ U
and E* \ V, and it follows from (6.9) that UV (= <C, F>) is pronormal in G. □
(6.11) Lemma. Let T. be a Hall system of G, and let U and V be pronormal subgroups
of G into which E reduces. If there exist x, yeG such that UxVy = VyUx, then
UxVy = (UVy for some g e G, and UV = VU.
Proof. As in the proof of the preceding Theorem 6.10, a Hall system E9 of G can be
found which reduces into both Ux and FL Since by hypothesis E9 reduces into both
Ue and F9, by (6.6) we have Ue = Ux and F9 = Vy. Thus UxVy = (UVf, and evidently
UV = (Uxl/yf 1 is a subgroup of G. □
(6.12) Theorem (Fischer unpublished). Let Ht,.... H„be pronormal subgroups of a
group G, into each of which a given Hall system E of G reduces. Furthermore, assume
that there exist elements gt, ...,g„e G such that Hf,..., H’* are pairwise permutable.
Let J = <,Н2,...,Н„>, and let L= H<f... H°". Then
(a) J = Lx for some xeG,
(b) J = HlH2...H„, and
(c) J and L are pronormal in G.
Proof. We shall prove all three statements simultaneously by induction on n. They
certainly all hold when n = 1. Let IV = H, H2. By (6.11) we have IF9 = Wj’H92 for
some g e G. Furthermore, IF is a subgroup of G and is pronormal in G by (6.10). Since
the (n - 1) groups in the set T = {IF9, Hf Hf ] are evidently pairwise per-
mutable and because E \ IF by (4.22)(b), we may apply the induction hypothesis
to T. Conclusions (a), (b), and (c) all follow directly, once we observe that J =
<fF, H„..., H„> and that L = IF’H?’... H9". 1=1
(6.13) Definitions, (a) A subgroup U of a group G is said to be locally pronormal in
G if each Sylow subgroup of U is pronormal in G. .
(b) Two subgroups U and V of a group G are said to be locally conjugate in G 1,
for each prime p, a Sylow p-subgroup of U is conjugate in G to some Sylow
P-subgroup of F. (By Sylow’s theorem this definition is independent of the chosen
Sylow subgroups of U.)
246
I. Introduction to soluble groups
(6.14) Theorem (Chambers [1]). If U is a locally pronormal subgroup of a finite
soluble group G, then U is pronormal in G.
Proof. Let Bo be a Sylow basis of U. This generates a Hall system Zo of U, and by
(4.16) this Zo extends to some Hall system Z of G. Then each subgroup in Bo has Z
reducing into it and is pronormal in G by hypothesis. Therefore by (6.9) the subgroup
U = (H: H e Bo> is pronormal in G. □
(6.15) Remarks, (a) Theorem 6.14 is false for insoluble groups. Let К = PSL(2, 7),
the simple group of order 168. Then К possesses two conjugacy classes of subgroups
isomorphic with Sym(4) and has an automorphism a of order 2 which interchanges
these classes (see Huppert [5], II, 8.18 and 8.19). Let G denote the semidirect product
[K] <a>, and let U denote one of the subgroups of К isomorphic with Sym(4). Then
U is not conjugate to C’“ in <1/, = K. However, the Sylow subgroups of U are
Sylow subgroups of К and are therefore pronormal in G. Therefore U is locally
pronormal but not pronormal in G.
(b) A subgroup of Sym(4) isomorphic with Sym(3) (for example, the stabilizer of
the symbol 4) is easily seen to be pronormal but not locally pronormal in Sym(4).
Therefore pronormality is a more general concept than local pronormality within
the class of finite soluble groups.
(6.16) Theorem (Chambers [1]). Let U be a locally pronormal subgroup of a finite
soluble group G, and let V be a subgroup which is locally conjugate to U in G. Then V
is conjugate to (J in G.
Proof. Let Z be a Hall system of G reducing into U, and let Bo = ..., H„} denote
the Sylow basis of In I/. Then, for 1 < i < n, we have Z \ H(, and also H, pr G
because U is locally pronormal in G. Since the subgroup V is locally conjugate
to U in G, it has a Sylow basis of the form {Щ',..., H“"} for suitable elements
3),..., g„eG; in particular, the subgroups Нр,...,Я’" are pairwise permutable.
Therefore Theorem 6.12 is applicable, and from Conclusion (a) of that theorem it
follows at once that U is conjugate to V. □
Remark. The statement of Theorem 6.16 is no longer true if the hypothesis about U
is relaxed from “locally pronormal” to simply “pronormal”. The outline of a counter-
example is given in Exercise 17 below.
(6.17) Proposition (Lockett [2], Lemma 4.9). Let К be a normal subgroup of a finite
soluble group G, and assume that G/K is nilpotent. Let H be a locally pronormal
subgroup of G such that H n К = 1, and let Z be a Hall system of G reducing into H.
Then
П < Ng(T,).
Proof. Let D = NG(Z). The inclusion I) < NG(H) is a direct consequence of (6.8) and
(6.14). To prove that H < D we use induction on |G|. Without loss of generality we
6. Pronormal subgroups
247
may clearly suppose that H is a p-group for some prime p and, in particular that H
is consequently pronormal in G.
First observe that if К = 1, by (5.7) we have D = G. and there is nothing to prove
Next let N = Op.(K) and suppose that N * 1. Clearly HN/NcK/N = 1, and by
(6.3)(c) we have HN/N pr G/N. Furthermore, by (4.13)(a) and (4.17)(a) the set LN/N
is a Hall system of G/N which reduces into HN/N. Hence by induction and (5.8) we
have HN/N < Ncin(LN/N) = DN/N. Since N is a p'-group, H is a Sylow p-subgroup
of HN. If p e £ n Sylp(G), by (5.4)(b) we have P n D e SyI„(D) c Syl„(DN), and there-
fore P n D = P n DN. Because £ \ Я, it follows that H < P r, DN, and so H < D, as
required.
Let R = Op(K). We may now suppose that Op(K) = 1 and hence that R = F(K).
Since HR is a p-group, we have H sn HR, and consequently H < HR by (6.3)(d).
Hence [H, R] < H n R = 1, and so H is a subgroup of the group C = Cc(R). Then
С о К = CK(F(K)) = Z(R) by A, 10.6(a), and so C/Z(R) CK/K, which is nilpotent
by hypothesis. Since [Z(R), C] = 1, it follows that C is a nilpotent normal sub-
group of G containing H. Therefore H sn G. and we have H < G by (6.3)(d) again.
Because G/K is nilpotent, we know that Kr+1(G) < К for some r e M, and therefore
[W, G, .Г., G] < H n К = 1. We conclude that H < Z00(G), which is contained in D
by(5.9)(b). □
Finally we mention briefly the concept of abnormality, an embedding property
closely related to pronormality but with a narrower range of application.
(6.18) Definition. Let H be a subgroup of a finite group G. Then Я is said to be
abnormal in G (written H abn G, or sometimes H x G) if g e (H, H’> for all geG.
(6.19) Illustrations, (a) A maximal subgroup M of G is abnormal in G if and only if
M £ G.
(b) A subnormal subgroup К of G is abnormal if and only if К = G (see (6.2O)(c)).
(c) We shall see in Chapter III that each finite soluble group G has precisely
one conjugacy class of abnormal nilpotent subgroups, namelv the Carter subgroups
of G.
The following observations are immediate consequences of the definitions.
(6.20) Let H be an abnormal subgroup of a group G. Then
(a) H pr G,
(b) H = NG(H), and
(c) if H < L< G, then H abn L and L abn G.
(6.21) Lemma. Let H be a subgroup of G.
(a) If H pr G, then NC(H) abn G;
(b) H abn G if and only if H pr G and H = NC(H);
(c) If £ is a Hall system of G reducing into H and if H abn G, then Nc( ) -
Proof, (a) Let g e G and H pr G. Put N = NC(H) and J = <N, N’>. Then He = H*
for some x e Jo = <H, H’>. Thus gx~' e N, and therefore g e NJ0 < NJ J-
248 I Introduction to soluble groups
(b) This follows from Part (a) in one direction and from (6.20), Parts (a) and (b), in
the other.
(c) In view of Part (b) this is a corollary of (6.8). □
Our last result in this section describes a criterion for abnormality formulated in
terms of the reducibility of Hall systems and therefore analogous to the criteria for
pronormality given in (6.6).
(6.22) Theorem. Let H be a subgroup of a group G, and let £ be a Hall system of G
reducing into H. Then H is abnormal in G if and only if the following two conditions
are satisfied:
(i) NG(£) < H, and
(ii) if £* is a Hall system of G reducing into H, then £*=£*' for some h e H.
Proof. First suppose that H abn G. Condition (i) follows at once from (6.21)(c). To
prove that (ii) holds, we first observe that £* = £e for some g e G by (4.11). Therefore
£ \ H9 ', and we conclude from (6.6) that g e NC(H) because H pr G. By (6.20)(b) we
have NG(H) = H, and so Condition (ii) is established.
To prove their sufficiency, now suppose that Conditions (i) and (ii) are fulfilled by
H. If £ \ H9 for some g e G, then £9 ' \ H, and it follows from Condition (ii) that
£» ' = £’’ for some he H. Thus hg e NG(£) < H by Condition (i), and therefore де H.
In particular, H9 = H, and we have shown that £ reduces into a unique conjugate of
H. Thus H pr G by (6.6). Finally let g e NG(H). Then certainly £ \ H9, and by the
above argument we again have g e H. Hence H is a self-normalizing pronormal
subgroup of G and, as such, is abnormal in G by (6.21) (b). □
Exercises
1. Show that H pr Sym(4)ifand only if )H| e {1, 3,4, 6, 8, 12, 24}. Deduce that each
of the following statements is false:
(a) If U and V are pronormal in their permutable join UV = VU, then U n V
pr UV.
(b) If U char V pr G, then U pr G.
(c) If U pr G and U < V < G, then V pr G. (Compare with the corresponding
situation for abnormal subgroups in (6.20)(c).)
(d) If U < G, let 3?(U) = {H: U pr H < G}, partially ordered by inclusion. Then
the maximal elements of !^(U) belong to a single conjugacy class of G.
(e) If U is a system permutable subgroup of G. then U is pronormal in G.
(f) If U is locally pronormal in G, then U is system permutable in G.
2. Show that there exists a primitive group G = NH with N = Soc(G) of order 49
and with H = Sym(3). If U e Syl3(G), show that G has a subgroup V of order 21
such that V = (Fn N}U. Let 1 n e Vn N. Conclude that
(i) V = <G, [/">, the join of two pronormal subgroups of G, and
(ii) V is not pronormal in G. (Compare with (6.9) and (6.10).)
3. Let U pr G. If UU9 = U9U for some g e G, prove that g e blG(U).
4. Show that U pr G if and only if g e NC(U){U, U9} for all g e G.
6. Pronormal subgroups
249
5. Use the example described in (6.15)(a) to show that Theorem 6.10 fails to hold
when the hypothesis that G is soluble is dropped.
6. Let G = NH be a primitive group for which N = Soc(G) is a p-group and
H n N - 1. Let S e Hallp.( W) and put L = NC(S). Show that L = (Ln N)(Ln H)
7. Let U < К < G. Show that the following statements are equivalent'
(a) U pr G;
(b) U pr К and G = NG(U)K.
8. Let P be a p-subgroup of a group G, and define the following associated sets of
subgroups:
^(P) — {X: P < X and X is a p-subgroup of G}, and
®*(P) = {S: P < S e Syl„(G)}.
Prove the equivalence of the following statements taken in pairs:
(a) P pr G; (b) P < NG(X) for all X e W(P); (с) P < WG(S) for all S e ®*(P).
9. (Fischer—unpublished) Let U < G, and let Z be a Hall system of G which reduces
into U. Define the reducer PG(U) of U of G as follows:
RC(U) = <xeG:Z*\U>.
Prove that:
(i) PG(U) is independent of the choice of L reducing into U.
(ii) The following statements are equivalent in pairs:
(a) U pr G; (b) U pr PG(U); (c) RC(U) = NG(U).
10. (Fischer—unpublished) If U < G, let 2(U) denote the set {F: V < UandFprG},
partially ordered by inclusion. Prove that the maximal elements of i’(L’) form a
single conjugacy class of U.
11. Theorem 4.11 states that G acts transitively by conjugation on the set H(G). If
U < G, let HG(U) denote the set
Hc(U)={ZeH(G):Z\U}.
Let U pr G and put N = NG(U). Show that HG(U) is an orbit of N, in other words,
a transitive constituent of H(G)N.
12. If (J < G, let y»(U, G) denote the set {H: U sn H < G}, partially ordered by
inclusion. If №(U, G) has a unique maximal element, it is called the subnormahzer
of U in G. Show that:
(a) If U pr G, then U has a subnormalizer in G.
(b) In general a subgroup does not possess a subnormalizer.
13. (Wood [3]) Let V be a subgroup of G with the following property:
If V < H < G, then V pr H.
Show that V pr G if and only if NC(H contains a system normalizer of G.
250
1. Introduction to soluble groups
14. (Mann [1]) If U < G, a subgroup К of G is called a strong subnormalizer of U in
G if К is a subnormalizer of U and if |K : 17| = z0(l7) (for the definition of z0, see
Exercise 11 of Section 4 of this chapter). Show that U pr G if and only if NB(U)
is a strong subnormalizer of U in G.
15. Let E denote the non-abelian group of order 27 and exponent 3 (see A, 20.5).
Show that £ possesses an automorphism a of order two such that |[£, a] | = 9.
Let G = [£] <a>, the semidirect product of £ with <a), and let U = Cc(a). Prove
that
(i) U is a cyclic subgroup of order 6,
(ii) U is abnormal in G, and
(iii) U is not locally pronormal in G.
16. (P. Hall, Cambridge lectures) Let U be a subgroup of a finite group G. Show that
the following two conditions are together both necessary and sufficient for U to
be abnormal in G:
(i) Every subgroup of G containing U is self-normalizing;
(ii) U is not contained in two distinct conjugate subgroups of G.
Prove that Condition (i) is already sufficient if G is soluble. (It is unknown whether
Condition (i) alone is also sufficient for arbitrary finite groups.)
17. (Losey and Stonehewer [1]) Let G = GL(2, 3), and recall that G = QU, the
semidirect product of a normal quaternion subgroup Q of order 8 with a sub-
group U isomorphic with Sym(3). Let <t) = T e Syl2( 17) and К e Syl3( 17). Show
that
(a) 17 pr G,
(b) Q has an element x of order 4 such that t~lxt = x3,
(c) 17 is locally conjugate in G to the subgroup V = K<tx2), and
(d) U is not conjugate in G to V
(Thus the hypothesis of local pronormality in (6.16) cannot be weakened to
pronormality.)
18. (Peng [1]) Let G be a soluble group, and let 17 < G. Then the following are
equivalent:
(a) 17 pro G;
(b) Whenever U < N < V < G, then V = NV(U)N.
(It is not known whether the hypothesis of solubility is necessary.)
7. Normally embedded subgroups
In this section we investigate a property called “normal embedding”, a concept due
to Fischer. It is a stronger embedding property than pronormality, and has special
relevance to Chapter IX, where injectors associated with certain Fitting classes are
shown to possess it.
£7.1) Definitions. Let 17 be a subgroup of a finite group G.
| (a) If p is a prime, we say that 17 is p-normally embedded in G if a Sylow p-subgroup
f 17 is a Sylow p-subgroup of some normal subgroup of G, and we write 17 p-ne G.
7. Normally embedded subgroups
251
(b) U is said tc.be normally embedded in G (or sometimes strongly pronormal) if U
is p-normally embedded in G for all primes p. We shall denote this by U ne G A Hall
subgroup of a normal subgroup of G is a typical example of a normally embedded
subgroup of G.
(7.2) Remarks, (a) If U < G and P e Sylp(G), it is clear that P e Syl„(N) for some
N < G if and only if P is a Sylow p-subgroup of its normal closure <PC). This
observation provides an alternative formulation of Definition 7.1(a).
(b) If P e Sylp(N) for some N < G, then P pr G. Hence a normally embedded
subgroup of G is locally pronormal and therefore, if G is soluble, certainly pronormal
in G by (6.14).
(c) In the group G = [PSI.(2, 7)] <a> described in (6.15)(a), the subgroup U is an
example of a normally embedded subgroup of an insoluble group which is not
pronormal.
(d) The subgroup <(1 2 3 4)> of Sym(4) is locally pronormal but not normally
embedded in Sym(4).
We now stipulate that for the rest of this section the hypothesis of solubility will be
tacitly assumed whenever it is required.
(7.3) Lemma. Let V be a p-normally embedded subgroup of a group G. Let К < G
and H < G. Then '.
(a) If U < H, then U p-ne H;
(b) UK p-ne G and UK/K p-ne G/K;
(c) If К < H and H/K p-ne G/K, then H p-ne G;
(d) U n К p-ne G, and if К is a p-group, then U г\ К < G;
(e) U either covers or avoids each p-chief factor of G.
Proof. Let P e Sylp(U) n Sylp(JV), where N < G.
(a) This follows from the observations that N ryH < H and P e Sylp(N n H).
(b) Let P < P* e SyIp(l7K). Since P*K/K contains PK/K e Syl/l/R/R), we have
P*K = PK. Thus | NK:P*K | = |NK:PK| = |JV:P(NnK)|,andthisisap'-number
because P e Sylp(7V). Therefore |JVK : P*| = |NK: P*K||P*K : P*l is also a
p'-number since P* e Sylp(P*R). Hence P* is a Sylow p-subgroup of NK < G, and
consequently UК p-ne G. The rest is now clear.
(c) Let Q e Sylp(H). Then QK/K e Sylp(R/K) for some R < G. Since Q e Sylp(QK),
it follows that |R : Q| = |R : QK\\QK : Q| is a p'-numberand hence that Q e Sylp(R).
(d) By A, 6.4(a) we have P n К e Sylp(G n K) n Syl„(N n K). Hence U n К p-ne G.
If К is a p-group. we have U г. К sn G and U г. К pr G, and so from 6.3(d) we
conclude that U n К < G.
(e) If H/K is a p-chief factor of G, we have H r. UK < G by Parts (b) and (d), an^
the conclusion follows.
(7.4) Lemma. Let H be a normally embedded subgroup of a group G. Let L
subgroup of G, and assume that there is a Hall system XofG reduang tnto both H and
L. Then
252
1. Introduction to soluble groups
(a) Hr L is normally embedded in L, and
(b) n L| < |H n L| for all geG.
Proof. Let p e P, and let P be the Sylow p-subgroup in X. By hypothesis we have
P г\ H e Sylp(H) n Sylp(JV) for some N < G, and therefore Pr.H = P nN. By I,
4.22(a) the subgroup P л (H n L) is a Sylow р-subgroup of Hr.L. However,
P n H n L = P n N r, L, and this is a Sylow p-subgroup of NnL because
PnLe Sylp(L) by hypothesis and Nn L < L. It follows that HnL p-ne L for all
p e IP, and Statement (a) is proved.
For Part (b), let R e Sylp(H“ n L). Since the Sylow p-subgroups of H, and hence
those of He, are contained in N, it follows that Pisa p-subgroup of N n L. But, as
we showed above, PnHnL = PnNnLe Sylp(NnL), and therefore we have
|H’n L|p = |R| < |NnL|p = \PnNnL\ = \PnHnL\ = |HnL|p for all p e IP.
This implies Statement (b). □
(7.5) Proposition (Chambers [ 1 ]). Let U he a normally embedded subgroup of G. Then
(a) U is a CAP subgroup;
(b) If V < G and V covers and avoids the same chief factors of G as U, then U and
V are conjugate in G.
Proof, (a) This follows at once from (7.3) (e).
(b) Let P e Sylp(l7) and P* e Sylp(F). Since P e Sylp(JV), where N < G, it follows
that P covers all p-chief factors of G below N and avoids all factors above N. Since
by hypothesis P* does the same, we infer from A, 3.7 that P* e Sylp(/V) and hence
that P is conjugate to P*. Thus V is locally conjugate to 17. As remarked in (7.2) (b),
the subgroup 17 is locally pronormal in G. Therefore V is conjugate to 17 in G by (6.15).
□
Our next goal is to show that the set of normally embedded subgroups of a group
G into which a given Hall system reduces forms a sublattice of the subgroup lattice
of G. To this end we need the following two lemmas.
(7.6) Lemma (Lockett [1]). Let P, and P2 be subgroups gf a Sylow p-subgroup P of
a group G, and assume that P, and P2 are normally embedded in G. Then P, P2 = P2P2,
and both Pt P2 and P, n P2 are normally embedded in G.
Proof. By hypothesis there exists a normal subgroup of G such that P; e Sy 1р(Л/,).
i = 1,2, and by A, 6.4(a) we have P( = Pn Nt. Also by that result we have P, n P2 =
P n (Nj n N2) e Sylp(/V, n N2), and therefore Pj n P2 ne G. Furthermore, we know
by A, 6.4(b) that (P n Nt)(Pn N2) is a Sylow p-subgroup of Nt N2; thus the product
PjP2 is a subgroup and is also normally embedded in G. □
(7.7) Lemma (Lockett [ 1 ]). Let X be a Hall system of a group G,let p and q be distinct
primes, and let P and Q be respectively a p-subgroup and a q-subgroup of G. Assume
that both P and Q are normally embedded in G, and that X ' P and X Q. Then
PQ = QP, and PQ is normally embedded in G.
7. Normally embedded subgroup!
253
Proof. If n s P, denote the group in X n Hall.(G) by G„. Then the hypotheses imply
' f G-P G P '>Gp *and G SylP (<6C>G, )- Thus the following {p, q}-subgroup
L = GlP.,l^<-PC>Gp.n<G«>G,.,
has order at most |P||Q| — |PQ|. But /.contains PQ because by hypothesis P < G <
G|p.e, n G,- and Q <Gq< G।pq। n Gp.. It follows that L = PQ, and the rest is now
clear. г-,
(7.8) Theorem (Lockett [1 J). Let U and V be normally embedded subgroups of G into
which a given Hall system X of G reduces. Then UV = VU, and UV and U r ,V are
normally embedded subgroups of G into which L reduces.
Proof. Let a(G) = {Pi,...,p„}, and let {1, P1;..., P„} be the Sylow basis of X with
P; e Sylp.(G). For i e {1,..., n} put U, = U n Pf and Ц = V n Pf, so that by hypothesis
the sets Вц = {1, Ut,..., l/„} and BK = {1, Vt,..., t^J are Sylow bases of U and V
respectively. By (7.6) and (7.7) each group in B^ permutes with each group in Br, and
therefore
UV = ( П ut II П vi = П (ВД = TG.
Thus UV is a subgroup of G, moreover, UM e Sylp(I7F), and L'jf' ne G by (7.6).
Therefore UV ne G.
By (4.22)(a) we have Ut n Ц = Pt n (I/ n V) e Sylp (O' n V), and by (7.6) also (7, n V:
ne G. Hence U n V ne G. □
The following theorem is an immediate corollary of (7.8). It is an unpublished result
of B. Fischer and was also proved independently by Lockett [1], Corollary 3.2.6, and
by Ti Yen [2], Theorem 2.
(7.9) Theorem. Let X be a Hall system of a group G. Then the following set of
subgroups
,4i(X) = {17 < G: U ne G and X 17}
forms a lattice whose join and meet operations are respectively “permutable product
and “intersection" of subgroups.
Since X £ , 4i(X), we have the following corollary.
(7.10) Corollary. A normally embedded subgroup of G is system
ticular,. 1 >(X) is a sublattice of .^(X). (The reader is referred to (4.24) and (4.29) for
the context of this result.)
254
I. Introduction to soluble groups
(7.11) Corollary (Schaller [3]). If U and V are normally embedded subgroups of G,
there exists an element geG such that U permutes with Vs.
Proof. If X e H(G) and X \ 17, by (4.16) we have X ч Vе for some geG. Since 17,
Vе e .Л7(Ц the result follows at once from (7.9). □
Next we prove a sequence of criteria for a p-subgroup of a soluble group to be
normally embedded. Criterion (b) is due to Schaller [3]; Criterion (c) is Theorem 1
of Ti Yen [2]; Criteria (d) and (f) are proved by Lockett [1] in Theorem 3.3.4 of his
thesis and are attributed to B. Hartley. Criterion (e) appears as Lemma 3 in Hartley
[1], and finally, Criterion (g) is an unpublished result of B. Fischer.
(7.12) Proposition. Let P be a p-subgroup of a soluble group G. Then the following
statements are equivalent in pairs:
(a) P is normally embedded in G;
(b) P permutes with a Sylow p-complement of G and is normalized by a Sylow p-
subgroup of G (this criterion should be compared with an analogous criterion for
E-permutability stated in (4.27));
(c) P eSy\p((P, P*)) for all g e G;
(d) P is normal in every p-subgroup that contains P, and P satisfies the following
condition:
(7.a) G = KNe(PnK) for all К < G.
(e) P centralizes every p-chief factor that it avoids and satisfies (7.a);
(f) P is pronormal in G and satisfies fl.a);
(g) P is a pronormal CAP subgroup of G.
Proof, (a) =>(b): By (7.10) the normally embedded subgroup P permutes with a
Sylow p-complement. Also, P e Sylp(K) for some К < G. Hence P = P* n К for some
P* e Sylp(G), and then P < P*.
(b) = (c): Let P < P* e Sylp(G), and assume that P permutes with S e Hallp.(G).
Since G = P*S, for g e G we may write g = xy with x e P* and у 6 S. Then P9 =
Py < PS. Since P is clearly a Sylow p-subgroup of PS, it is certainly a Sylow
p-subgroup of the subgroup <P, P9) of PS.
(c)=>(d): If P is contained in a p-subgroup P*, and if x e P*, then <P, P*> is a
p-group, and by assumption P = <P, P*>. Therefore P = P\ whence P < P*. Next
let К <1 G, let g e G, and put L = <P, P9 ). Since P e Sylp(L) and L n К < L, we have
Pr>K = PnLnKeSylp(LnK).
Similarly (P n K)9 = P° n К e Syl„(L n K). Therefore (P n K)e = (P n Kf for some
x e К ri L, and consequently gx~l e NG(P гу K). Hence g e NC(P n K)K, and it fol-
lows that (7.a) is fulfilled.
(d) =>(e): Let H/K be a p-chief factor of G avoided by P. Let P* be a Sylow
p-subgroup of PH containing P. Since PH/K is a p-group, we have PH = P*K and
7. Normally embedded subgroup;
255
therefore PK < PH because P <g P* by supposition. Therefore [P, Hl < ГРК Hl <
PK с. H < К because P avoids H/K. In other words, P centralizes H/K
(e) =>(f): We prove by induction on |G| that if (e) holds, then P is pronormal in G
ft! ’ ‘ K/A ~ G/A' (PA/A} П (К/Л) = <P n K)A^ which is nor-
malized by Ne(Pn K)A/A. Therefore (7.a) holds for the p-subgroup PA/A of G/A
and clearly so does the rest of Statement (e>. Therefore, if g e G and J = <P, p»> by
induction we have ’ ’ 3
VP)
PeA = PXA
for some x e J. If A is a p'-group, we have P9, Px e Syl„(P’/1), and by Sylow’s theorem
P3 = (Px)r for some у e <P9, P*> < J. Thus P9 is conjugate to P in J.
Therefore we can suppose that 0p.(G) = 1. Then F(G) = 0p(G), and by A, 10.6(c)
the section F(G)/4>(G) is the self-centralizing socle of G/4>(G). If P n F(G) = 1, then P
avoids, and hence centralizes, the chief factors of G between F(G) and 0>(G), and so
P < Cc(F(G)/4>(G)) = F(G). Then P = 1, which is certainly pronormal. Therefore
suppose that the subgroup Po = PnF(G) is nontrivial, and let R = <Pq > < F(G).
Then Pn R < PnF(G) = Po, and consequently P n R = Po. By assumption (7.a) we
have
G = RJVc(PnR) = Nc(P0)R,
and therefore R = <P„ ). Since R is a p-group, we conclude that Po = R < G. In this
case we can take A < Po < P, and the induction step (7./i) yields P’ = PeA = PXA =
Px, as required.
(f) => (g): Let H/K be a chief factor of G. Since H/K is abelian, H normalizes
K(P n H), and clearly so does NG(P n H). Therefore К (P n H) < HNe(P n H) = G
by (7.a). The definition of a chief factor then forces K(P n H) = H or K. Thus P has
the cover-avoidance property (see A, 10.8).
(g)=>(a): We prove this implication by induction on |G|. Let К — <P°}, and let
N-< G. By (6.3)(c) the quotient PN/N is a pronormal subgroup of G/N and is
obviously also a CAP subgroup of G/N. Then by induction PN/N is a Sylow
p-subgroup of <(P/V//V)<i"'i> = KN/N, and it is easy to check that P e Sylp(K) if one
of the following situations occurs:
(i) N is a p'-group;
(ii) No К = I;
Therefore suppose that OP (G) = 1; that N < K; and that N £ P, in which case
PciN = 1 by hypothesis. Since P pr PN, which is a p-group, we have P < PN
by (6.3)(d) and therefore [P, N] < Pn N = 1. Since Cc(lV) < G, we then conclude
that К < Cr(N). The Sylow p-subgroup PN of К splits over N, and there ore у
Gaschiitz’s Theorem A, 11.1 the group К splits over N. Thus К = H x N for-some
H < K, and in particular K' n N = 1. Our suppositions lead to this conclusion
256
I. Introduction to soluble groups
all N < G. Since K' char К and hence K' < G, it then follows that K' = 1. Therefore
P sn G. and by (6.3)(d) we have P < G; in particular P ne G. □
As a corollary of (7.12) we obtain the following
(7.13) Theorem (Fischer—unpublished). Let U be a subgroup of a soluble group G.
The following statements are equivalent in pairs:
(a) U is normally embedded in G.
(b) V is a locally pronormal CAP subgroup of G;
(c) U is locally pronormal and system permutable in G.
Proof. It is clear from (7.2)(b) and (7.3)(e) that Statement (a) implies (b). That
Statement (b) implies (a) follows from the implication: (g) => (a) of (7.12).
The implication (a)=>(c) is contained in (7.2)(b) and (7.10) Finally, to see that
(c)=>(a), observe that a Sylow p-subgroup P of U permutes with a Sylow
p-complement of G. Also because P pr P* for some P* e Sylp(G), we have P < P* by
(6.3)(d). Hence P satisfies Statement (b) of (7.12), and therefore Statement (a) of this
theorem holds. □
We conclude this section with an example of a group G having a CAP subgroup 17,
which is pronormal, but not system permutable, in G, and hence by (7.10) not
normally embedded in G. This example shows that the implication ‘(g) => (a)’ of (7.12)
is not valid without the assumption that P is a р-subgroup, and also that the word
‘locally’ cannot be omitted from (7.13)(b).
(7.14) Example. Let T be an extraspecial group of order 27 and exponent 3. By A,
20.11 the group Aut(T) contains an element a of order 8 such that, if Z(T) = <z),
then z“ = z-1. Put A = <a>, and let H denote the semidirect product
H = [Г)Л.
Let [> = a2, and put В = and L = ТВ. Then clearly L = CH(Z(T)). We shall
proceed with the construction in steps, the first goal being to classify the faithful
irreducible modules for L over a suitable field K.
Step 1: Let К be a field containing a primitive 24th. root of unity, t, say, and let
co = £8. Let 2 be the faithful linear representation of Z(T) defined by
2: z1 -»co1, i = 1, 2, 3.
By B, 9.16 the extraspecial group T has a faithful simple module over К uniquely
determined by the condition that the composition factors of (Ул)г(Г) afford 2; further-
more, the module has dimension 3 over К and is absolutely irreducible. Hence by
B, 7.12 (see also Remark (a) after the statement of B, 9.18) there exists a module Ft
which extends F< to KL, and by B, 9.18 there exists a <5 = + 1 and a 1-dimensional
КВ-module У such that symbolically we have
257
7. Normally embedded subgroups
(PjaS KBQSY.
Since F has dimension 3 and KB dimension 4, we conclude that 6 =
that
and hence
(F)„@ YSKB.
Let <7 = f6, and let Y, denote the 1-dimensional KL-module such that (i)
T < Ker(L on Y() and (ii) (YJg affords the linear representation v, of В determined by
MF) = <Л
for i = 0, 1,2, 3. Let V? = F, ®k Yt. Since (Yf)T = KT, we have V? (<<), = V2,
hence, in particular, each is absolutely irreducible. Choose Ye {Yo,..., Y3} such
that Y®K Y= Yo = KL. Then, by replacing by Yj®K Y if necessary, we may
assume that (Fj)„ © KB = KB and hence that
(7.У)
i = 0, 1,2,3.
Thus we have obtained four pairwise non-isomorphic, faithful, absolutely irre-
ducible modules for L over K. By substituting ц = A-1, we similarly obtain a further
four such modules i = 0,..., 3}. Since Z(T) is the unique minimal normal
subgroup of L, there exists a bijection between the isomorphism classes of irreducible
modules for L/Z(T) and those for L which are not faithful. Therefore, if d denotes the
sums of the squares of the degrees of the faithful irreducible representations of L, by
the formula derived from the Wedderburn-Artin Theorem B, 4.4(c) we have
d = |L| - |L/Z(T)| = 8-32.
Since £?=0 [Dim(FAw)2 + Dim^10)2] = 8 - 32, it follows that {Ия<0, Г„<п}?=0 is a com-
plete set of representatives for the faithful irreducible modules for L over K.
Step 2: Let W and W' be irreducible КН-modules, faithful for H. Our aim in this
step is to show that WA (and similarly WA) is a sum of six pairwise-distinct L
dimensional К A-modules. Since К certainly contains a primitive 8th root of unity,
the regular module KA contains eight pairwise-distinct direct summands, and from
this it will follow that WA and have at least four summands in common. Let
wZ{T) = w1
where the W are homogeneous components. Since Z(T) has just two faithful irre-
ducible representations over K. we have I < 2, and without loss of generality we can
suppose that the composition factors of Wt afford the representation 2 of Z(T) U
iv e IV,, we have (wa)z = w(aza'‘)« = “ '('«<), and so Wta affords copies of e
representation p. Thus t = 2 and W2 = Furthermore, by B, 9.15(b) the inertia
subgroup fc( Wt) of is CH(z) = L. By Clifford’s Theorem B. 7.3 the component ,
258
I. Introduction to soluble groups
is a simple К Л-module, clearly faithful for L, and, furthermore, W s (И7!)". But by
Step 1 the simple KL-module IT, is isomorphic with Vj" for some i e {0, 1, 2, 3}, and
so by Mackey’s Theorem B, 6.20 we have = ((IT])ьл)л = ((IT,)в)л. Hence from
(7.y) we obtain
WA © ((Х)в)л = ((K“’)b © (Х)»)л S (KB)A = KA.
From the fact that (X)b's 1-dimensional, it is easy to see that ((У()в)л is the sum of
the two irreducible (1-dimensional) КЛ-modules whose restriction to В is (F)n,
whence we conclude that WA is the sum of the remaining six irreducible КЛ-modules,
thus completing Step 2.
Step 3: Now let p be a prime such that 24 |(p — 1) (e.g. p = 73) and set К = Fp. Fix
the notation
W = (Ff01)",
and recall from the preceding step that IT is a 6-dimensional irreducible KH-module
such that WA © (Кв)л = KA; in particular, we have
(7.5) Сж(4) = 0.
Form the Hartley group P = H(W. IT), and recall from B, 12.17 and the remark
following B, 12.18 that H acts as a group of operators on P in such a way that
P/P' = IT © W and P' = W ® IT Let we Ikj, a homogeneous component of ITZ(r),
and let wz = 2(z)w. Then (w ® w)z = A(z)2(w ® w) = p(z)(w ® w), and so the H-
invariant subgroup Cr(Z(T)) is a proper subgroup of P'. Let R be a maximal
Я-invariant subgroup of P' containing CP(Z(T)), and set Q = P/R and X = P'/R. By
A, 12.1 we have CX(Z(T)) = 0, and therefore X, viewed as a KH-module, is both
simple and faithful for H. Evidently H acts as a group of automorphisms of Q in such
a way that Q/Q = IT @ IT and Q г X. Form the semidirect product
G = [Q]H.
Step 4: Next we show that Q has an Л-invariant abelian subgroup E with the
following properties:
(7.e)
'(а)
- (b) EQ/Q' is a chief factor of G, isomorphic as KH-module with IT;
(c) Tdoes not normalize E.
Let D be an H-invariant subgroup of Q containing X such that D/C maps to a
submodule IT under the natural homomorphism Q -» Q/X. The structure of H(IT, IT)
implies that D is elementary abelian, and viewing D as a KH-module, by Maschke’s
Theorem A, 11.5 we can find a KH-isomorphism
259
7. Normally embedded subgroups
ф-. D -> W © X.
By Step 2, the modules and X certainly have a common composition factor, and
appealing again to Maschke s theorem, we can find decompositions
Wa — and
XA — (n2) © M2
such that the map ni -» n2 extends to а К/1-isomorphism between the 1-dimensional
КЛ-modules Oh) and Oh)- Let J denote the submodule
J = Oh + «2> ©Mi
of (IV © X)A, and let £ = Obviously we have IT© X = J © X, and it is at
once clear that £ is an Л-invariant abelian subgroup satisfying (a) and (b) of (7.r).
Suppose that T normalizes £. Then J = ф(Е) is a T-submodule of W@ X, and
therefore M, = J о W is a submodule of WT in this case. But IVT is isomorphic with
кд © ЬУ Step 2 and therefore has only two proper non-trivial submodules, each
of dimension 3. Since Dimj-fMJ = 5, we have a contradiction, and so finally we
conclude that (7.r)(c) holds.
Step 5: Let U = EA. The last step is to show that U is a pronormal САР-subgroup of
G which is not system permutable.
First we verify the pronormality. It is clear that Q'U is pronormal in G because it
is the product of the normal subgroup Q'E with a Sylow 2-subgroup A of G. Therefore
by (6.4) it is enough to show that V is pronormal in NG(Q'EA). Since Q'E is a normal
2-complement of Q'EA, we have NG(Q'EA} = Q'ENe(A). Moreover, because A has a
normal complement QT in G, we may write А'(,(Л) = ЛС0Т(Л). Now A acts fixed-
point-freely on each factor of the partial chief series
Q E < Q < Qzm < QT
of G; this is because T/Z(T) and Z(T) are non-trivial irreducible F3 Л-modules, and
because Q/Q'E S ITand CW(A) = 0 by (7.<5). Since (|A|, IQT|) = 1, we conclude from
A, 12.1 that Cqt(A) < Q'E and therefore that Ne(Q'EA) = Q'EA. Since £ о Q'EAand
A e Syl 2 (Q‘ £Л), we have EA pr Q'EA. It is clear at last that V is pronormal in
Ne(Q'EA\ and the assertion that V pr G is proved. Moreover, because (i) G has a
unique minimal normal subgroup Q' and Q' n V = 1, (ii) Q'E/Q < G/6, and (iii)
A e Syl2(G), it is evident that U is also a cover-avoidance subgroup of G.
Finally it remains to show that V is not system permutable in G. Suppose it
were the case that U T = TU. Then (£Л) T = E(A T) would be a subgroup of G, and
we could deduce that E = (EA T) n Q < EA T, which would violate (7.r.)(c). Hence U
does not permute with the Sylow system X whose Sylow basis is {1, Q. T, A}. Since
U pr G and clearly X ч V, it follows from (6.7) that U is not system permutable
in G. U
260
I. Introduction to soluble groups
Exercises
1. Show that each of the following statements is false:
(a) If U pr G and U satisfies Condition (7.ct), then U is locally pronormal in G.
(b) If U is a CAP p-subgroup of G normalized by a Sylow p-subgroup of G, then
U is normally embedded in G.
(c) If U is a CAP p-subgroup of G which centralizes every p-chief factor that it
avoids, then U is normally embedded in C.
2. Let S = Sym(4), and let V be an irreducible FjS-module faithful for S. Let
Ге Syl2(S). Then VT = V, ф P2 with f< irreducible and Dim(j (L<) = i. Set C =
[F]S, and let C = f'2T < G. Show that U is a pronormal, system permutable
subgroup of G which is not a CA P-subgroup, and hence, in particular, not
normally embedded. (Thus the word ‘locally’ cannot be omitted from Assertion
(c) of Theorem 7.13.)
3. (Schaller [1]) Let U < C.Then U ne G if and only if the following two conditions
are satisfied:
(a) U satisfies condition (7,a);
(b) For all peP and R e Sylp(C), there exists a P e Sylp(G) such that R <
Pn(Rr') (see A, 14.13(a) for notation).
4. (Fischer—unpublished) Let u(G) = {pj,..., p„}, and let / be a map
f: {рп ..., p„} -» {A: N < G}. Let f< e 5у1р.(/(р;)) for i = 1, ...,n. Then the
minimal elements of the set
partially ordered by inclusion, are normally embedded in G. Every normally
embedded subgroup of G arises in this way. (Hint: Use (7.8).)
5. (Schaller [3]) Let U be a locally pronormal subgroup of a soluble group G.
Assume that the Sylow subgroups of U are cyclic and that either
(a) U has odd order, or
(b) G has no section isomorphic with Sym(4).
Then U ne G.
6. (Wood [3]) Let ре P, and let G be a soluble group. Show that the following
statements are equivalent in pairs:
(a) The maximal subgroups of G are p-normally embedded;
(b) The Sylow p-subgroups of the maximal subgroups of G are pronormal in G:
(c) G has p-length at most one.
Prove that if all the pronormal subgroups of a soluble group G are locally
pronormal, then G has p-length at most one for all ре P. Finally, investigate
which of the preceding assertions are false without the assumption of solubility.
7. (Wood [1]) Show that the statement
“A subgroup U of a group G is pronormal if and only if it is normally embedded”.
is true in each of the following situations:
(a) U is a p-subgroup and G is p-soluble of p-length at most one;
(b) G is metabelian and has abelian Sylow subgroups.
7. Normally embedded subgroups
261
Describe a metabelian group G which has a pronormal subgroup U that is not
normally embedded in G. {Note-. Metabelian groups have p-length at most 1 for
all primes p.)
8. Show that the following statements about a soluble group G are equivalent in
pairs:
(a) All subgroups of G are pronormal in G;
(b) All subnormal subgroups of G are normal in G;
(c) If N is the smallest normal subgroup of G with G/N nilpotent, then
(i) N is an abelian group of odd order and G/N is hamiltonian,
(ii) (|G:N|,|N|) = 1, and
(iii) for all g e G and all x e N there exists a natural number n(g) such that
x« = x"<9>;
(d) All subgroups of G are normally embedded in G.
(A hamiltonian group is a non-abelian group whose subgroups are all normal;
these groups turn out to have the form Q x A x B, where Q is a quaternion group
of order eight, A is abelian of odd order and В has exponent at most two—see
Huppert [5], Satz HI, 7.12.)
9. Let G be a soluble group such that all its subgroups are either subnormal or
pronormal. Show that the following statements hold:
(a) G has p-length at most one for allpe P;
(b) If P e Sylp(G), then P/Op(G) is hamiltonian;
(c) If the quotient G/F(G) has odd order, then its Sylow subgroups are abelian;
(d) G/F(G) has the properties described in Exercise 8(a)-(d);
(e) If R is a p-subgroup of G such that R i Op(G). then R n Op(G)a G.
10. (Doerk). A subgroup U of a group G is said to be subnormal/}’ embedded in G
(written U se G) if each Sylow subgroup of U is a Sylow subgroup of some
subnormal subgroup of G.
If U is a pronormal subgroup of G, show that its subnormal hull У is
normal in G. Deduce that U ne G if and only if (i) U se G and (ii) U is locally
pronormal in G. (Compare with Theorem 7.13.)
Chapter II
Classes of groups and closure operations
1. Classes of groups and closure operations
We often need to discuss collections of groups distinguished by some special property,
for example abelian groups defined by the additional axiom of commutativity. Since
set theory does not admit “all groups with property г?” as a set, we use the word
“class” instead.
(1.1) Definitions. A class of groups is a collection X of groups with the property that
if G e X and if H = G, then H eX. The isomorphism class (G) of a group G consists
of all groups isomorphic with G. We will often use the term X-group to describe a
group belonging to X
Notation. With the exception of the symbol 0 (denoting the empty class of groups),
we will always use the Fraktur (Gothic) font when a single capital letter is used to
denote a class of groups. If if is a set of groups, we use (,'/’) to denote the smallest
class of groups containing У, and when У = {G}, a singleton, we write (G) rather than
({G}).
We will reserve fixed Fraktur letters for certain frequently cited classes; among
these are the following:
0 denotes the empty class of groups;
91 denotes the class of finite abelian groups;
91 (91J denotes the class of finite nilpotent groups (of nilpotency class at most c);
U denotes the class of finite supersoluble groups;
S denotes the class of finite soluble groups;
ф denotes the class of all primitive groups in the universe under consideration;
93" denotes the class of all groups G in with Soc(G) a л-group;
3 denotes the class of all finite simple groups;
G denotes the class of all finite groups.
Although some authors (e.g. Hall and Hartley [1]) require that a class of groups
contain groups of order 1, we make no such proviso, as is clear from the examples
0 and 4$ just defined. If X is any class of groups and л a set of primes, we will use
X„ to denote the class of all groups in X whose orders involve only primes in л, and
if л = {pj, we write Xp rather than Xlp|; in particular therefore, Gr = lZr is the class
of finite p-groups, and Sp. is the class of soluble groups whose orders are prime to p.
1. Classes of groups and closure operations 263
(1.2) Definitions. Let C be a finite group and X a class of groups
(a) We define '
<r(C)= {p-.pe Pandp||G|}, and
<r(3E) = U {<r(X): X e £}.
(b) We also define
Char(JE) = {p: p e P and Zp e JE}, and call Char(iE) the characteristic of 1.
Clearly Char(JE) £ <r(JE). If £>" denotes the class of л-perfect groups in some universe
® thus:
£1" = (G e ®: 0”(G) = G),
then Char(G₽ ) = p, whereas rr(©₽ ) = P when ® is large enough (for example when
e s «).
(1.3) Definitions (Products of classes). IfiE and?) are classes of groups, we define their
class product .t?) as follows:
X?) = (G: G has a normal subgroup N e 1 with G/N e ?)).
If X = 0 or ?) = 0, we have the obvious interpretation I?) = 0. For powers of a
class, we set JE° = (1), and for n e M make the inductive definition
F = (I”-’)X
A group in JE2 is sometimes called for example, groups with abelian derived
groups are called metabelian, and a group G is said to be metanilpotent if G/F(G) is
nilpotent.
This product of classes is not associative; in fact, there are easy examples to show
that in general (JEJE)JE 3E(3E3E) (see Exercise 1.2). However, it is obvious that
(La) *(?)3) S (*2))3.
and in (1.10) below we give a sufficient condition for equality to hold. In subsequent
chapters we shall define other types of products for classes of groups, which will have
special relevance to the study of formations and Fitting classes, and which will be
distinguished from the class product by special notation (see IV, 1.7, [formation
product] and IX, 1.10 [Fitting class product]).
A map which sends a class of groups to a class of groups will be called a class map,
among class maps are the so-called closure operations, which play an important ro e
in studying properties of group classes.
264
II. Classes of groups and closure operations
(1.4) Definitions, (a) A class map c is called a closure operation if for all classes I
and 9) the following three conditions are satisfied:
CO1: X £ cX (we say c is expanding);
CO2: cX = c(cX) (we say c is idempotent);
CO3: If X S '!), then cX S c'l) (we say C is monotonic).
(b) A class X is said to be c-closed if X = cX. (If c is a closure operation, it is clear
from CO2 that c'l) is c-closed for any class 9).) We adopt the convention that the
empty class 0 is c-closed for every closure operation c.
(c) The product ab of two class maps is defined by composition thus:
(ab)X = a(bX)
for all classes X.
(1.5) The following list contains some of the most frequently used closure operations.
For a class X of groups we define:
sX = (G: G < H for some H e X);
qX = (G: 3H g X and an epimorphism from H onto G);
s„X = (G: G sn H for some H e X);
R0X = (G: 3Nf < G (i = 1,..., r) with G/7V( e X and Q N( = 1);
ii
n0X = (G: 3K, sn G (i = 1,.... r) with К, e X and G = ..., Kr));
dcX = (6: G = H, x x Hr with each Ht e X);
eX = (G: 31 = Go <i G[ <i • • <i G„ = G with each С(/С(_] e X)(= (J Xr);
r=l
EZX = (G: 37V < G with N < Z„/G) and G/N e X);
ЕфХ = (G: 37V < G with N < <D(G) and G/N e X);
pX = (G: Q(G) n'J £ X), the class of all groups (in some fixed universe) all of
whose primitive epimorphic images are in X.
(1.6) Lemma. With the exception of p, the class maps defined in the list (1.5) are all
closure operations. (The properties of p are discussed in III, 2.5.)
1. Classes of groups and closure operalions
265
Proof It is obvious that all the maps on the list (with the exception of p) are both
expanding and monotonic. It is also immediately evident from the definitions that s
Q, s„ No, Do and E are idempotent. To prove the lemma it remains to show the
following:
(1) R(; = Ro: Let G e Then G has normal subgroups /Vb /у such tpia[
G/N, e R„Xand N, = 1. Therefore each group G/N, has normal subgroups K /N,
(j= !,...Л say) such that QJl, Ktj = N, and G/Кд s {G/N^/Nf e X. Sinre
I “ I liNi - Ь “ follows that GerJ and hence that rJX c rox But since
Ro is expanding and monotonic, we have R()X s Rg(r0X) = R;X, and therefore
Ro = Ro.
(2) Ej = Еф: Let G e EjX. Then G has a normal subgroup К < Ф(6) such that
G/K e ЕФХ. Consequently G/K possesses a normal subgroup L/K < ®(G/K) such
that (G/K)/{L/K) e X. Thus G/L e X, and since 0>(G/K) = 0>(G)/K by A, 9.2, we have
L < O(G). Therefore G e ЕфХ, and since Еф is expanding and monotonic, it follows,
as in the case of Ro, that Еф is idempotent.
(3) e| = Ez: If К < Z„[G), then clearly Z^fG/K) = Z^Gf/K, and so the proof that
Ez is idempotent is identical to that for e*. □
Further examples of closure operations are to be found in III, 2.5 (pq), IV, 1.12 (sw),
IV, 2.2 (Ro), and in IX, 3.5 (sr).
(1.7) Remarks, (a) Let c be a closure operation. If {Хд}ДеЛ is a set of c-closed classes
it follows easily from the definition that QZeAX is also c-closed. Furthermore, if X
is an arbitrary class, then cX coincides with the class Q {9): !'£?) = c9)}. Thus a
closure operation c is determined by the c-closed classes.
(b) A closure operation is called finitary if it is determined by its effect on finite
classes, by which we mean that
cX = (J {C9): 91 is a finite subclass of J}
for all group classes X (We remark that all the operations in the list (1.5) are finitaiy.)
Thus, if c is a finitary closure operation, and if {Хд}ДбЛ is a directed set ofc-closed
classes, directed in the sense that for all Л, p e A, there exists v e A such that
X^X.sX,,
then it is straightforward to verify that ЦсЛ X, is again c-closed. In particular, this
is true when {Хд}ЛеЛ is totally ordered by inclusion.
(c) It is sometimes helpful to describe closure properties in words instead of
symbols. For example, we might refer to sX as the subgroup-closure ofX; if X - OX
we could call X quotient-closed, or if X = D„X, say that X is closed under forming direct
products.
Next we discuss some elementary closure properties of class products.
(1.8) Lemma. If X and 9) are classes of groups, then 9)(oX) £ 0(9)X).
266 II. Classes of groups and closure operations
Proof. Let G e '2)(q£). Then G has a normal subgroup N e ')) with G/N e q£. Thus
there exists an £-group X and an epimorphism с: X -» G/N. Consider the subgroup
S = {(x, g): x e X, g e G, and c(x) = gN}
of the direct product X x G. If К = Ker(c), then S contains К x N. Since S/(l x N) s
X 6 X, it follows that S e 4)X, and therefore G, which is isomorphic with S/(K x 1),
belongs to Q(£9)). □
(1.9) Lemma. If c is any one of the closure operations Q, s, or s„, then X'2) is c-closed
whenever X and ')) are c-closed.
Proof. Suppose that £ = q£ and ')) = Q'l), and let G e £')). Then G has a normal
subgroup N eX with G/N e ')). Let К < G. We must show that G/K e £'2).
To this end, consider the normal subgroup NK/K of G/K. Certainly we
have NK/K s N/(N n К) e q£ = X Moreover, (G/K)/(NK/K) s G/NK s
(G/N)/(NK/N) e q(G/N) £ q'2) = ')). Therefore G/K e XI), and it follows that X4J is
Q-closed.
Next suppose that I = s„£ and 9) = s,'2). We show that if К < G e £'2), then
К e £')). Let N < G with Nel and G/N e ')). Since К n N <1 N, we have К r> N e
s„£ = £, also К ЦК r. N) ~ KN/N < G/N. Hence КЦК r> N) e S„9), and we conclude
that К e X>2). Thus X'2) is S„-closed.
The proof of the lemma for c = s is similar. □
We now return to the question of the associativity of the class product.
(1.10) Lemma. If X, '2), and 3 are classes of groups, then each of the following two
conditions is sufficient to ensure that £(9)3) = (£?))3-
(a) £ = n0£ and 9) = q'2);
(b) £ = s„£ and 9) = r09).
Proof. By (La) it will be enough to show that (£'2) )3 £ £(?)3)-
Case (a): Let G e (£')))3- Then G has a chain of subgroups
(l.jB) 1 < К < L < G
such that G/L e 3, L/K e '2), and К e X
If g e G, clearly Ke < L, and therefore the normal closure N of К in G, defined thus:
N = <№ : g e G>
is a subgroup of L belonging to n0£ = £. Furthermore, the group G/N has a
normal subgroup L/N e q(L/K) £ <2'2) = '2), and (G/N)/(L/N) = G/L e 3- Therefore
G/N e '2)3, and we have shown that G e £('2)3). as desired.
Case (b): Again suppose that G has a chain (l.fi) of subgroups with the stated
properties, and let
1. Classes of groups and closure operations
267
N = Coreo(K) = Q {K«: 0 e G}.
Then N e s„(K) £ s,X = X, and since L/K«s L/K e 9), we have LIN e R 7) = 4)
Thus G has a normal X-group N such that G/N e 9)3, which proves that (X9)| t c
X('I)3) in this case also. “
(1.11) Definition (A partial order on closure operations). If a and в are class maps (in
particular, closure operations), we say “a is contained in в” (and write a < в) if
aXebX
for all group classes X.
Remarks, (a) It is straightforward to verify that “<” is a relation of partial order on
class maps and hence on closure operations.
(b) It is obvious from the definitions that s„ < s. Do < Ro, and oc < n0.
The partial order just defined can be characterized as follows.
(1.12) Lemma. Let a and в be closure operations. Then a < в if and only if every
в-closed class is A-closed.
Proof. First suppose that a < B, and let X = bX. Then X £ aX £ вХ = X since A is
expanding, and therefore X = aX.
Now assume that every в-closed class is also А-closed, and let X be any class of
groups. Since в is expanding and idempotent, we have X £ bX, and bX is в-closed.
Since A is monotonic, it follows by assumption that aX £ a(bX) = bX. Hence a < B.
□
(1.13) Definition (The join of a set of closure operations). Let {c,: z e Л} be a set of
closure operations.
We define their join c = :zeA) by
cX = П {?) :3E £ ?) = сд?) for all ze Л}
for any class X of groups.
(1.14) Lemma. Let {c:, : A e A } be a set of closure operations, and let c = <c, J. E A}
be their join.
(a) The class map c is a closure operation;
(b) If X is a class of groups, cX is the smallest class containing X which is simul-
taneously c,-closed for all A e Л; .....>
(c) In the partial order on closure operations defined in (1.11) the join c is the le .
upper bound of the set {с,:1еЛ).
Proof, (a) It follows immediately from the definition that c is expanding and mono-
tonic. It is also clear that for any class X we have {9): X £ 9) - c,l) or a
268 II. Classes of groups and closure operations
{?): cX £ ?) = c2?) for all ЛеЛ}, and therefore c(cX) = cX. Thus c is also
idempotent.
(b) Since <X is the intersection of enclosed classes, it is enclosed by (1.7)(a) for all
z e Л. On the other hand, any class ?) such that 'll = c2?) 2 X for all Л e Л certainly
contains cX by definition, and therefore Statement (b) is true.
(c) This assertion follows at once from Lemma 1.12 and Part (b). □
(1.15) Lemma. Let c,, c2,..., c„ be closure operations, and let X be a class of groups.
Then X = c,c2... c,X if and only if X = <c,, c2,..., c„)X.
Proof. The sufficiency of the condition is clear. To prove its necessity, suppose that
JE = c,c2...c„X, and let 1 < i < n. By (1.14)(b) it will be enough to prove that
JE = c,X. Since cl+1, ..., c, are expanding, we have JE £ c.,, c,,2...c,JE, and hence
c,JE £ C,CW1... C,JE because c, is monotonic. However, c,, c2,..., c,_, are also expand-
ing and monotonic, and therefore
C,X £ C,C2...C,_,(C,JE) £ C, . . ,.C,X).
Hence c.JE £ c,c2... c„X = JE, and consequently I = c,X, as required. □
(1.16) Proposition. If a and в are closure operations, any two of the following state-
ments are equivalent:
(a) The class map ab is a closure operation;
(b) ba < ab;
(c) ab = <A, B>.
Proof, (a) => (b): If X is a class of groups, from the expanding and monotonic proper-
ties of A and в we have
baX £ ba(bX) £ a(ba(bX)) = (ab)2X.
If ab is a closure operation, then (ab)2 = ab, and hence baX £ abX. Thus ba < ab.
(b) => (c): Let X be a class of groups, and let X £ ?) = a9) = B?). Then abX £
ab?) = ?), and so abX £ <A, B>X by definition of <A, B>. On the other hand, abX is
an А-closed class containing X, and since b(abX) = (ba)bX £ (ab)bX = abX, the class
abX is also в-closed. Therefore (a, b>X £ abX, and Statement (c) holds. Since (a, b)
is a closure operation by (1.14)(a), it is clear that (c) => (a), and the circle of implications
is complete. □
We now describe some examples of pairs of closure operations whose products are
again closure operations. For this we introduce another closure operation E„, asso-
ciated with a prime p, which is defined as follows:
t„X = (G: ЭК < G with К < 0p(G) such that G/K e X).
It follows easily from this definition that Ep really is a closure operation.
1. Classes of groups and closure operations
269
(117) Lemma, (i) о s < so0; (ii) d„e, < u,)( (Ui) рЕф < ад. (jv) £ s sE .
(v) EPNO < N(1E„. Thus by (1.16) each of the following products is a closure operation-
SDO, Еф1)„, Ефр, SE,„ and N„E,. '
Proof (i) Let G e dosX. Then there exist X-groups G,, G2,.... G„ with subgroups
.... H„ respectively such that G S H, x x H„. Since H, x-xfl can be
identified with the obvious subgroup of G, x • • x G„ e D„X, we conclude that G e
SD0X.
(ii) Let G e о(,ЕфХ. Then there exist groups H,,..., H„ with normal subgroups К,
K2,..., K„ respectively, satisfying К, < Ф(Н,) and HJKi e X for i = I,..., n and such
that G = H, x -xH,. Since Ф(6) = Ф(Н,) x • • • x Ф(Н„) by A, 9.4, and since
G/(Ki x ••• x K„)s(H,/K1) x x we have Gee,!),,.!', and therefore
Assertion (ii) holds.
(iii) Let G e ОЕФХ. Then G s H/N, where N < H and H has a normal subgroup
К such that К < Ф(Н) and H/K e 3c. Thus
(H/N)/(KN/N) H/KN s IH/K)/(KN/K) e q3c,
and KN/N < 4>(H)N/N < 4>(H/N) by A, 9.2(e). Therefore H/N e e»qX. and Asser-
tion (iii) now follows.
(iv) Let G e ErsX. Then G has a normal p-subgroup К such that G/K = H < X e 3i.
By A, 18.9 there exists a monomorphism from G into and by A, 18.8(a)
there exists a monomorphism from К 1ireg H into К X. Since К Qjrej X e E„X, it
follows that G e se„X. Hence Ers < sep and (iv) is proved.
(v) Now let G e epnC1X. Then G has a normal p-subgroup К such that G/K is
generated by subnormal X-subgroups Nt, .... Nr. Since N^LJK, for suitable
L, sn G, we have L, e E„X and therefore G = <L„ ..., Lp> e noepX. Hence epn0 <
N0Ep, as required. □
(1.18) Lemma. Let X be a class of groups.
(a) A group G belongs to R0X if and only if G is isomorphic with a subdirect subgroup
of a direct product of a finite set of X-groups.
(b) R„Q < qr0. whence QRC is a closure operation.
(c) Ro < sd0, whence every SD^-closed class is R0-closed.
Proof, (a) If Ge R„X. then G has normal subgroups 4 such that G/NfeX
for all i = 1,..., r and Q;=1 = 1. From A, 4.17 we conclude that G is isomorphic
with a subdirect subgroup of (G/N\) x • x (G//Vr). Conversely, if
p: G - H, x • • x H,
is a monomorphism with p(G) subdirect and each H, e X, then the
the homomorphisms я.р: G - Ц (i = 1...D =
Ker(p) = 1. (Here л, denotes the projection onto the rth component of the direct
product.) Thus G e R0X.
270 II. Classes of groups and closure operations
(b) Let G e R„qX. Then by Part (a) there exist groups Ht, ..., H, e qX and a
monomorphism
p: G -> D = Hj x • x Hr
such that p(G) is subdirect in D. For i = 1,..., r let Hi = GJNi with N, < G, e I, and
let 0 be the standard isomorphism from D to the group
W = (Gi x x Gr)/(Nt x x Nr);
finally let v denote the natural homomorphism from G, x x Gr onto W. If
J = 6fi(G), and if L denotes the inverse image of J under v, it is straightforward to
check that L is subdirect in G, x • - x Gr. Hence L e R„X, and since
L/(Nt x x Nr) = 6p(G) S G,
we conclude that G e qr0£. Thus rcq < QR<„ and by (1.16) the class map qr0 is a
closure operation.
(c) This follows at once from (a) and (1.12), bearing in mind that sd0 is a closure
operation by (1.17). □
Remark. In III, 2.5 we shall show that pq is also a closure operator and further in
III, 2.10 that в» < pq.
We end this section by describing a useful exponential notation for unary closure
operations.
(1.19) Definitions, (a) A closure operation c is called unary if cJE = JJ {c(G): G e .1}
for all classes X. (Thus q, s, s„ are examples of unary operations whereas d0, r0 and
n0 are not.)
(b) If c is a unary closure operation and X a class of groups, we define
3EC = (G: c(G) S ЭЕ).
It follows easily from these definitions that Xе is the unique largest c-closed class
contained in X and that, in particular, c(£c) = Xе.
(1.20) Lemma. Let c and D be closure operations and X a D-closed class of groups.
Assume that c is unary and that CD < DC. Then Xr is n-closed.
Proof. We have CD.tc s dcXc = D.V s d.I = X, and therefore d.T' is a c-closed
subclass of X. Consequently D.V c £c; and it follows that X' is D-closed. □
Exercises
1. Let a and в be closure operations such that ba < ab. Show that if a class X is
в-closed, then so is лХ.
2. Some special classes defined by closure properties 271
2‘ 1т»ДтГтт51’8У1П(3)) an<1 G = ^^AiwSymU). Show that Ge
3. Show that Do < e and that Ro jt e.
4. Show that eq is a closure operation and that Nc < eq
5. Define c0X = (G-.G = Н,Н2...Ц, with [W(>«.] = 1 and Hj e X for 1 < i
Show that c0 is a closure operation and that Dc < c0 < qd0.
6. Show that neither s„n0 nor nos„ is a closure operation.
7. Show that if the domain of the class map p defined in (1.5) is restricted to Q-closed
classes, then p is expanding, monotonic, and idempotent.
2. Some special classes defined by closure properties
Classes of groups which satisfy certain closure properties—for example, Schunck
classes, formations, and Fitting classes—are a central theme of this book. In this
section we define some of these special types of class and describe some of their closure
properties.
We begin by confronting a dilemma we have met about the status of the empty
class. Since the empty class is by convention closed under all closure operations, it
will always appear in any family of classes which is specified by a list of closure
properties. However, there are certain advantages of exposition if classes of a certain
type are defined to be non-empty. For example, if a Schunck class is deemed to be
non-empty, one obtains a clean bijection between Schunck classes and their boun-
daries; furthermore, projectors then always exist in every soluble group. Again, if
Fitting classes are decreed to be non-empty, then the associated radical is always
defined and injectors always exist.
On the other hand, there are also situations where the empty class must be allowed
if the theory is to run smoothly. For example, in the theory of local formations 0
has to be an allowed value for a formation function, and in the Schunck class context,
insisting that boundary classes are non-empty leads to clumsiness in the statement
results. We have tried various schemes for designating certa n types of class non-
empty, but all led to difficulties: ambiguity, inelegant formulations, ponderous proofs.
Therefore, to avoid excessive pedantry and clumsiness, we have settled for the
following compromise.
Conventions about the empty class
1. The empty class is closed under all closure operations.
2. All classes defined by closure properties may be empty; there are no formal
exclusions. . ...,
3. If the non-emptiness of a class is an essential part of a result or a proof, it will be
explicitly stated, particularly if failure to do so may lead to confusion.
4. Elsewhere, it will be left to the reader to decide from the context whether a class
under consideration needs to be empty or not. Thus, for example, if we say le
G e S”, or if we refer to the radical or the injectors of a Fitting class 5,
be an implicit assumption that 3 *s not emPty-
272
II. Classes of groups and closure operations
(2.1) Definitions, (a) A Q-closed class is called a homomorph.
(b) An Еф-closed class is called saturated.
(c) A class X satisfying I = pl is called a Schunck class. Thus X is a Schunck class
if X comprises precisely those groups whose primitive epimorphic images are all in
X It follows easily from this description that Schunck classes are saturated homo-
morphs. We defer a more detailed study of the properties of Schunck classes until
the next chapter.
(2.2) Definition. A formation is a class of groups which is both Q-closed and Ro-
closed. We shall sometimes write form(X) instead of <Q, Ro>X for the formation
generated by X By (1.18)(b) a formation is precisely a QR0-closed class, and by (1.18)(c)
classes which are simultaneously closed under s, q, and d0 are formations. Thus 9(
(abelian groups), 91 (nilpotent groups), and 6 (soluble groups) are all examples of
formations, whereas the class of finite cyclic groups is not.
(2.3) Definition. The next result includes the definition of the X-residual G x of a group
G; it always exists if the class JE( f 0) is R0-closed and it is epimorphism-invariant
when X is a formation.
(2.4) Lemma. Let X be an RQ-closed class and G a finite group. Then the set
y = {N <G: G/N e X},
partially ordered by inclusion, has a unique minimal element, denoted by G1 and called
the X-residual of G. It is a characteristic subgroup, and if X is a formation and
e: G -»e(G) is an epimorphism, then c(G)x = e(Gx).
Proof. Let R = Q {N: N e if}. Since the set У” is invariant under Aut(G), evidently
R char G. Since G is finite, so is if, and therefore the group G/R has a finite set of
normal subgroups
{N/R -.Neif}
with trivial intersection such that (G/R)/(N/R) e X. Thus G/R e R0X = X and it
follows that R belongs to if. Therefore R is the desired smallest element of if.
Let e: G -» e(G) be an epimorphism. Let R = Gx, T = e(G)x, and N = c~'(T), the
inverse image of T in G. Then by the isomorphism theorem we have G/N = c(G)/T e
X, and therefore R < TV; hence e(R) < c(N) = e(e-1(T)) = T. On the other hand,
e(G)/e(R) e q(G/R) £ of = X and therefore T < e(R). Thus T = e(R), as claimed.
□
The following elementary facts will be useful in establishing the structure of minimal
counterexamples in proofs of theorems involving q- and R0-closed classes.
(2.5) Proposition. Let X and ?) be classes of groups.
(a) Let X = qX, 9) = ro9), and let G be a group of minimal order in X\fl). Then G
is monolithic. If, in addition, 9) is saturated, then G is primitive.
2. Some special classes defined by closure properties
273
jVr Gh a, cZP Order in Th™ G has normal subgroups Nt
and N2 such that G/Nt e X for i = 1, 2 and N, n N2 = 1. if I = QX, then к a',d N\
can be chosen to be minimal normal subgroups of G.
Proof, (a) If G has distinct minimal normal subgroups Nt and N2, then G/N e qX =
X, and therefore G/N,e Q) (i = 1, 2) by the choice of G. Therefore Gep/i = ')), a
contradiction. Therefore G has a unique minimal normal subgroup, N say, and
G/N g 9).
Let 41) = eoQ). If N < <D(G), then G e = Q), contrary to the choice of G. There-
fore G = MN for some M < G and Corec(M) = 1, that is to say G is primitive.
(b) Since the group G is in R0X, it has a set JT = {K„ ..., K,} of normal subgroups
Kt satisfying
(2.a)
(i) G/K, e X for i = 1,..., r, and
(ii) IX. K, = 1.
Without loss of generality we can clearly assume that for all proper subsets JZ of
we have
n K#1.
Ke S
If r = 1, then Kt = 1, and therefore Gel, contrary to hypothesis. Therefore r > 1,
and the subgroups N, = Q'Z,1 K, and N2 = Qj=2 K, are non-trivial normal sub-
groups of G such that N, r> N2 < К, = I. Since (G/NJ/fK./N,) G/K, e I for
i = 1,..., r — 1, it follows that G/N, e R0X, and hence that G/N, e X by the choice
of G. Similarly G/N2 e X, and the first conclusion of Part (b) holds.
Now let X = qX, and, as above, suppose that Jf has been chosen as small as
possible. Again we have r > 1, and this time we take minimal normal subgroups N,
and N2 of G contained in QS K, and Qj=2 K, respectively. Then (G/N,)/(K,/N,) s
G/K, e X for i = 1,.... r - 1, and (G/N, )/(KrNi/M) = G/Krh\ e Q(G/Kr) S qX = X.
Moreover,
fl K‘N> = ( .0 к<икЛ1 =
N, = N„
and so {K N,/N,} is a set of normal subgroups of G/N, satisfying (2.a). In
other words, G/N, e R„X, and therefore G/N, e X by the minimality of G. Similarly
G/N2 e X, and the final assertion of Part (b) is justified. О
As an application of (2.5)(b), we obtain the following criterion, which provides a
simple test for R„-closure.
(2.6) Proposition. A class X is ^-closed if and only if it satisfies the following
criterion-.
274
II. Classes of groups and closure operations
(2./J) Whenever a group G has normal subgroups Nt and N2 such that G/Д', e 3t( i = 1,2)
and N} c\ N2 ~ 1. then Ge£
Furthermore, if X is a homomorph, the word “normal” in Condition (2./J) can even
be replaced by “minimal normal".
Proof. It is obvious from the definition of r0 that (2./J) holds when X = R()X. Con-
versely, suppose that (2./J) holds and that I Rt,X. Then R(lX\X is non-empty and
by (2,5)(b) contains a group G which has normal subgroups Nt and N2 such that
G/Nj e X(i = 1, 2) and Nt r> N2 = 1. But then G e X by (2./J), and we have a contra-
diction. When X = qX, the assumption that the modified form of (2./J) holds and that
X R0X leads to a similar contradiction by the final assertion of (2.5) (b). Hence, in
either case, X = R0X. □
(2.7) Lemma. Let g be a formation.
(a) nP«Pcpepg = 9ig.
(b) For each л £ P, we have Qpe„(Cp.g r> (£„) e g.
Proof, (a) It is clear that 91g e(£p<3pg for all primes p. To prove the reverse
inclusion, let G e (£p.(Spg for all primes p. Then GR e йр.6р, and so if Q e Syl?(G")
for some prime q, we have Q < 0p(GR) for all pe P\{<?}. Since Qpib<IOp.(GR) =
0,(G R), it follows that Q = 0,(G R) and hence that G R e 91. Thus G e 9lg, as desired.
(b) Let pen, and let G e ep.g n e„ £ (£„. (p, g. Then G/O„4p)(G) e Qg = g, and
since Qp<?„O„xlp)(G) = 1, we conclude that G e Rog = g. Part (b) therefore holds.
□
We now turn our attention to the closure operations s„ and No, which may be
regarded as “dual” to q and r0 respectively. From this viewpoint the dual of a
formation is an <s„, N0>-closed class.
(2.8) Definitions, (a) A Fitting class is a class of groups which is both s„- and No-
closed. We shall sometimes write Fit(X) for <s„, n0)X, the Fitting class generated
by X. (Fitting classes are named after H. Fitting [1], who first showed in 1938 that
the class of nilpotent groups is <s„, N0>-closed.) Since Do < Nc and r„ < SD0, it
follows that a subgroup-closed Fitting class is Ro-closed. Whereas 91 and 6 are
examples of Fitting classes, the classes 91 (abelian groups) and U (supersoluble groups)
are not.
(b) Corresponding to the concept of an X-residual Gx when X is a formation, we
have the dual concept of an X-radical Gx of G when X is a Fitting class; its definition
appears in the statement of the following lemma.
(c) The cosocle of a group G (denoted by Cosoc(G)) is the intersection of the
maximal normal subgroups of G. The head of G is G/Cosoc(G).
(2.9) Lemma. Let X be an no-closed class and G a finite group. Then the set
У = {NsnG:NeX),
2. Some special classes defined by closure propenies
275
partially ordered by inclusion, has a unique maximal element, denoted by Gx and called
KsnG^henK - К n ^haraCter,Slic Sub°rouP °f G’ V * « « Fining class and
л. sn о, теп к x = к C\ Gj.
Proof. Let R <N: N e iff Since N sn R when N e if and since .7' is finite, we have
К e N0X X. Since У" is obviously left invariant by automorphisms of G the sub-
group R is evidently characteristic in G. Therefore R e if, and the first part of the
statement holds.
Now suppose that X is a Fitting class. Since Kx < К sn G, we have Kx e if, and
hence К x < R = Gx. On the other hand, К r> R sn R e I, and so К n R e s„X = X
Since К r> R < K, we conclude that К n R < Kx, and therefore К r> R = Kx. □
As a dual to Lemma 2.5 giving structural information about minimal counter-
examples, we have the following result.
(2.10) Lemma. Let X and V) be classes of groups.
(a) Let X = s„X, V) = n„Q), and let G be a group of minimal order in X\'l). Then G
has a unique maximal normal subgroup, namely Cosoc(G), and Cosoc(G) e V).
(b) Let G be a group of minimal order in N„X\X. Then G has normal subgroups Nt
and N2 belonging to X such that G = N2N2. If X = s.X, then Ni and N2 can be chosen
to be maximal normal subgroups of G.
Proof, (a) Let M be a maximal normal subgroup of G. Then M e s„X = X and so
M e 4) by the choice of G. If G had a second maximal normal subgroup /V, distinct
from M, then N e V) and so G = MN e n0^ = V). contrary to hypothesis. Hence
M = Cosoc(G) e V).
(b) If G e N0X. then G has subnormal X-subgroups Kt, .... K, such that G =
(Kj,..., Kf), and without loss of generality we can suppose that G is not generated
by a proper subset of {K„..., Kr}. If r = 1, then G = K, e.f, contrary to hypothesis.
Therefore r > 1. Let Lt = <K2,..., Kr> and L2 = <Kt,.... Kr_t). Then L,6N„X
and L, f G by supposition: hence LteX (i = 1, 2) by the hypothesis that |G| is
minimal. For i = 1, 2 set Nt = <L? >. Since L, is proper and subnormal in G by A,
14.4, we have G f e N„X because the G-conjugates of L, are certainly subnormal
in N,. Therefore IV, e X by the minimality of G. and since Nt N2 contains K,, K2,...,
K,_: and K„ evidently G = NtN2.
Finally, suppose that X is s„-closed. Since G = N, N2 with Nt and N2 proper normal
X-subgroups of G, it follows that Nt r>N2 f Nfi = 1,2). Hence we can find a maximal
normal subgroup 1W( of /V, containing N, n N2 and thereby obtain maximal normal
subgroups Nf = MtN2 and N* = M2N, of G. Since M: < N,X = s.X, we have
M, e X; therefore N* e N0X, and hence Nf e X by the minimality of |G|. Since G -
/V* Nf, the final assertion of Part (b) is justified. □
Part (b) of Lemma 2.10 yields a useful test for N„-closure included in the next result.
(2.11) Proposition, (a) AclassXiss.-closexlifamlonlyifNeXwheneverN^GeX.
(b) A class X is ^-closed if and only if the following conditton >s satisfied.
II. Classes of groups and closure operations
(2.)') Whenever a group G has normal X-subgroups Nt and N2 such that G = N2 >
then G e X.
Furthermore, if X is s„-closed, the word “normal’' in Condition (2.y) can even be re-
placed by “maximal normal".
Proof, (a) If normal subgroups of ^-groups are in X, it is clear by induction on the
length of a subnormal chain that subnormal subgroups of ^-groups are also in X.
(b) It is obvious from the definition of N0-closure that (2.y) holds when X = N0X
Conversely, suppose that (2.y) is satisfied. If X N„.t, then a group of minimal order
in satisfies the first assertion of (2.10)(b) and therefore violates (2.y). Thus X
is N0-closed in this case. If further X = s„X, the final conclusion of (2.I0)(b) yields a
similar contradiction to the hypothesis that n0X\I is non-empty when the modified
form of (2.y) holds. □
The obvious analogy between Proposition 2.6 and 2.11 (b) lends support to the case
for a duality between formations and Fitting classes. We shall pursue this fruitful idea
more closely in later chapters (in particular, see Chapter VIII, Section I).
Terminology. A Fitting class which is also a formation will be called a Fitting
formation.
(2.12) Lemma (Lockett [1]). Let g be a Fitting formation, and let G = N2N2, where
N, and N2 are normal subgroups of G. Then
Gs = (Nt)s(N2)e.
Proof. Let {i, j) = {1,2}. Then А;/(А; n G5) =e e s„(G/G®) <= s„g = g, and
consequently
(2.6) (NJS < N, r> G®.
On the other hand, we have (A.-A,-®)/^® A;®) S А|/(А(n NpN?) =
r> Nf) e QfNJNfy Qg = g, and since G/N?NjS is the normal product of
N2Nf/N?Nf and A2 A»/A®A2®, it follows that G/NfNf e Nog = g. Hence G® <
N2N^, and in view of (2.6) equality now follows. □
(2.13) Example. Let G be a non-abelian simple group. Then form(G) = Fit(G) =
D„(G).
Proof. It will suffice to show that d0(G) is a Fitting formation. We appeal to A, 4.13(b).
If N < D = Gj x • • • x G„ with G, = G, then A is a direct product of a subset of the
direct components G( (identified with 1 x x 1 x G,- x 1 x x 1), and further-
more D = N x Cd(N), where CD(N) is the direct product of the complementary subset
of direct components. From this it is immediately clear that d0(G) is both Q-closed
and s_-closed.
2. Some special classes defined by closure properlies
277
To prove that D (G) is R -closed, let N, and N2 be normal subgroups of a group
= H/N‘£ °o(G) f°r * = ’’ Z If K is a simple component
of N„ then KN2 < H because H/N2 e d0(G), and so К = N, n KN2 < H. By (2.6) it
will be sufficient to show that H e d„(G) under the assumption that N2 and N
are minimal normal subgroups of H since X = Q.X. But under this assumption we
have К = Nt and Aut„(K) s Aut„(KW2/N2) = AutK(K/V2/N2) = AutK(K) because
KN2/N2 is a simple direct component of H/N2. Thus, setting C = CH(K), we obtain
W = C x К since C r> К = 1. But C s H/K e d0(G), and consequently H e d0(G), as
required.
Finally, to show that D0(G) is NfJ-closed, let A, and N2 be normal subgroups of a
group H = N,N2 with N; e d0(G) (i = 1, 2), and let M = Nt r> N2. If Q = CW,(M),
it is clear that Ct nC2 < См(М)= 1 and hence that |C,C2| = |Л\ ;M||A2: M| =
: M|. It follows that C„(M) = CtC2 = (N1/M) x (N2/M) ed0(G), and therefore
that
H = M x CH(M) e d„(G).
We can now conclude from (2.11 )(b) that D0(G) is indeed N0-closed.
□
We end this section with some basic definitions and a few elementary facts from the
theory of varieties of groups. The most comprehensive source for this subject is Hanna
Neumann’s book [1]. In this discussion, of course, the universe is the class of all
groups (finite and infinite).
A word is an element in the free group Xa_ of countably infinite rank in variables
x,, x2,... . A variety 93 is a class of groups defined by laws, that is to say, by a set of
words
»' = {w2}26A
such that the equations
w2 = 1 (2 e A)
are all identically satisfied by each group of 93 and by no other group. (An equation
w(X],..., x„) = 1 is identically satisfied by a group G if w(g,......g„) = lc for all
substitutions x, -> gt with g2, ..., g„ e G.) Thus, ^example, the variety 91 of aU
abelian groups is defined by the single word (law) Xj x2x1x2(—I).
The verbal subgroup 93(G) of an arbitrary group G is defined to be the subgroup
generated by all elements of the form
where x, -»<?, is an arbitrary substitution of group elements G for theenables
x„ ..., .г, involved in the word w2. Thus, when 93 = 91 and И7 = {[x xJ}.ta
23(G) = G-, the derived subgroup of G. For an arbitrary variety 93 weobv.ously have
G e 93 if and only if 93(G) - 1.
278 II- Classes of groups and closure operations
Furthermore, if Xr = <xn..., xr> denotes a free group of rank r, the group
Fr(®) = Xr/S(Xr)
is called the relatively free group of rank r in ®. It turns out that every group in 'll
which can be generated by r elements is an epimorphic image of Fr(tB).
A variety is said to be locally finite if it consists entirely of locally finite groups,
namely groups whose finitely generated subgroups are all finite. Evidently, a variety
is locally finite if and only if its relatively free groups of finite rank are all finite. It
turns out (see Hanna Neumann [1], Theorem 15.71) that if G is a finite group, the
relatively free group of rank r in the variety generated by G has order at most |G||cr,
and this has the following consequence.
(2.14) Proposition. The variety generated by a finite group is locally finite.
The class of finite groups in a variety ® (namely the class й n ®) is obviously a
subgroup-closed formation. It is therefore not surprising that the theory of varieties
plays a part in the study of formations of finite groups. (For more on this connection,
the reader is referred to Brandl [1], [2].) The following celebrated theorem of Oates
and Powell [1] has an analogue in the theory of formations.
(2.15) Theorem. If G is a finite group, then the variety Var(G) which it generates can
be defined by a finite set of laws.
The analogous conjecture states that a formation generated by a finite group contains
only finitely many subformations. This has been proved for soluble G by Bryant,
Bryce and Hartley [1], and their theorem will be proved in Chapter VII, Section 1
without recourse to the deeper theory of varieties. There is, however, one concept
from the theory of varieties which seems indispensible in the sequel, namely that of
a verbal or varietal product. This construction is used by Bryce and Cossey in their
analysis of metanilpotent Fitting classes with additional closure properties, and we
shall give the details in Chapter XI, Section 2.
Exercises
1. Let X be a class of groups, and assume that for all G ей the set of normal
subgroups N of G with G/N e X has a unique minimal element. Then JE is
R0-closed.
2. Let T = Ro.F, and assume for all N < G e й that (G/N)x = GXN/N. Then T is
O-closed.
3. Let л £ P, and for a class X of groups, define
e,1E = (G: ЭК < G, К < O„(G) such that G/K e IE).
Show that e, is a closure operation.
Chapter III
Projectors and Schunck classes
1. A historical introduction
Undoubtedly the most important basic result in finite group theory is the theorem
of Sylow ([1], 1872) which states (i.a.) the following:
Existence: Every finite group G possesses Sylow p-subgroups;
Conjugacy: The Sylow p-subgroups of G form a conjugacy class of G;
Dominance: Each p-subgroup of G is contained in a Sylow p-subgroup.
In Chapter I we saw how, for soluble groups, P. Hall was able to extend the scope
of Sylow’s theorem from p-groups to л-groups for a set л of primes. The first evidence
that further extensions were possible came in 1961 with the following discovery of
Carter [2]:
Every finite soluble group has self-normalizing nilpotent subgroups (or ‘Carter sub-
groups’ as they became known) and these form a characteristic conjugacy class of the
group.
The discovery of Carter subgroups aroused considerable interest, not least because
they invite analogy with Cartan subalgebras, which play a part in the classification
of finite-dimensional, semisimple Lie algebras. However, not all finite groups possess
Carter subgroups, and no counterpart has been found for a general insoluble group.
Nevertheless, Carter’s discovery gave a considerable fillip to .he subsequent develop-
ments in soluble groups.
The significance of the three stated parts of Sylow’s theorem depends upon the
chosen definition of a Sylow p-subgroup. If it is defined in the usual way to be a
subgroup of G of order |G|P, then of course each part of the theorem needs justifying.
But if it is defined as a maximal Sp-subgroup, then the properties of existence and
dominance are given free, and the burden is to prove conjugacy and to compute the
order. Whereas each of these alternative definitions extends naturally to Hall sub-
groups, a moment’s reflection shows that neither suffices to characterize Carter
subgroups when the class 91 of nilpotent groups is substituted for the class Sp of
p-groups. This is because, on the one hand, there is no general relationship between
the order of a Carter subgroup and that of its parent group (see Exercise 1 below),
and, on the other hand, the only groups which possess a unique conjugacy class of
maximal nilpotent subgroups are the nilpotent groups themselves. We shall now
describe the evolution of a unifying definition, which leads to an elegant and far-
reaching generalization of Hall and Carter subgroups.
280
III. Projectors and Schunck classes
The most important landmark in this development, and the source of inspiration
for much subsequent work, is the seminal paper of Gaschiitz [8], published in 1963
and entitled “Zur Theorie der endlichen auflosbaren Gruppen”. There he presents
the following remarkably fruitful extensions of Carter’s ideas:
1. In place of the class of nilpotent groups he considers a general <Q, Ro, Eo>-closed
class g, that is a saturated formation. (Originally, Gaschiitz’s concept of saturation
was not defined by Enclosure, but in joint work with his student Lubeseder [1] he
later showed that the two definitions are equivalent within the soluble universe.)
2. He defines the concept of an ‘g-Untergruppe’ (which in English became ‘g-
covering subgroup’), and he shows that they exist and form a conjugacy class in each
finite soluble group. By specializing g to Sp, S„, and 91 in turn, one obtains the
Sylow, Hall, and Carter subgroups respectively.
3. He shows furthermore how a rich supply of examples of saturated formations
can be obtained from the construction of local formations, including the already
well-known classes of p-groups, nilpotent groups, л-groups, supersoluble groups,
Sylow tower groups, and groups of a given p-length. (It should be remarked that the
formation character of special classes of groups (e.g. nilpotent and supersoluble
groups), as well as their saturation and local characterization, had been previously
observed and studied by different authors (see, for example, Wielandt [5], Huppert
[2], and Baer [1]), but of course without the formation terminology.)
Let g denote a class of finite groups. According to Gaschiitz’s definition, an
^-covering subgroup of a group G is a subgroup E of G which belongs to g and covers
each g-quotient of every intermediate group; thus it is defined by the properties:
(i) E e g, and
(ii) whenever E < H < G and К < H such that H/K e g, then H = EK.
Although in a soluble universe the g-covering subgroups satisfy the existence and
conjugacy properties when g is a saturated formation, they fail to satisfy the
property of dominance (namely that every subgroup of G belonging to g is
contained in an g-covering subgroup of G), except when g = S„. This is be-
cause a saturated formation of characteristic n contains all nilpotent л-groups, and
so a dominant g-covering subgroup would have to contain a Sylow p-subgroup
of G for each pen. However, it is clear from the definition that g-covering
subgroups have the following important property of ‘persistence in intermediate
groups’:
Persistence. If E is an g-covering subgroup of G and if E < H < G, then E is an
g-covering subgroup of H.
This property, together with conjugacy, shows at once that g-covering subgroups
are pronormal.
The next significant step in this development of a generalized Sylow theory was
prompted by the question: For which classes § do ^-covering subgroups exist in each
finite soluble group? A complete answer to this question was given by Schunck in his
Kiel Dissertation, written under the direction of Gaschiitz, and published by Schunck
[1] in 1967. He showed that these classes can be elegantly characterized in terms of
their primitive groups and that they form a considerably larger family of classes than
281
1. A historical introduction
the saturated formations. They eventually became known as Schunck classes and are
the main concern of this chapter.
It was known well before the publication of Schunck’s paper that Gaschfitz’s local
method of constructing saturated formations is, in fact, completely general; in 1963
Lubeseder [1] proved that every saturated formation is a local formation. Thus the
local definition became the basic tool for the solution, and even the formulation of
questions specifically related to saturated formations. No comparable local or induc-
tive method was available for the study of Schunck classes in general, and there was
little progress here until 1974 when Doerk [4] broke new ground with the publication
of “Uber Homomorphe endlicher auflosbarer Gruppen” based on material from his
Habilitationschrift (Mainz, 1971). In this work he introduces the simple, but extreme-
ly fruitful, concept of the ‘boundary’ of a Schunck class, roughly analogous to the
set of critical groups of a variety. There is a bijection between Schunck classes and
their boundaries, and many questions about Schunck classes can be more readily
resolved when translated into questions about their boundaries. From this time the
theory of Schunck classes developed its own special character and has since pro-
ceeded along somewhat different lines from the theory of formations. Of course,
saturated formations of finite soluble groups are special cases of Schunck classes, and
much of the foundation material in this chapter may be regarded as part of the basic
theory of saturated formations, which we shall draw upon for examples and illustra-
tions from time to time. Nevertheless, the techniques and motivating questions in the
two areas are so different, both at the basic and advanced levels, that they require
quite separate treatment in separate chapters.
By the time of the appearance of Schunck’s paper an investigation into a dual
theory was under way. This was initiated by Fischer in his Habilitationschrift
(Frankfurt, 1966) and was subsequently elaborated and relined by him and by others.
The key concepts in this dualization are those of an ‘^-injector’ and a ‘Fitting class’.
In a fundamental paper entitled “Injektoren endlicher auflosbarer Gruppen”, Fischer,
Gaschiitz, and Hartley [1] prove that g-injectors exist in each finite soluble group
if and only if g is a Fitting class, and that they share with covering subgroups the
properties of conjugacy and persistence. Thus the analog}' between covering sub-
groups and Schunck classes on the one hand, and injectors and Fitting classes on the
other, is pleasingly close. However, when the definition of an injector used by Fischer,
Gaschiitz, and Hartley is itself dualized, it gives rise not to a covering subgroup, but
to the following concept, called by analogy a ‘projector’. If i is a class of groups, an
X-projector of a group G is a subgroup £ of G satisfying: EN/N is ЗЕ-maximal in G/N
for all N < G. (A subgroup X of a group Z is said to be X-niaximal in Z if (i) X e 3£
and (ii) whenever X < Y < Z with У e it, then X = У.)
For general 3£ and G, the 3£-projectors of G may differ from 3£-covenng subgroups,
but if i is a Schunck class and G is soluble, then they coincide. Therefore, since 1969,
when this fact was proved by Gaschiitz [10] in his Canberra notes, the term projector
has been widely adopted in this context in preference to ‘covering subgroup.
Since Gaschiitz’s 1963 paper the theory of formations has been very thoroug у
explored and richly generalized; its scope has been extended to the umverse of a
finite groups by Baer [6], P. Schmid [1], [2], Erickson [2], P- Forster [11] [13]
[14], Baer and Forster [1], and others, and we shall give details of many
282 П1. Projectors and Schunck classes
developments in the later chapters on formations. The first serious attempt to
broaden the study of general Schunck classes and their projectors and take it outside
the confines of soluble groups was made by Forster [11], [13] and [14]. It is largely
on this approach that we base the next two sections, in which we lay the foundations
of the theory of projectors. The completely general setting thus adopted requires that
we first establish in Section 2 some preliminary machinery whose motivation may
not become evident until it is applied to projectors in Section 3; but we hope that this
brief historical introduction will help to tide the reader over. In addition to its greater
generality, Forster’s approach has the advantage of simplifying and clarifying the
basic concepts and proofs. We pursue it for as long as these benefits can be reaped,
and when this generality begins to obscure the view with increasing complexity and
case-by-case arguments, then we return to the quieter pastures of a finite soluble
universe, where the theory assumes its most elegant form and where to date it has
made the greatest impact.
Exercises
I. Let |G| = 6, and let C be a Carter subgroup of G. Prove that |C| = 2 or 6 and that
both possibilities occur.
2. Let G be a group which has a Carter subgroup of order 2. Prove that G is
metabelian.
3. Show that Alt(5) does not have self-normalizing nilpotent subgroups, whereas
Sym(5) does.
4. If G e 3£, show that G is the unique I-covering subgroup of G.
5. Let C denote the class of п-perfect groups. Describe the set of C-covering
subgroups of G and show that they have the dominance property.
2. Schunck classes and boundaries
This section contains some of the basic facts about Schunck classes. The results are
elementary, but they establish the standpoint for the rest of the chapter. We introduce
the important concept of a boundary and prove the fundamental bijection between
Schunck classes and their boundaries. This correspondence plays an essential part
in the subsequent development of the theory of projectors and covering subgroups,
the characteristic conjugacy classes of subgroups which motivate the study of Schunck
classes.
Throughout this section we shall work within a fixed, but unspecified, universe,
which will be denoted by S3. Our only general assumption about S3 is that it is a
non-empty homomorph of finite groups. Thus
0 * S3 = qS3 £ (£.
Other closure properties may be imposed on S3 as the need arises. At several points
in the chapter it will be helpful to use the notation (q — 1), which is a class map defined
as follows:
1. Schunck classes and boundaries
283
1) (Q - 1)0 = 0,
2) (q — 1)(1) = (1), and
3) if A /0 or (1), then (q - 1 )3E is the class generated by the proper epimorphic
images of the groups in JE.
Thus, if G * 1, we have (q - 1)(G) = (G/N: 1 #W< G), and for any class JE, clearly
qJE = JE и (q - 1)JE.
(2.1) Definitions, (a) A class ® of groups is called a Q-boundary if
(Q — 1)!Вс,!В= 0.
From this definition it is clear that 0 is a Q-boundary, that a Q-boundary never
contains groups of order 1, and that a subclass of a Q-boundary is again a Q-boundary.
(b) The map h: For a subclass JE of the universe 93 define
/i(3E) = {Ge93:Q(G)nJE = 0).
Thus b(JE) consists of all ‘J£-perfect groups’ in 93, namely all those groups which have
no JE-groups among their epimorphic images. This definition implies that
h(0) = 93, and
h(JE) = 0 whenever 1 e JE.
Furthermore, we have
(2.a)
JE n h(X) = 0.
(c) The map b: For a subclass 9) of S3 define
h(9J) = (G e 93\9): (Q - 1)(G) E ?)).
The class b(?)) is called the Q-boundary of?). Observe that (Q - 1 )b('2)) E ?), whereas
b(?)) £ 93\9); hence b( V)) is certainly a Q-boundary in the sense of (a) above. Also note
that evidently
b(0) = b(93)=0,
and that b(( 1)) is the class of simple groups in 93.
(2.2) Remarks, (a) Let § and Я be homomorphs contained in 93 If C- eS\S and
(q - 1) (G) S S, then G e b(§); in particular, a group of minimal order in Я\» belongs
to b(f>).
Proof. Since G e 93\§ and (q - 1)(G) £ & we have G e b(S>) by definition of IШ ff
G is of minimal order in Я\§, then G 1 as 1 e If 1
Я, and therefore G/N e S by choice of G. Consequently (Q 1)( ) - »
?84
III. Projectors and Schunck classes
(b) Let ® and G be Q-baundaries such that 93 £ G £ ®. Then
G\® £ h(®)\h(G).
Proof. Let G e G\®. Since G is a Q-boundary, we have (q - 1 )(G) n G = 0. Hence
q(G) n ® = 0, and therefore G e Л(®).
(c) Let he a non-empty homomorph, and let G e ®\§. Then G has an epimorphic
image in b(§).
Proof. Simply apply Remark (a) above with Я = q(G).
(2.3) Proposition. The maps h and b defined in (2.1) induce mutually-inverse bijections
thus:
3.
ь
between the set 3 of non-empty homomorphs and the set 3f of Q-boundaries within the
universe ®.
Proof. First we show that the maps b and h have the stated targets. Since ® is Q-closed
by assumption, it is obvious from the definition of h that h(X) is Q-closed for any
subclass X of ®. If, in addition, S is a Q-boundary, then 1 ф X by (2.1)(a), and
so h(3E) is not empty; thus h(JE) is a homomorph. Let 9) £ ®, and suppose that
b(9)) 0- (Note that b(9)) 0 when 0 9) = q9) c ® by (2.2) (a).) If G e b(9)),
then
(Q - 1) (G) n h(?)) £ ?) n (®\9)) = 0.
Hence (q — 1 )b(9)) n h(9)) = 0, and h(9)) is indeed a Q-boundary.
Next we prove that h о b is the identity map on 3. Let 0 1 § = q§ £ ®. Then
no epimorphic image of a group in § can belong to b(§), and so § £ h(b(§)). Suppose,
by way of contradiction, that § 4 h(b(§)), and let G be a group of minimal order in
h(b(!r>))\£. Since h(b(§)) is Q-closed, by (2.2)(a) we have G e />(§), and therefore setting
£ = h(§) in (2.a), we conclude that G e 0, a contradiction.
Finally, we show that b ° h is the identity map on ^Q. Let ® be a Q-boundary in
®, and let Ge®. Then 1 # G ф h(®). Furthermore, (q — 1) (G) £ ®\®, and it follows
easily that (q - 1)(G) £ h(®). Hence G e b(h(®)), and so ® £ b(h(®)). Since h(®) e 3
and h о b is the identity map on 3 by the previous paragraph, we have h(b(h(®))) =
h(®). Therefore, applying (2.2)(b) with b(h(®)) substituted for G, we conclude that
b(h(®))\® £ h(W)\h(b(h(<8))) = 0,
and hence ® = b(h(®)), as desired. The conclusions of the proposition now follow
easily. □
"1. Schunck classes and boundaries
285
51’ The uaS'S map P: Der,ne p0 = 0’ and for апУ non-empty class
X _ 53 let pX denote the following class:
pX = (G g 53: q(G) n c j).
Tbus pX consists of those groups all of whose primitive epimorphic images belong
to X. Since S3 is supposed to be a homomorph, this definition is unchanged if the
class 'fl of all primitive finite groups is replaced by '(1 n S3. Henceforth we therefore
adopt the following convention:
The letter 'fl will always denote the class of those primitive groups that lie in
the universe under consideration in the given context.
(2.5) Lemma, (a) The таре isidempotent,monotonic, but not necessarily expanding.
(b) P = qp and Q < PQ.
(c) The class map pq is a closure operation.
Proof. We will prove Part (b) first. Let X £ S3. Certainly pX £ qpX because q is
expanding. If G e qpX, we have G = L/N for some L e pX, and consequently
q(G) n 'fl £ q(L) ri 'fl £ X. Hence G e pX, and we have shown that qpX £ pX. It
follows that p = qp, as claimed. Next let G e qX. Then q(G) £ qX, and in particular
q(G) n 'fl £ qX; thus G e pqX, and so q < pq.
To prove Part (a) we have only to show that p2 = p; for it is obvious that p is
monotonic, and it is also clear that p need not be expanding. By Part (b) we have
pX = q(pX) £ pq(pX) = p(qp)X = P2X.
On the other hand, if G e p2X, we have q(G) n 'fl £ pX. Let H e q(G) n 'fl. Since
H e pX, we have q(H) n 'fl £ X; but He q(H) r, 'fl, and so H e X. Therefore
q(G) n 'fl £ X and hence G e pX. This proves that P2X £ pX, and consequently P is
idempotent.
To prove Part (c), observe that by Part (b) we have X £ qX £ pqX, and therefore
PQ is expanding; furthermore, PQ is obviously monotonic. Finally, to see that it is
idempotent, we use Parts (a) and (b) to deduce that
(PQ)2 = p(QP)Q = p2Q = p°-
(2.6) Definitions, (a) A non-empty PQ-closed class contained in S3 is called a 93-
Schunck class, or simply a Schunck class if the universe S3 is understood. It is clear
from (2.5)(b) that a Schunck class is always a homomorph.
(b) A Q-boundary contained in 'fl is called a ('B-)Schunck boundary. (Recall that
S3 £ S3 by convention.) Note that a subclass of a Schunck boundary is again a
Schunck boundary.
(2.7) Theorem. Let 0 # & £ S3. Then the following assertions are equivalent each to
the other:
286
III. Projectors and Schunck classes
(a) § is a Schunck class;
(b) & = P§;
(c) § is a homomorph and b(Sj) £ ф.
Proof. If § = P§, then by (2.5)(b) we have = qp§ = ₽§ = §• Therefore pq§ =
p§ = J), and the equivalence of Assertions (a) and (b) is now clear.
Next we show that Assertion (c) follows from (a). If Sj = PQ§, then certainly
§ = q§. Let G e b(5y), and suppose that G f 23- Then Q(G)n 2$ £ (q — 1 )(G), which
is a subclass of by definition of b(f>). Thus G e P§ = §, which contradicts the fact
that G lies in b(f>). Therefore Ge® and Assertion (c) holds.
It remains to show that (c) implies (a). Therefore suppose that Sy is a homomorph
whose boundary consists of primitive groups, and observe that PQ§ is a homomorph
containing f>. If PQ§\& is non-empty, it contains a group G of minimal order, which
by (2.2)(a) belongs to b(5y) and hence to ®. Since G e pq§ = pSv, it follows that G e A
by definition of p. This contradiction forces the conclusion that § = pq§. □
(2.8) Corollary. For an arbitrary class ЭЕ the class Pjt‘ is a Schunck class, and if X is a
homomorph, it is the smallest Schunck class containing A.
We now come to the 'fundamental bijection’ mentioned at the outset of this section.
Let Ж and 36 denote respectively the sets of Schunck classes and Schunck boundaries
in 23. If § e Ж, then b(§) £ ® by the implication: (a) = (c) of (2.7), and so b(f>) e 36.
On the other hand, suppose that ® e Л. Then by (2.3) we have 23 = b(§) for some
§ e Ж Therefore by the implication: (c) = (a) of (2.7), the set Ж contains § =
^(b(£>)) = b(®). Hence b restricted to Ж maps to Si, and h restricted to Si maps to
Ж and therefore Proposition 2.3 yields the following theorem.
(2.9) Theorem (Doerk [4]). The maps h and b induce mutually-inverse bijections
between the set of ‘D-Schunck classes and the set Si of 4i-Schunck boundaries thus:
Ж 3d.
h
Remark. It is obvious that an inclusion ®, c ®2 between Schunck boundaries gives
rise to the reverse inclusion h^Bf) 2 h(®2) between their associated Schunck classes.
There is no corresponding preservation of inclusions by the map b in the reverse
direction. Indeed, it is possible to have a pair of Schunck classes £ f>2 f°r which
h(&i) b(§2) = 0 (see Exercise 3 below).
We have chosen to put the ideas of this section straight to work and to postpone
their illustration with examples until the theory is more fully developed and its
purpose is clearer. Nevertheless, a reader so inclined could profitably glance ahead
to the examples described in (3.12), or to those scattered throughout Section 4, before
launching into the study of projectors in the next section. In order to complete our
preparations, we need just one more elementary result.
(2.10) Lemma. If ® = еф®, then a D-Schunck class is r.„-closed.
2. Schunck classes and boundaries
287
Proof Suppose that ft = P§ £ ®, and let G e Eoft. Then G e Еф® = S. By definition
of Еф there exists a normal subgroup, К of G such that К < <D(G) and G/K e ft Let
G/N e Q(G) n ф. By A, 15.4(b) the normal subgroup, N is the core of a maximal
subgroup, of G, and therefore <D(G) < N. Consequently G/N e q(G/K) s Qft = ft and
it follows that G e Pft = ft. Thus E,ft £ ft, and therefore § is Enclosed. ' ’ □
Finally we record the following elementary but useful consequence of (2 2)(a) and
(2.7), (a) => (c).
(2.11) Remark. Let ft be a Schunck class, and let Я be a homomorph such that
Я\б # 0- Then a group of minimal order in ft\ft is primitive.
Exercises
1. If X is a class of groups, show that pqS is the smallest Schunck class containing JE.
2. Let 3E £ ®, and let ® = (G e T : (q — l)(G)r> JE = 0). Show that (i)® is a Schunck
boundary, and (ii) if ®* is a Schunck boundary such that Л(®*) = fi(3E), then
®* =®.
3. Show that 91 and 912 are Schunck classes such that (i) 91 £ 9l2 and (ii)
b(91)nb(912) = 0.
4. Let {§л}ЛеЛ be a set of Schunck classes with the property that for all 2, p e A there
exists an element v e A such that
&toft„£ft,.
Show that (JieAftA is a Schunck class. Does this observation generalize to all
classes that can be defined by closure operations?
5. Let 0 # n £ P. Show that S, contains uncountably many Schunck classes if
|tt| > 2 and only two if |zr| = 1.
6. Let ft be a Schunck class p>roperly contained in S. Prove that there exists a
Schunck class Я such that ft i fi i to.
7. Let ft be a Schunck class, and define <DP0(ft) = (G e ft: whenever X £ ft = pq(G, X),
then ft = pqJE.)
(a) If there exist maximal sub-Schunck classes in ft, then <I>PQ(ft) is the intersection
of all maximal sub-Schunck classes ft. Otherwise ФР0(§) = ft-
(b) Let 3 = n (Q - 1 )(ft n ®)- Then ®r„(ft) = PQ3. and in particular ФР0(91") =
91”-1 for all ne hJ.
8. Let p be a prime, and let ft be the Schunck class of p'-perfect soluble groups. Then
Char ft = {p} с P = ff(ft).
[Unless otherwise specified in the above examples, assume a general universe
which satisfies any closure properties that may be needed.]
288
III. Projectors and Schunck classes
3. Projectors and covering subgroups
(3.1) Definition. Let £ be a class of groups. A subgroup U of a group G is called
X-maximal in G provided that
(a) V e £, and
(b) if U < V < G and V e £, then U = V.
Consider the case where £ = the class of finite soluble л-groups. The S„-
maximal subgroups of a soluble group are precisely its Hall л-subgroups. Moreover,
if U is a Hall л-subgroup of G and К a normal subgroup, the UK/K is a Hall
л-subgroup of G/K, and U К is a Hall л-subgroup of К (see I, 3.2). Thus a Hall
л-subgroup U of a finite soluble group is characterized by each of the following
properties:
(1) VK/K is Q^-maximal in G/K for all К < G;
(2) U r K is Q„-maximal in К for all К sn G.
This suggests the following problem:
Problem A. Which classes £ can one substitute for Sn in the Statements (1) and (2)
above and still retain the essential features of the theory of Hall subgroups'!
The property described by (2) leads to the theory of Fitting sets and classes, which
will be dealt with in Chapters VIII, IX, X and XI. For the rest of this chapter and
beyond we shall be concerned with the property described in (1), which we now
formalize with an appropriate definition.
(3.2) Definition. Let £ be a class of groups. A subgroup 17 of a group G is called an
X-projector of G if
VK/K is £-maximal in G/K for all К < G.
Thus the Hall л-subgroups of a soluble group are the ^„-projectors in this ter-
minology. We shall use Projj(G) to denote the (possibly empty) set of £-projectors
of G.
(3.3) Remark. If a is a homomorphism of a group G and if U e Projj(G), it follows
easily from the above definition and the isomorphism theorems 17“ e Projj(G“). In
particular, Projj(G) is a union of G-conjugacy classes.
Problem A evolved during the 1960’s, mainly under the influence of Gaschiitz and
his school, and for the universe S of finite soluble groups a comprehensive theory
was worked out. (For an elegant presentation of this material in a considered form,
the reader is referred to Gaschiitz’s Canberra notes: these notes were prepared from
lectures which were given by Gaschiitz at the Ninth Summer Research Institute of
the Australian Mathematical Society, held in Canberra in 1969. In 1979 Gaschiitz
undertook to bring these notes up to date, a task which entailed considerable revision,
and they have now been published by the Australian National University Press (see
Gaschiitz [15]).
3. Projectors and covering subgroups 289
Given this satisfying answer to Problem A for the universe 3, it is natural to ask
the following question:
Problem B. Can the universe for which the theory of projectors works be enlarged
beyona the class (S?
In certain cases this is certainly possible. For example, we know by Sylow’s theorem
that ^-projectors exist, and are conjugate, in each finite group. As we mentioned in
Section 1, Forster has made some general progress with Problem B, and we shall
largely follow his approach, by working, to begin with at least, within a fixed, but
unspecified, universe S3. Then, when situations arise which are beyond the scope of
this book, we specialize S3 to 3 and derive the soluble theory of projectors alluded
to above as a special case. Therefore until further notice in this section we make the
following general assumption.
(3.4) Hypothesis. AU groups under consideration belong to a fixed universe S3, which
is a non-empty class of finite groups closed under the operations s, q, and Еф.
(3.5) Definitions, (a) A subclass § of S3 is called ‘il-projective if Proj6(G) 0 for all
G e S3. (The prefix ‘S3’ is usually omitted if the universe is understood.) Thus projective
classes are those for which the existence of projectors is guaranteed in all groups
under consideration.
(b) Let ЭЕ £ S3. An X-covering subgroup of a group G is a subgroup E of G with
the property that
E e Projj(H) whenever E < H < G.
The set of S-covering subgroups of G will be denoted by Cov x(G). Thus S-covering
subgroups are X-projectors with the property of persistence described in Section 1,
and, in particular, we have
Covj(G) £ Proj^G).
(c) A subclass ЭЕ of S3 will be called a Gaschiitz class for S3 if Covx(G) / 0 for all
G e S3. Clearly Gaschiitz classes are projective.
(3.6) Remarks, (a) The following observations about an ЭЕ-covering subgroup E of a
group G are easy consequences of the definition:
(i) If E < H < G, then E e Cov^H); , i. f ic\
(ii) If a is a homomorphism of G, then E‘ e Cov^G); in parttcular, if the set Covx(G)
is non-empty, it is a union of G-conjugacy classes. , nr r,
(b) If 3E = q3E, it is an easy exercise to characterize an 3E-covenng subgroup of G
as a subgroup E satisfying the following two conditions:
csi whenever £ < H < G and К < H such that H/K e X, H
Thus S-covering subgroups are I-subgroups which cover each X-quotient of every
intermediate group, which explains the terminology.
290 III. Projectors and Schunck classes
When X-covering subgroups (X-Untergruppen) were first introduced by Gaschiitz
[8] in 1963. he used Conditions CS1 and CS2 to define them, and he proved that
saturated formations are, in our terminology, Gaschiitz classes for <5. In 1967 Schunck
[1] characterized theGaschiitzclasses for Sas the PQ-closed subclasses of S, in other
words, the S-Schunck classes, and he showed that these form a larger family of classes
than the saturated formations. The X-projector concept makes its first appearance
in 1969 in Gaschiitz’s Canberra notes, where it plays the fundamental role and
displaces the X-covering subgroup. However, nothing is lost, because in those notes
the universe is <5, and Gaschiitz proves that if X is an 3-Schunck class and G g 6,
then Proj i(G) = Cov j(G)(an observation that had been made by Hawkes [3] in the
case where X is a saturated formation). In other words, Gaschiitz shows that for the
universe of finite soluble groups the concepts of‘projective class’ and ‘Gaschiitz class’
are equivalent. However, in Forster’s more general setting they no longer coincide;
here X-projectors need no longer have the persistence property, even for projective
classes X. The key to Forster’s approach is to show that the questions of existence
and conjugacy of X-projectors and X-covering subgroups can usually be resolved
simply by reference to groups in the boundary b(X).
(3.7) Proposition. Let S> be a homomorph (contained in 93). Let a denote a function
which assigns to each group G g 93 a possibly empty set j(G) of subgroups of G. If j
is either of the functions Proj6( ) or Cov6( ), then it satisfies the following two
conditions:
f (i) G e j(G) if and only if G e Sy,
{(ii) Whenever N < G,N < V < G,U e j(V),and V/N e a(G/N),then V e -j(G).
Remark. In the hypothesis of Condition (ii) we have implicitly used the <q, s>-closure
of 93 in supposing that a(F) and n(G/N) are defined.
Proof. It is obvious from the definitions that Condition (i) is fulfilled in both cases.
We show first that Condition (ii) is satisfied when a = Proj6( ). Let V be a subgroup
of G satisfying the hypotheses of Condition (ii). We must show that, if К < G, then
VK/K is ^-maximal in G/K. Let VK/K < T/K e Since V e Proj6(F) and V/N e
Sy, we have V = UN; moreover, because V/N e Proj6(G/7V), it follows from (3.3) that
VK/NK g Projg(G/7VKj. But then
VK/NK = UNK/NK < NT/NK =< T/(Tn N)K e q(T/K) <= f>,
and we conclude that VK = NT by the §-maximality of VK/NK in G/NK. In
particular, we have T = T n VK = (T n V)K, and it follows that (T n V)/(K n V) s
T/K e Sy. Since U e Proj&(P), and because U(Kn V)/(Kn V) is therefore an S>-
maximal subgroup of V/(K n V) contained in (T n V)/(K n V), we conclude that
U(K n V) = Tn V Hence T = V(K n V)K = UK, and so UK/K is ^-maximal in
G/K, as desired.
Finally, we verify Condition (ii) when л = Covg( ). Assume that U g Covg(F),
where V/N g Covg(G/7V) for some normal subgroup N of G contained in V. If
3. Projectors and covering subgroups
291
V < L < G, we must show that U e Projb(L). From
tion of a covering subgroup we have
(i) U e Proj&(Ln P), and
our assumptions and the defini-
(ii) V/N e Projj.UJV/TV),
Since V = GW, we have (L n V)N = LN nV =
from LN/N to L/(LciN) transforms V/N to (L
(ii), yields
V. Thus the standard isomorphism
n V)/(L n N) and, when applied to
(n>) (L n V)/(L n N) e Proj&(L/(£ n N)).
By the result already proved for о = Proj6( ), we conclude from (i) and (iii) that
V e Projs(£). Q
(3.8) Proposition. Letobeafunctionwhichsatisfiesthetwoconditionsof(3a}forsome
non-empty homomorph §. Then j(G) / 0 for all G e ® if and only if a(G) / 0 for all
G e b(§).
Proof Suppose that a(X) / 0 for all X e b(§), and let Ge®. Proceeding by induc-
tion on |G|, we may clearly suppose that G £ § (in particular that G #1) and that
a(I-) # 0 for all groups L e ® such that |£| < |G|. Let 1 / N < G. By induction
a(G/N) contains a subgroup, V/N say, and if |P| < |G|, there exists a subgroup
17 e a( V). Then from (3.a)(ii) we conclude that V g j(G). There remains the possibility
that j(G/N) contains G/N, which implies that G/N e § by (3.a)(i). But if G/N e for
all non-trivial normal subgroups N of G, we have G e b(§), and then a(G) / 0 by
assumption. The induction argument is therefore complete. □
The upshot of (3.7) and (3.8) is that the question of the universal existence of projectors
and covering subgroups for a non-empty homomorph now reduces to an examina-
tion of its boundary groups. The next result is designed to facilitate this task. Recall
from A, 15.2 that there exist 3 distinct types of finite primitive groups and that
accordingly we write
ф = ф10ф20ф3.
The socles of groups in and ®2 are minimal normal subgroups, abelian for type
1 and non-abelian for type 2, whereas the socle of a group in ®3 is a direct product
of two isomorphic non-abelian minimal normal subgroups. Also recall the conven-
tion that we consider only those primitive groups lying in the fixed universe, in other
words, ф is the class of all primitive groups in ®.
(3.9) Lemma. Let Sy be a homomorph, and let G e b(§).
(a) If Projs(G) is non-empty, then G is primitive.
(b) If G is primitive, then the following statements are true:
(i) If Ge ®. и®3, then Cov6(G)and Proj6(G)both coincide with the non-empty
set comprising those subgroups of G which are complements in G to each minimal norma
^(i^If Gel and then Proj6(G) is non-empty and consists of all Sy-
maximal subgroups of G which supplement Soc(G) in G.
292 III. Projectors and Schunck classes
Proof, (a) Let 17 e Projf,(G). Since G (f f>, we have U < G. Let U < M < G, and let
A = Corec(A/). If К / 1, then G/K g § by definition of the boundary, and therefore
bv definition of an Svprojector we have G = UK < M < G. This contradiction shows
that К = 1 and hence that G e ф.
(b) First we make the obvious remark that, because G g b(§), a subgroup H is an
y>-projector of G if and only if
J (i) H is f>-maximal in G, and
(3 {(ii) HN = G for all N •< G.
For the purposes of this proof only, put
,/(G) = {S < G: SN = G for all N <G},
and observe that Projs(G) £ .‘/(G). For G g 33t и 'Тз we know that У(С) / 0 by A,
15.2, and that furthermore, each S g У(С) is a maximal subgroup of G which not only
supplements, but actually complements, each minimal normal subgroup N of G.
Hence in this case S = G/N e f>, it follows that S is ^-maximal, and therefore
S e Proj6(G) by (3./7). Since an ^-projector which is a maximal subgroup is obviously
also an ^-covering subgroup, we have therefore verified Statement (i) of Part (b).
Now let G e $2, and let N be the unique minimal normal subgroup of G. To justify
Statement (ii) we need only show that Projg(G) =A 0, since the rest follows from (3./7).
Let Ho be a minimal supplement to N in G. By A, 9.2(c) we have Ho n N <, Ф(Н0);
since H0/(H0 nN) = H0N/N = G/N e f>, it follows that Ho g Eef> = by hypo-
thesis. If H is an ^-maximal subgroup of G containing Ho, a further appeal to (3./?)
shows that H e Proj6(G). □
We can now state and prove Forster’s characterizations of projective classes and
certain Gaschiitz classes in the framework of a general <s, q, E4>-closed universe ®.
(3.10) Theorem. A class is ®-projective if and only if it is a fR-Schunck class.
Proof. Let § be a 'Л-projective class, and let G g f>. (Since 93/0, clearly § / 0.)
Since Projg(G) / 0, it follows from the definition of a projector that Proj6(G) = {G}
and hence that for all TV < G the quotient GN/N is ^-maximal in G/N, in other words,
that G/N e therefore § = Q§. By (3.9)(a) we have b(f>) s ф, and consequently §
is a '-B-Schunck class by (2.7).
Conversely, if § is a 93-Schunck class, by (2.7) we have b(§) £ 33, and therefore,
bearing in mind that § = еф§ by (2.10), by (3.9)(b), (i) and (ii), we have Proj6(G) / 0
for all G g b(f>). Finally, by (3.7) and (3.8) the class § is '-B-projective. □
An identical argument using Lemma 3.9(b) (i) yields the following result.
(3.11) Theorem. A ‘R-Schunck class whose boundary contains no primitive groups of
type 2 is a Gaschiitz class.
3. Projectors and covering subgroups
293
This concludes our discussion of existence questions in this general setting Before
we move on to an investigation of conjugacy, we pause briefly to take an informal
look at a few examples, which may help to illuminate the theory so far. We shaU
discuss many concrete examples of Schunck classes later in this chapter, but as these
will usually be restricted to the universe S, the ones we are about to describe are
analysed in the universe G of all finite groups.
(3.12) Examples. Let ® = G throughout.
(a) Let IB be a class of finite simple groups, and let G.a denote the formation of
finite groups all of whose composition factors are in ®. Clearly S is a Schunck
boundary, and the G-Schunck class Л(й) is the class C® of «-perfect groups, viz.
groups G which have no quotients in the class ®.
Of course, by (3.10) each finite group has D®-projectors. although in general they
are not easy to characterize. For example, if ® contains only one finite simple group,
S say, then obviously every maximal subgroup of S is a C®-projector of S. However,
if ® has the property that every simple section of a S-group is again in ®, then 1 is
the unique E®-projector of each Be®, and from Theorem 3.19, proved later in this
section, we can deduce that each finite group G has a single conjugacy class of
C®-projectors. In fact, it is easy to check independently that in this case the nor-
mal subgroup 0®(G) is the unique Q®-projector of each group G; indeed, it is the
unique largest C®-subgroup of G and is therefore also a C®-covering subgroup
of G. Thus, apropos of (3.11), we see that a Schunck class may still be a Gaschiitz
class, even when its boundary contains primitive groups of type 2. Schunck
classes whose projectors are always normal subgroups will be discussed in the next
section.
(b) The purpose of this example is to show that a projective class need not be a
Gaschiitz class.
Let л £ P, and consider the class 'Ji, of nilpotent л-groups. Since ® n 91, =
(Z : p e л), a group in p'Ji, has the property that all its maximal subgroups are normal.
But a group with this property is nilpotent by A, 8.3. It follows easily that p'Ji, = 'Ji,
and hence by (2.7) that 91, is an G-Schunck class; in particular, 91,-projectors exist
in each finite group. It is easy to verify that b(9l„) consists of all those primitive groups
В of types 1 or 2 for which B/Soc(B) e 91, and the minimal normal subgroup Soc(B)
is either non-abelian or a (/-group for some prime q not dividing |B: Soc(B)|.
We assert that if {2, 3, 5} £ л, then 91, is not a Gaschiitz class. To see this we shall
quote without proof some elementary properties of the group A = Alt(5). For r - 2,
3, 5 let P e Syl,(A), and let N, = NA(P,). Then S Alt(4), N3 = Sym(3), and N, s
Dih(10). Furthermore, every maximal subgroup of A is conjugate to one of the /V, s,
and it follows that
{Pr“: r = 2, 3, 5 and a e A}
is a complete list of the maximal 91,-subgroups of A Since P, is not an 91,-covering
subgroup of Nr for any r, we conclude that Cov« (Л) - 0-
,e) tim .« .KI.
2, 3, 5) whose intersection is not a Gaschutz class, ano у
294
III. Projectors and Schunck classes
classes cannot be characterized by means of closure operations. (Naturally, Schunck
classes, which can be so characterized, are closed under intersections.)
Keeping to the notation of Example (b) above, we set
®, = (-4, ty, N.k where {r, s, t} = {2, 3, 5}.
Clearly ®r is a Schunck boundary, and so the class = h(®r) is a Schunck class.
Furthermore, Proj6r(/1) = {7V/1}, while the ^-projectors of Ns and Nt are the com-
plements of their socles. Thus the f>,-projectors of each of the three groups in b(Jjjr)
form a single conjugacy class of maximal subgroups and are therefore f>,-covering
subgroups. Consequently is a Gaschiitz class by (3.7) and (3.8). Now let § =
§2 n S3 n Ss- Since the class ® = (A, N2, N3, N5) is a Schunck boundary and § is
a Schunck class, it is not hard to verify that § = h(®). Therefore ®|2 3 5| £ 5, and
by the argument used in Example (b) above, we conclude that § fails to be a Gaschiitz
class.
Now we turn to the conjugacy question. As before, the object is to try to show that
it can be resolved by examining the groups in the boundary. This approach works
well for covering subgroups as the next result demonstrates.
(3.13) Theorem (Forster [II]). Let f> be a homomorph, and consider the following
condition for a group G;
(3.y) Covg(G) is either empty or a single conjugacy class.
Then this condition is satisfied for all G e SB if and only if it is satisfied for all G e b(§).
Proof. We argue by contradiction, assuming that (3.y) is satisfied for all G e b(SB) but
not for all Ge®. Then there exists a group G of minimal order in ® containing
non-conjugate ^-covering subgroups, U and V say, and it is clear that G </ §. If
1 # N < G, by (3.6)(a) the group UN/N and VN/N are ^-covering subgroups of G/N,
and therefore UN = (P/V)s for some g e G by the minimality of |G|. But then U,
Vе e Covf,(UN), and if | <7/V| < |G|, it follows, again from the choice of G, that U is
conjugate to Vе and hence to V. This contradiction means that G = UN. Therefore
G/N s U/(U n N) e = f>, and this holds for all non-trivial normal subgroups N
of G. But then G e b(Sj}, and so U is conjugate to V by hypothesis. This is the final
contradiction. □
Now we turn our attention to the conjugacy problem for projectors, and here the
situation is not so clear cut. The main result, Theorem 3.19, gives only a qualified
answer, because primitive groups of type 3 have to be excluded from boundaries of
the Schunck classes under consideration. The analysis leading up to the proof of this
theorem, due in large part to Forster, is also not so tidy, and requires separate
pleading to deal with abelian and non-abelian socles. The abelian case is included in
the following result.
Projectors and covering subgroups
295
(3.14) Lemma (Gaschiitz). Let § be a (^Schunck class. Let N be a nilpotent normal
Then X ProfTn G’ and et H be an ^"'ax,mal sub9rouP °f G ^h that G = HN.
1 fit. fl П с г rojftiwl.
Proof. We argue by induction on |G|. IfG e we have H = G e ProL(G). Therefore
XP°St^at РУ (2'2)(C) there eX‘Sts a normal subgrouP К of G such that
G/K 6 b(§), and clearly NfK because G/N = H/(H nN)eQf> = ^ Thus G/K has
a non-trivial nilpotent normal subgroup NK/K and so is a primitive group of type
1 by (2.7) and A,15.2. Hence NK/K = F(G/K} = Soc(G/K), and it follows that the
S-subgroup HK/K is a maximal subgroup of G/K complementing NK/K' con-
sequently HK/K e Proj&(G/K). Since H(HKcsN) = HK by the Dedekind law, H
is an ^-maximal subgroup of HК supplementing its normal nilpotent subgroup
HKc.N, and therefore by induction H e Proj6(HK). Hence by (3.7) we have
HeProj6(G). ‘ (j
The next lemma and its corollary describe a minimal configuration often encountered
in the study of Schunck classes, and although only a part of the lemma is required
for our immediate purposes, we state it in its fullest form for future reference.
(3.15) Lemma. Let § be a Schunck class such that b(§) Let X e ®\§,
and assume that X has distinct minimal normal subgroups M, N such that X/M and
X/N are in f). Let H be a proper subgroup of X such that X = HM = HN; for example,
let H be an ^-projector of X. Then the following assertions are true:
(a) M and N are abelian;
(b) H n M = H n N = 1;
(c) MN/M and MN/N are Frattini chief factors of X;
(d) If T = Ф(Х) n MN, then 1=£T < X; moreover X/T<£ §, and M ф T / N;
(e) T = HcyMN-sX;
(f) M = T ~ N as X-modules;
(g) Х/Согел(H) e b(§) n ф,.
Proof, (a) Since X <£ by (2.2) (c) there exists а К < X such that X/K e b(§). If
M < K, then X/K e Q(X/M) S = S, which is not so. Therefore M n К = 1 and
[M, K] = 1. Furthermore, M = MK/K, whence MK/K<i X/K, and since the pri-
mitive group X/K has type 1 or 2 by hypothesis, it follows that MK/K = S«WK)-
Similarly, NK/K = Soc(X/K), and so. in particular, M < NK. Since [M, NJ <
M rs N = 1 we conclude that [M, M] < [M, NK] = 1, and evidently Assertion (a)
is true.
(b) Since НМ = X and M is abelian, we
proper subgroup of X, we must have H r\ M
have H n M < НМ = X. But as H is a
< M, and therefore H n M = 1 because
“(c) LaR = MNnK;thenR < X. and MW = MNrsMK = MR =
thlt the chief factor MN/N is not Frattini; since it is atelian, it tten has^romfdement
in X/N, say L/N. It follows that LR = LM = X and thatLo M 1Ln MNnM _
N n M = 1 Hence Ln R = 1 because |R| = |M|, and consequently X/R = L _
296
III. Projectors and Schunck classes
X/M e But then we have X/K e Q(X/R) s f>, which contradicts the choice of K.
Hence Assertion (c) holds.
(d) Clearly T < X, and in view of Part (c) it follows from A,9.11 that T / 1. If
X/T e f>, then X g EeS = by (2.10), contrary to hypothesis. Therefore X/T ф § and
Assertion (d) is now clear.
(e) Parts (a) and (b) imply that H < X and therefore that T < H n MN. Since MN
is abelian, H n MN is normal in X, and because T # 1 and MN has X-composition
length two, it then follows that H n MN must equal either T or MN. But the second
possibility would imply that M < H, contrary to Assertion (b); therefore Assertion
(e) is true.
(f) Again because MN has X-composition length two, we have MN = TN, and
hence M = MN/N = TN/N = T. Similarly N = T.
(g) Since HK/K is an ^-projector of X/K e b(§), it follows that НК < X. But
H is a maximal subgroup of X; hence К < H, and consequently К = CoreA(H).
Finally, we know from Part (a) that Soc(X/K) is abelian, and so X/CoreA(H) =
Х/КеЦ^Ц,. □
The information given in the above lemma comes in useful when analysing Schunck
classes § which are not formations, for reasons suggested by the following corollary.
(3.16) Corollary. Let § be a Schunck class, and assume that Rr,f>\f> is not empty. Then
Rr,f>\f> contains a group X such that
(q - 1)(X)or„§ s §.
Furthermore, X has two distinct minimal normal subgroups M and N such that X/M
and X/N are in Not only are Assertions (a)-(g) of Lemma 3.15 therefore satisfied,
but in addition the following statement holds:
(h) If p is the prime divisor of |MN|, then OpfX) = 1.
Proof. A group X of minimal order in Rof>\f> obviously fulfils the condition
(q — l)(X)nR„§c§. Now such a subgroup X has t normal subgroups K, =
K, K2,..., K, such that X/K, g § and Q'-, X, = 1. By omitting any redundant terms
from this intersection, we may suppose that the group L = QJ=2 K, >s non-trivial.
But then X/L g (q — 1 )(X) n Rof>, and therefore by hypothesis X/L g note that we
also have К n L = 1. Let M be a minimal normal subgroup of X contained in K.
Since К n LM = M, the group X/М has two normal subgroups К/M and LM/M
with trivial intersection and with quotients in Therefore X/M g (q — 1)(X) n R„f>,
and by hypothesis X/M e f>. Similarly if N < L, N-< X, then X/N g f>. Thus the
first part of the corollary is clear, and it only remains to check that Assertion (h)
holds.
Let R = Op-(X). Since the ^-projector H of X has p-power index, then R < H.
Moreover, because M is the unique Sylow p-subgroup of MR, it follows that MR/R
and NR/R are distinct minimal normal subgroups of X/R. If R # 1, we therefore have
X/R g (q — 1)(X) n Rof> S anc] jn this case X = HR = H g §, a contradiction.
□
297
3. Projectors and covering subgroups
We now return to the preparations for the proof of Theorem 3.19.
(3.17) Proposition (Forster [11]). Let § be a Schunck class such that b(f>) e
Let A and В be normal subgroups of a group X such that
(i) A n В = 1,
(ii) X/B e f>, and
(iii) X has an ^-maximal subgroup H such that X = HA.
Then X = HB.
Proof. We proceed by induction on |X|. If A = 1, or В = 1, or X e §, the result is
clear. Let R be a non-trivial normal subgroup of X contained in either A or B. For
any Y < X, let Y* denote YR/R. Let Ht/R denote an ^-maximal subgroup of X*
containing the f>-group H*. Then it is straightforward to verify that Statements
(i) (iii) remain true when Ht is substituted for H and stars are applied. There-
fore by induction X* = (HffB*, whence X = HtB. Suppose that H, < X. Since
Я(Я1 n A) = Ht n HA = Ht and Hj/fH, n B) s HXB/B = X/B e §, in this case we
can apply the induction hypothesis to H, to conclude that H(Ht n В) = HL. Then
X = HtB = H(Hi n B)B = HB, as required. Thus we can assume that = X, and
hence that
(3.<5) X/R e § whenever 1 / R < G and R < A or R < B.
Let M, N < X with M < A and N < B. Since X/M and X/N are in § by (3.<5) and
since we can suppose that X it follows from (3.15) that M and N are abelian and
isomorphic with each other as A"-modules; in particular, CX(M) = CX(N). Since
[A, N] < A n В = 1, it therefore follows that A centralizes M. Since § = Еф§, we
have M f Ф(Х), and so there exists a complement, L say, to M in X. In this case we
can conclude from A, 1.3 that L n A is a normal subgroup of X complementing M
in A. If L n A # 1, let Mo be a minimal normal subgroup of X contained in L n A.
By (3.<5) we have X/Mo e f>, and so (3.15) applies (with Mo substituted for the N
of that lemma). From (3.15)(d) we deduce that Ф(А')пЛ/1, in which case
Х/(Ф(Х) n A) e f>, and therefore we have X e = §, contrary to supposition.
Hence L n A = 1 and A = M e 81. But then by (3.14) we have H e Projg(X), and
consequently X = HB. □
The next result is the analogue of (3.14) for a non-abelian socle.
(3.18) Lemma (Forster [11]). Let § be a Schunck class such that b(£>)
Let G = HN, where N is a direct product of non-abelian simple groups and is normal
in G, and where H is an ^-maximal subgroup of G. Then H e Projg(G).
Proof We shall prove that H has the defining property of a projector, namely that
iTK < G, then HK/K is ^-maximal in G/K. Since NnK< G, by A,46 we have
N = No x (N n K), where No < G. Suppose that HK/K < L/K e f>- Then
L = HNcL= H(N nL) = H(N0 nL)(Nn K).
298
III. Projectors and Schunck classes
Put X = H(N0 n L), and note that L = XK. Therefore X/(X nK) XK/K g
Now apply (3.17) to X, with No n L and X n К in place of A and В respectively, to
conclude that X = H(X n K), and hence that L = H(X n K}K = HK. □
We are now ready to prove the main conjugacy theorem for projectors.
(3.19) Theorem (Forster [11]). Let § be a %}-Schunck class such that b($f) c u®2.
Then the statement :
(3.e) “Projg(G) is a conjugacy class of G"
is true for all Ge ® if and only if it is true for all G e b(§).
Proof Supposing that (3.e) holds for all G e b(§), we argue by induction on |G| that
it holds universally. Let Ge®, and let TV-< G. By (3.10) Proj6(G) 0; therefore let
Hl, H2 e Projg(G). Since HtN/N e Proj&(G/TV) for i = 1, 2, by induction we have
Pf N/N = (PP2N/N)9fl for some g e G; hence Pf and PJ2 are ^-maximal subgroups of
HtN = H%N. From (3.14) if N is abelian, and from A, 4.20 and (3.18) otherwise, we
conclude that If and Hf are ^-projectors of PfN. If |H, /V| < |G|, by induction Pf
is conjugate to f/f and hence to H2. We can therefore suppose that HtN = G and
hence that G/N = Pf/(Pf n N) e Qf> = § for all N < G. Thus, either G and then
Pf = G = H2, or G e b(f>) and Pf is conjugate to H2 by hypothesis. □
(3.20) Remarks, (a) In fact, our Theorem 3.19 is only a part of what Forster proves
in this direction in his cited work. For ® = G he also shows that if a Schunck class
§ contains a primitive group of type 3 in its boundary, then there exists a group with
at least two conjugacy classes of ^-projectors. Thus, for the universe of all finite
groups, the following condition:
(3-0 b(§)c«p1U«p2
is a necessary condition for the conjugacy of ^-projectors in all finite groups, and,
when (3.f) holds, their conjugacy in the groups of b(§) implies their universal
conjugacy.
(b) In the same work Forster indicates that the maps Projg( ) and Covg( )
coincide on G if and only if (i) they coincide on b(f>), and (ii) Condition (3.£) holds.
(c) Of course, Theorem 3.19 applies in a soluble universe, where Condition (3.£)
always holds. Insoluble examples of a situation where Theorem 3.19 applies are
provided by (3.12)(c). There it is observed that each group in the boundary of the
Schunck class f>r contains a unique conjugacy class of §r-projectors; furthermore,
these boundary groups are either simple or soluble and are therefore of type 1 or type
2. From Theorem 3.19 we can therefore deduce that each finite group has a unique
conjugacy class of f>r-projectors for r = 2, 3, and 5.
We now wish to interpret the foregoing results in a soluble setting, and therefore
we stipulate that
for the rest of this section the universe is 6.
3. Projectors and covering subgroups 299
Part (a) of the main ‘soluble’ theorem which r„u
(3.21) Theorem. Let § be an Q-Schunck class, and let Get. Then
(a) Cov6(G) is a conjugacy class of G, and
(b) Cov&(G) = Proj6(G).
In particular, § is a Gaschiitz class for <5.
Proof. Since the primitive groups in S are of type 1. by (3.11) we have
0 / Cov6(G) c Projg(G).
If В e b(§), by (3.9) (b) (i) the ^-projectors of В are the complements in В of Soc(B),
and by A, 15.6 these form a conjugacy class of B. Hence by Theorem 3.19 ProjJG)
is a conjugacy class of G. Therefore by (3.6)(a)(ii) we have Covg(G) = Proj6(G), and
both parts of the theorem are now clear. q
Now we state two obvious, but important, consequences of this theorem.
(3.22) Corollary. Let § be a Schunck class, and let H be an ^-projector of a soluble
group G. Then
(a) if H < L < G, then H g Projg(L), and
(b) H is pronormal in G.
The following closer analysis of the behaviour of ^-projectors in an 'Jift-group will
be useful later.
(3.23) Proposition. Let Sy be a Schunck class. Let N be a nilpotent normal subgroup of
a group G (g<5), and let Lbe a supplement to N in G. Assume that G/N e f). Then:
(a) The Sy-maximal supplements to N in G coincide with the Sy-projectors of G, and
hence form a conjugacy class-,
(b) If Ее Projg(G), then there is a unique Sy-projector of G containing E;
(c) If the Sy-projectors of G avoid N, then Projg(L) £ Proj6(G).
Proof, (a) This follows immediately from (3.14) and (3.21).
(b) Since L/(L n N) G/N e Sy, we have F.(L cN)=L and hence EN = G. Thus
it will suffice to prove (by induction on |G|), that given an ^-subgroup E supplement-
ing N in G, there exists exactly one «(-projector containing it. By Part (a) there is at
least one such «.-projector. Suppose that E < If n H2 with and H2 in Projg(G).
Since we may clearly suppose without loss of generality that G£H, there exists a
normal subgroup К of G such that G/K e b(f>). Let S* denote SK/K for all S < G.
Then evidently E* is an ^-subgroup supplementing the nilpotent normal subgroup
IV* of G* and contained in the «.-projectors H* and H2*. If К = 1, then G it1 primitive
and so N = Soc(G). In this case we obviously have E — — H2. ,
induction hypothesis yields Я.К = HiK < a Since H' and aK &-ProJectors of
300
III. Projectors and Schunck classes
Я,К containing the supplement E to the nilpotent normal subgroup HtK n N of
/7, Л', the induction hypothesis again applies and gives H, = H2.
(c) Let Ее Proj6(L) and let E < H e Proj&(G). Then, as above, EN = G, and
therefore H = El n EN = E(H c.N) = E, since H n N = 1 by hypothesis. □
In the sequel we shall also need the following technical lemma.
(3.24) Lemma (Forster [1]). Let f> be a Schunck class. Let G = NL be a semidirect
product in which the normal subgroup N is nilpotent, and let H be an ^-maximal
subgroup of G. Then
H = [H c. N)[H r. L")
for some g e NH.
Proof Let Lo = NH n L. Then NL0 = NH, and therefore Lo = NH/N s
H/(N ri H) e q£> = У>. By (3.23) (a) we have H e Proj b(NH), and by the same result
Lo is contained in some conjugate of H. Hence L90 < H for some g e NH. It fol-
lows that L$ = H ci Lq = H с (NH)9 rL9 = H c\L9, and consequently that H =
H NL90 = (H N)Lq = (H n N)(H L9). □
(3.25) Concluding Remarks. The following elementary, but important, observations
for the soluble universe are implicit in what has gone before and are stated here for
emphasis. In the sequel they will often be used without explicit reference. As usual,
§ is a Schunck class.
(a) Let G ei₽. Any two of the following statements are equivalent:
(1) G e b(&);
(2) Soc(G) is complemented by an ^-projector of G;
(3) Soc(G) is complemented by all ^-projectors of G.
(b) Let N < G. If U e Proj 6( UN) and UN/N e Proj&(G/A), then U e Proj&(G).
(This is a restatement of a part of Proposition 3.7)
(c) If N < G and U/N e Proj6(G/A), then U = HN for some H e Proj &(G). (This
is a consequence of (3.7) together with the fact that an ^-projector covers U/N e У>.)
(d) Let H be an ^-projector of a primitive group G. Then by (3.24)
H = (H c Soc(G))(H n L)
for some complement L to Soc(G) in G.
Postscript
Our main theme in this section has been Forster’s generalization of the theory of
projectors and Schunck classes to arbitrary finite groups. Other ways of extending
the soluble theory have also been looked at. Schmid [2] and Erickson [2], for
example, have studied ^-projectors in the universe Sf> and have found conditions
on the (not necessarily soluble) class § for their existence, conjugacy and persistence
in that universe.
3- Projectors and covering subgroups
301
Another approach is due to Salomon [1] and takes as its starting point the
observation that in a soluble group every subgroup either covers or avoids all simple
sections of the groups. He makes the following definition*
Let G be a finite group. A subgroup U is well-embedded in G if every simple section
H/K oi G is either covered or avoided by G (that is to say if Hr, U < К then
H < (H n G)K). * ’
The well-embedded subgroups (WE-subgroups, for short) form a lattice with
respect to intersection and the supremum join, though not in general a sublattice of
the usual subgroup lattice; their composition factors are evidently a subset of those
of the parent group. Salomon proves, furthermore, that when their composition
factors are all non-abelian, then they are subnormal. The WE-subgroups of Sym(n)
(n > 5) are just the normal subgroups together with the subgroups generated by the
odd permutations of order 2. Of course, in a soluble group every subgroup is
well-embedded.
With such concepts as primitive group, projector, and Schunck class aptly defined
in terms of WE-subgroups (and agreeing with the usual definitions in S), Salomon
proves the existence and persistence of‘well-embedded’ projectors for ‘well-embedded’
Schunck classes in every finite group. Moreover, their conjugacy is universally
guaranteed if they form a conjugacy class in each group of the boundary.
By way of application consider, for a given prime p, the subgroups G of a finite
group G which are minimal subject to the conditions
(1) G is well-embedded in G, and
(2) p||G: G|.
Salomon’s theory shows that such ‘well-embedded’ Sylow p-subgroups are projectors
of G for a suitable ‘well-embedded’ Schunck class, and furthermore that they form a
conjugacy class of G.
Exercises
1. Show that Sym(4) has two conjugacy classes of «-projectors and no «-covering
subgroups. Find a subgroup of Sym(4) that has no «-projectors.
2. Let & be a Schunck class, and let M be a maximal subgroup of a soluble group
G. If M is in f>, is M necessarily contained in some ^-projector of G?
3. Let E be an ^-maximal subgroup of a soluble group G. Is either of the following
conditions sufficient to ensure that E e Proj?,(G)?
(a) EN/N is ^-maximal in G/N for all N < G;
(b) E/V//V e Proj(,(G) for all < G. UL.1CI (1й11.и«
4 Let ®, denote the class of finite simple groups S such that |S| e t2 60), let
(respectively «,) consist of all finite primitive groups G of type 1 and 2 such that
‘the quotient G/Soc(G) is non-trivial and has all its composition factors isomor-
phic with Z2 (respectively Alt(5)). Let S = U?=i let the unlverse be ®’and
= h(®). Then prove the following assertions:
(i) 93 is a Schunck boundary; h that
(ii) If A = Alt(10), then A has a subgroup S isomorphic with Sym(5) such tha
c.(S) = i; r.
(iii) If |S: H| = 2, then H is an ^-projector ol A,
302
III. Projectors and Schunck classes
(iv) H is not an ^-covering subgroup of Л;
(v) If В g 23 and T e Syl2(B), then T e Cov&(B).
Deduce that § is a Gaschiitz class for G such that Proj&( ) Covg( ).
5. (Forster [11]) Let § be a 21-Schunck class, where 2< satisfies (3.4). Show that if
b(f>) contains groups of type 3, then the assumption: "0 * Projs(B) = Cove(B)
for all В e b(S>)” does not imply that Proj6(G) = Cov6(G) for all G e 23.
6. Theorem 3.13 suggests the following proposition: “If each group in b(£>) has at
most n conjugacy classes of ^-projectors, then so does each group in the universe
23”. Show this to be false when n > 2.
7. (Forster [11]) Let § be a 21-Schunck class. Prove the equivalence of each pair
of the following three assertions:
(i) b(£>) £
(ii) For all В e b(£>), the following set:
{H < В: H is ^-maximal in В and HN = В for some N < B},
coincides with Proj j,(B);
(iii) For all G e 23, the following condition:
'‘HN/N is ^-maximal in G/N for all terms N of a given chief series of G”
implies that H e Projg(G).
8. Let 23 be an <s, q, i^)-closcd class containing S. Show that if every group in 23
has a unique conjugacy class of S-projectors, then 23 = S.
9. Find an example to show that (3.24) fails to hold when the hypothesis: “H is an
^-maximal subgroup of G” is replaced by: “H is an (p-subgroup of G”.
10. (a) Let N < G e S with N e 9L Let § be a Schunck class, and let H be an f>-
maximal subgroup of G such that HN/N e Proj f/G/N). Prove that H e Proj &(G).
(This has a dual in the Fitting class case—see IX, 1.6).
(b) Use Part (a) to deduce that if Я is a subgroup of G with the property that
HNi/N; is ^-maximal in G/N, for each term Nt of some normal series
1 = No < N, < ••• < Nr = G
of G with Ni/Ni_l e 91 for i = 1,..., r, then H 6 Proj6(G).
4. Examples
Initially we work within a general universe 23, tailored to the requirements of each
result. We recall from II. 2.2 and 2.1 that a formation is a class of groups closed
under forming quotients and residuals (q- and Ro-closed), and that a saturated class
is one closed under Frattini extensions (E^-closed).
4. Examples
303
(41) Proposition. Assume that ® is a non-empty saturated homomorph. Then a class
anTaUformahoned formation $ if it is a 33-Schunck class
Proof Let g be a saturated formation in <8. Since g = Qg, we have g <= Pg bv
(2.5)(c). Now let G g Pg. If M <. G, the group G/Corec(M) is primitive an! therlm
belongs to g. Because <I>( G) is the intersection of the normal subgroups CoreJM) as
M runs through the maximal subgroups of G, it follows that G/<I>(G) e Rog = g, and
consequently G e E^g = g. Therefore g = Pg, and g is a Schunck class by (2.7)’ The
reverse implication is clear because by (2.10) Schunck classes are always saturated.
□
This proposition puts a rich source of Schunck classes at our disposal. In Chapter
IV, Theorem 3.3 we shall prove that local formations are saturated. The classes: U
(supersoluble groups—see IV, 3.4(f)), 9Г (soluble groups of nilpotent length at most
г). £Р(Н (p-soluble groups of p-length at most r), and their intersections with the classes
G„ and S„, are among the many well-known examples of local formations and are
therefore Schunck classes. We shall now discuss some of these and other examples
of Schunck classes, describe properties and characterizations of their projectors, and
finally show how to identify their projectors in the subgroup lattice of a specific group.
Schunck classes with normal projectors
Schunck classes whose projectors are always normal subgroups were first studied in
the universe S by Blessenohl and Gaschiitz [1]; they characterize them as the classes
Q' of (soluble) л-perfect groups. Lafuente [1] later extended this investigation to the
universe G. We now apply the methods of Section 3 to characterize such Schunck
classes in the setting of a more general universe which satisfies (3.4) and the
additional condition 9!2 = 93. This analysis is a good illustration of the use of the
boundary concept as an effective and economical tool in the study of certain questions
about Schunck classes.
(4.2) Theorem. Assume that the universe 93 satisfies the conditions: 932 = 93 =
^S, Q, еф>93, and let S)be a 33-Schunck class. If one of the following assertions is true,
then they are all true. , n. m
(i) There exists a formation g S 93 satisfying g = sg = g such that Proj6(G) —
{G”} for all G e 33:
(ii) Proj s(G) contains a normal subgroup of G for all Ge 93;
(iii) b(&) is a class of simple groups such that every simple section of a group in h(ft)
also belongs to b(Sy).
Proof, (i) =>(ii): This is clear. . ,
(iij^(iii): Let В G b(&), and choose an H in Proj6(B) with H B=I # L th^_
BIHeto - l)(B)G.ft, and so theft-projector Я covers B/H. Thus В
i ж . «С \ ррлГЯ Hence H= 1 Let N be a minimal normal subgroup
which contradicts the choice of B. Hence n i. N d
of B. Since B/N g У>. it is covered by each H g Proj£,(B); therefore В - HN N, ana
we see that В is simple.
304
III. Projectors and Schunck classes
Now let T <) • S < B, let £ be a minimal supplement to T in S, and note that E 1.
Since В e b(S>) £ ®, we have S/Те qs® = ®, and hence A, 9.2(c) yields E e еф® = ®.
If S T g У>, then £ is in and hence in § by (2.10). But we have shown above that
the subgroup 1 is an ^-projector of B; in particular, 1 is ^-maximal in B, and therefore
£ = 1. From this contradiction we conclude that the simple section S/T is not in §
and therefore belongs to b(5).
(iii) => (i): Set ® = b(£>), and let ft be the class of all ®-groups whose composition
factors belong to ®. The condition ®2 = ® implies that ® = Do®; therefore by 11,
1.18(c) the class ® is a formation. It is then obvious that ft is a formation and that
ft2 = ft; furthermore, it is straightforward to verify that the requirement that ®
contains all simple sections of ®-groups implies that ft = sft.
Let G e ®, let H e Proj6(G), and let R = Gx. First we show that H < R. If not,
then H/(H R) is a non-trivial group, isomorphic with HR/R e sft = ft. Therefore
if КЦН r. R) < H/(H ci R), then H/K is a simple group in Qft = ft, and by assump-
tion belongs to b(£>). But this contradicts the fact that H/K e q?> = S), and so H < R.
Now we show that Ref). For, if R ф f>, then by (2.2)(c) there exists a T < R such
that R/Te b(S>) = ® £ ft, and consequently R" < T < R. But G/Rr e ft2 = ft, and
so RR = R, a contradiction. Therefore Re§, and so finally H = R. □
(4.3) Corollary, (a) If Sy satisfies one (and hence all) of the Conditions (i)-(iii) of
Theorem 4.2, then
Proj6(G) = Cov6(G)
for all Ge®.
(b) Theorem 4.2 remains true i/‘Proj6( )’ is replaced throughout by ‘Cov6( )’.
Proof, (a) Let G e ®, let H e Proj&(G), and let H < L < G. Since L/H e s(G/H) =
s(G/GR) £ sft = ft, it follows that H = Lf g Proj6(£). Hence H e Cov&(G). Since it
is always true that Cov6(G) s Proj6(G), we have equality.
(b) This is now clear from (a). □
A special case of Theorem 4.2 concerns the situation when the boundary of a Schunck
class Sy consists of abelian simple groups, for then it is clear that Condition (iii) of the
theorem is fulfilled. In this case we write
tt= {ре P : Zp e b(£>)},
and observe that the formation ft of Condition (i) is then the class S„ and that Sy
itself is the class Q" of all ®-groups which coincide with their ©„-residual; or,
equivalently:
Sy = (G e ® : G/G' is a л'-group).
In particular, this situation occurs when the universe is <5, and in this case we therefore
obtain the following reformulation of (4.2).
4. Examples
305
(4.4) Theorem (Blessenohl and Gaschiitz [1]). An ^-Schunck class ft has the propertv
that the ^-projectors of G are normal in G for all G e S if and only if =
class oj soluble n-per/ect groups, for some set n^P.In this case Proj JG) = {O"(G)}
Remarks, (a) Since a pronormal subgroup which is subnormal is already normal by
L 6.3(d), it follows from (3.22)(b) that Theorem 4.4 also gives a description of
^-Schunck classes with subnormal projectors.
(b) Forster [2] has characterized Schunck classes with normally embedded pro-
jectors, but as his proof requires deeper techniques, it is presented with the advanced
material (see Section 4 of Chapter VI).
The remaining examples that we shall discuss are most comfortably handled in a
soluble setting. We therefore specify that
for the rest of this section the universe is S.
Two facts particularly relevant to this universe are worth repeating. The first is that
by (3.21 )(b) projectors now always coincide with covering subgroups and therefore
enjoy the property of persistence. The second is that the ^-projectors, as maximal
to„-subgroups, are just the Hall л-subgroups Our next example has both historical
and structural importance for the theory of soluble groups. Its discovery had its roots
in Carter’s Cambridge doctoral dissertation, where he began a detailed investigation
into properties of system normalizers.
Carter subgroups
(4.5) Definition. A Carter subgroup of a group is a self-normalizing nilpotent
subgroup.
(4.6) Theorem (Carter [2], 1961). Each finite soluble group possesses exactly one
conjugacy class of Carter subgroups, namely the 9l-projectors.
Before we prove Carter’s theorem, it is convenient to settle first the general question
of when an ft-projector is self-normalizing.
(4.7) Lemma. Let n be a set of primes. A Schunck class ft of soluble groups has
characteristic n if and only if 9I„ £ ft £ О .
Proof. Set I = (Z : p e л), and suppose that Char(ft) - n. Let G e 9l„. By A 8.3
G in ft\Q”’ has a quotient isomorphic with Z, for some-q e n .the ex's'“‘“ =
a group would therefore -mply that Z.,^ = 5 an hence that^
which is impossible. Therefore ft £ £ . Inc necessuy o.
proved, and the sufficiency is obvious.
306
III. Projectors and Schunck classes
(4.8) Lemma. Let fybea Schunck class of characteristic n, and let H be an ^-projector
of a group G. Then NC(H)/H is a n'-group.
Proof. If NC(H}/H 4 <=„, then NG(H) has a composition factor, R/S say, such that
H < S and |R/S) e it. Since H e Projg(R) and R/S e £>, it follows that R = HS = S, a
contradiction. Therefore NC(H}/H e <5n.. □
If G e 9i„ and Char(£>) = n, then Projg(G) = {1}. Therefore from (4.7) and (4.8)
we can deduce the following description of Schunck classes with self-normalizing
projectors.
(4.9) Corollary. Let Sy be a Schunck class. The ^-projectors of G are self-normalizing
in G for all G e S if and only if 91 £ S>.
We remark in passing that (4.9) also gives a criterion for projectors to be always
abnormal subgroups; for by (3.22)(b) projectors are pronormal, and by I, 6.21(b) a
pronormal subgroup is abnormal if and only if it is self-normalizing. Thus, in
particular. Carter subgroups are always abnormal.
The proof of Theorem 4.6. By (4.9) an 91-projector is clearly a Carter subgroup. To
see the converse, Jet U be a self-normalizing nilpotent subgroup of a group G. By A,
8.3 the subgroup U is 91-maximal in G. Arguing by induction on |G|, we choose a
minimal normal subgroup N of G. If UN = G, then U e Proj^fG) by (3.14). Thus
we may suppose that UN < G and hence, by induction, that U e Projw(UN). Let
xN e Ne/fl(UN/N). Then Ux is a Carter subgroup of UN; it is therefore by induction
an 91-projector of UN and, as such, is conjugate to U. Thus Ux = Ur with у e UN,
whence we have xy-1 e NG(U) = U and x e UN. Hence we have shown that UN/N
is self-normalizing in G/N and is therefore a Carter subgroup of G/N. Thus, again by
induction, UN/N e Proj gjG/N). It now follows from (3.7) (in particular from the fact
that Proj9|( ) satisfies Condition 3.a(ii)) that U e Proj9,(G). □
Schunck classes described by properties of maximal subgroups
A group belongs to a Schunck class § if and only if its primitive epimorphic images
belong to S>. Thus is determined uniquely by its primitive groups, that is by the class
From the Q-closure of S>, it follows that Q(£»f) ф <= S ф.
(4.10) Definition. A class Я is called a Schunck basis if
(4.a) 0(Я) n 93 £ Я £ *p.
Thus, when § is a Schunck class, 5* is a Schunck basis; we call simply ‘the basis’
of ft (Gaschiitz [15] calls a class Я satisfying (4.a) a ‘primitive class’.)
4. Examples
307
We now record the following elementary consequence of this definition.
(4.11) Proposition. The class maps f and p induce inclusion-preservina mutual
inverse bijections serving, mutually-
between the sei JT of Schunck classes and the set Jf of Schunck bases. (Although
STk ,'aSS7 are "оп’етР1> ЬУ definition, the empty class is a Schunck basis; in
fact, the class (1) in JT corresponds to 0 in Ж At the other extreme, note that £ in
JT corresponds to in jf.)
Thus with each Schunck class there are associated two disjoint classes of primitive
groups; its basis and its boundary. It will become clear in Chapter VI that the
properties of the boundary hold the key to the deeper structural questions about
Schunck classes. Nevertheless, there are also situations where the basis is useful,
especially as a descriptive tool.
(4.12) Remarks and Examples, (a) Let I £ ^3. Evidently the class
(G e I: q(G) n 'J3 s I)
is the largest Schunck basis contained in I and is the basis of the Schunck class pX
(b) If § denotes the Schunck class generated by Sym(3), it must contain the
class R = Q(Sym(3))n ф = (Z2, Sym(3)). Since R is a Schunck basis, it follows that
& = pR. (f> can be described as the class of all G3S2-groups whose complemented
3-chief factors are cyclic and eccentric; such groups are supersoluble.) More generally,
if ЭЕ is an arbitrary class of groups, it is easy to see that is the basis of pqJE,
the Schunck class generated by X. If T contains only finitely many isomorphism
classes, so does qI n In this case there are only finitely many Schunck bases
contained in qI n and therefore by (4.11) a Schunck class generated by a finite set
of groups contains only finitely many Schunck subclasses. (The corresponding result
for formations of soluble groups is also true—see VII, 1.6.)
(c) For later reference we observe that the basis of the class is (Zp:pen);
also that by IV, 3.4(f) the basis of the class U of supersoluble groups consists
of all primitive groups with a cyclic socle. Whereas such classes as and U
are obviously more elegantly described by their bases than by their boundaries,
in the case of the ‘large’ class Q’ the boundary provides the more economical
description.
The bijection described in A, 15.7 between the primitive quotient groups of a
soluble group and its conjugacy classes of maximal subgroups lets us translate
statements about the primitive epimorphic images of a group into corresponding
statements about the behaviour of its maximal subgroups. For example, to say that
all the primitive epimorphic images of a group are cyclic is the same as to say that
all its maximal subgroups are normal; and the supersolubihty of a group translates
into the property of having all maximal subgroups of prime index (cf. Theorem VU,
2.2, (a) «-(c)).
308
III. Projectors and Schunck classes
(4.13) Deflnition. Let ЭЕ be a class of groups. A maximal subgroup M of a group G
is said to be X-normal if
G/Corec(Af) e 3E;
otherwise it is said to be H-abnormal.
(4.14) Remarks, (a) A maximal subgroup is I-normal if and only if it is (ЭЕ n 'bi-
normal.
(b) A maximal subgroup is X-normal if and only if it is (S\I)-abnormal.
(c) A maximal subgroup is normal in the usual sense if and only if it is 91-norrnal
in this terminology. However, a normal maximal subgroup, of index p say, is only
I-normal if Zp e I.
(d) Let ЭЕ be a class of groups, and let § denote the Schunck class p3E. Then, because
of the bijection mentioned above, just before (4.13), it follows easily that any two of
the following statements are equivalent:
(i) G e
(ii) All maximal subgroups of G are ^-normal;
(iii) All maximal subgroups of G are 3E-normal;
(e) If § is a Schunck class, by (2.2)(c) a group G is in § if and only if it has no
quotient group in b(£>). Therefore the following statements are equivalent:
(i) Ge&;
(ii) Every maximal subgroup of G is b(fj)-abnormal.
Describing projectors by properties of maximal chains
Let § be a Schunck class, let H be an ^-projector of a group G, and consider
a maximal subgroup M of G containing H. If К = Corec(Af), then HK/K is
an ^-projector of G/K contained in the complement M/K to Soc(G/K) in G/K.
Thus the primitive group G/K has the property that its socle is avoided by its
^-projectors.
(4.15) Definition. If £> is a Schunck class, define
a(S>) = (G e ф : H n Soc(G) = 1 for all H e Proj6(G)).
We shall call a(£>) the avoidance class of f). It is clear that b(§) £ a(£>) £ 'b\&- The
avoidance class, whose properties we are about to exploit, also plays an important
part in Chapter VI.
(4.16) Remark. Let A ea(£>), and let S be a stabilizer of A. Then S contains an
^-projector of A.
Proof. Let H e Proj 6(Л) and N = Soc(/1), and note that A = NS. Now apply (3.24)
with A and S in the roles of G and L respectively. Since H n N — 1, it follows that
H < Se for some geG. Therefore S contains the ^-projector H" 1 of A. □
4. Examples
309
Г т1₽Ь<!,ПГ' Ч' Ь£'r°nChUnCk daSS’ ,et M ° maximaI suhem“P °f a group
G. Then an) two of the following statements are equivalent-
(a) M is a($j)-normal in G;
(b) Proj6(M) £ Proj6(G);
(с) M contains an ^-projector of G.
Proof. The implication: (b) => (c) is obvious, and we have already indicated above in
anticipation of Definition 4.15, why Statement (c) implies (a). It remains to prove that
Statement (a) implies (b). Suppose that M is abnormal in G, and let К = Corec(M)
By (4.16) the complement M/К to Soc(G/K) in G/K contains an ^-projector of G/K
and by (3.25)(c) this has the form HK/K for some H e Proj6(G). Thus M contains H,
and Statement (b) now follows easily from the persistence and conjugacy of projectors.
□
(4.18) Proposition. Let 5) be a Schunck class, and let X be a class satisfying b(§) £ 3E s
a(Jj). Let &~(G) denote the set of all subgroups T of a group G satisfying the following
two conditions:
(i) There exists a chain
T=Mr< M,_, <
<Ml<M0 = G
such that Mj is an X-normal maximal subgroup of for i = 1, ..., r;
(ii) T has no X-normal maximal subgroups.
Then ,9fG]is precisely the set of fy-projectors of G.
Proof. Since Mj is certainly a(f>)-normal in repeated application of (4.17),
(a) =>(b), shows that Proj8(T) £ Proj6(G)forany T e ZT(G).lfT e.'F(G), then Thas
no I-normal maximal subgroups; in particular, it has no b(S5)-normal maximal
subgroups, and therefore T e by (4.14)(e). In this case TeProj&(T), and so
T e Proj6(G). We obtain all ^-projectors of G in this way by conjugating with the
elements of G. □
We now come to the main theorem characterizing projectors by means of maximal
chains.
(4 19) Theorem Let f> be a Schunck class, let X be a class of primitive groups, and set
X = ^\x.LetGbea group, and let :i (G] denote the set of subgroups H wluch satisfy
the following two conditions:
(i) If U < H, then U is X-normal in H:
(ii) If H < S < T <G, then S is X-abnormal in T.
Then T(G) = ProjJG) for all Ge & if and only if
(4.₽)
ф r, 5 £ I £ ф\а(£>)-
Proof. To prove the sufficiency of the condition assume that
H e Proj6(G). Since H e £>, by (4.14) (a) and (d) the subgroup H satisfies Condition
310 III. Projectors and Schunck classes
(i). Moreover, if H < S < T, then H e Projs(T) by persistence, and therefore S is
a(§)-normal in T by (4.17), (c) => (a). Since a(§) S X' by assumption. Condition (ii) is
also fulfilled by H. Hence Projs(G) S .'/(G).
Next, we use induction on |G| to prove that if H e SC(G), then H is an ^-projector
of G. First note that X' n = X' n ф n S X' n X = 0. Let N < G. Since Condi-
tions (i) and (ii) are obviously inherited by the subgroup HN/N in the quotient G/N,
the induction hypothesis gives HN/N e Proj4,(G/A) whenever 1 A A < G. Therefore,
to show that H, which belongs to § by Condition (i), satisfies the definition of an
.^-projector of G, it will suffice to show that it is Ji-maximal in G. But if not, we can
find subgroups H < V <• V 6 £> and derive the immediate contradiction that the
quotient I7/Core(,( F) belongs to S X by (4.[<) and to X' by Condition (ii).
Hence we have shown that T(G) s Projs(G) and thus that (4./J) is a sufficient
condition for the set ST(G) to coincide with Projs(G). It remains to prove that it is
also necessary.
Assume that the two sets of subgroups coincide, and let Geфг>$. Then
G g Proj6(G) and so G e 3~(G). Therefore, if V <• G, then V is X-normal in G, that is
to say G/CoreG((7) e X. Since G is primitive, it has a maximal subgroup U with
CoreG((7) = 1, and therefore G e X. Hence $ n £ X. Finally, let A e a(f>) and let
S be a stabilizer of A. By (4.17), (a) => (c), S contains an H in Proj s( A). Since H belongs
to ^(A) by assumption, the maximal subgroup S of A is X'-normal, and so
A = A/Corex(S) g X'. Therefore <r(§) S X', and consequently X £ <P\a(Jj). □
(4.20) Examples. We now give two illustrations of the preceding theorem.
(a) First let = 91, and set X = ф n 9? = (Zp: p g IP), the basis of 91. Then the term
‘X-normal’ means ‘normal’ in the usual sense, and ‘X-abnormal’ means simply ‘non-
normal’. Hence the Carter subgroups of a group G are characterized as those
subgroups H with the property that whenever
(4.y) 1 < U < H < S < T < G,
then U < H and S $}T.
(b) Next take = II and X = ф n II, the basis of II, which consists of all primitive
groups with a cyclic socle. In this case a maximal subgroup is X-normal if and only
if it has prime index. Therefore in each finite soluble group G the set of subgroups H
for which |H: 17| g IP and |T: S| $ IP whenever (4.y) holds, form a single conjugacy
class of G, namely the class of U-projectors of G.
Projectors described by numerical restrictions on indices
We have already encountered examples of projectors which are characterized by
arithmetical conditions on the indices and IT: Sf arising from (4.y). For
example, the requirement that \H: 171 should always be a л-number and |T:S| a
л'-number forces H to be a Hall л-subgroup of G. And again, if we demand that
IH : (7| and |T: S| are respectively prime and composite, then we obtain the super-
soluble projectors. Conjugacy classes of subgroups defined in this way have been a
recurring theme in the work of Gaschiitz (see particularly his [15] and [17]), and it
seems appropriate to make the following definitions.
4. Examples
311
(4'21ь Der,"“‘ons and Notation- 'a> The symbol P* will denote the set of all natural
numbers which are powers of some prime: thus P с P* c N. Let Q <= P* denote the
set N\Q by Q', and consider the following properties of a subgroup H of a group G.
GS1: If U <• H. then |H : t/| e Q;
GS2: If H < S < T < G, then |T: S| e Q'.
A group H satisfying GS1 is called an Q-group (or, in relation to G, an Q-subgroup)
A subgroup H satisfying both GS1 and GS2 is called a Gaschiitz Q-subgroup of G
(see Gaschiitz [17]). We shall denote the set of all Gaschiitz Q-subgroups of a group
G by Gaschn(G).
(b) The degree, dG, of a primitive group G is defined to be the degree of its unique
faithful primitive permutation representation; thus dG = |Soc(G)|. Further, if JE £ ^3,
set
dX = {dX: X e I}.
Clearly dX £ IP* in our soluble universe.
(c) If S £ M, let denote the set
*PS = {G e : dG e S}.
If T = S n IP*, solubility implies that ^3S = ^3T.
Remarks, (a) Hall л-subgroups are Gaschiitz Q-subgroups with Q={p":pen,
n e FJ j. Supersoluble projectors are Gaschiitz Q-subgroups with Q = P.
(b) To say that the index of a maximal subgroup V of H is in Q is the same as to
say that H/Core„( U) e Hence the class of Q-groups is precisely the Schunck class
(4.22) Lemma. Let H e Gaschn(G).
(a) If H <L< G, then H e Gaschn(L);
(b) If К < G, then HK/K e Gaschn(G/K).
Proof Assertion (a) is obvious. To prove Assertion (b) »et
?
1 < L/K <• HK/K < S/K < T/K < G/K.
The standard isomorphism from HK/K to H/(Hn K)maps L/K to(HnlJHnK),
and so H n L <• H. Therefore \HK/K: L/K\ = IH: HnL\
S < T, it is clear that |T/K :S/K| e Q’. Hence the subgroup HK/K of G/K satisfies
the requirements of a Gaschiitz О-subgroup of G/K.
w. г
* = Vn- Then Л ano " . IO . .hicrase the set ^(G) of that theorem
tions (i) and (ii), respectively, of Theorem 4.19. In this case the set ( )
312
III. Projectors and Schunck classes
is precisely Gaschn(G). Let § be the Schunck class p.f. It is obvious that 'F n § £ J,
and therefore, by Theorem 4.19, the condition: I £ 'Р\'Л4>). or the equivalent
condition:
(4.<5) Фп^о(5) = 0
is necessary and sufficient for the ^-projectors of a group to coincide with its
Gaschiitz П-subgroups. Thus Condition 4.Й implies the universal existence and
conjugacy of Gaschiitz П-subgroups. We shall prove that the converse is true.
(4.23) Theorem (Gaschiitz [16], Hawkes [13]). Let П £ P*, and let £> = рфп, the
Schunck class of El-groups. Then any two of the following statements are equivalent;
(a) Every soluble group has a Gaschiitz El-subgroup;
(b) Gaschn(G) = Projs(G) for all G eS;
(с) П n da(Sj>) = 0.
Proof. Since Statement (c) implies, and is implied by Equation 4.8, the equivalence
of Statements (b) and (c) is clear from the preamble to the theorem. Furthermore,
Statement (b) obviously implies (a). Therefore, to complete the proof, we now show
that Statement (b) follows from (a). Let H be a Gaschiitz П-subgroup of a group G.
We assert that H is Symaximal in G. For, if not, there exists an fvsubgroup L of G
properly containing H, and we can find a V such that H < V < L. But then |L: V\
belongs to П’ because H satisfies GS2 and to H because L is an П-group. This
contradiction therefore proves the assertion. If К <i G, then HK/K is a Gaschiitz
П-subgroup of G/K by (4.22). Hence HK/K is ^-maximal in G/K, and consequently
HeProjs(G). Thus Gaschn(G) £ Projs(G). Since the set Gaschn(G) is obviously
invariant under inner automorphisms of G, Statement (b) now follows from the
conjugacy of projectors. □
The proof of (4.23) also shows:
(4.24) Corollary. If Gaschiitz Q-suhgroups exist in a group G, then they form a
conjugacy class of subgroups.
Our next goal is to describe some useful sufficient conditions for the universal
existence of Gaschiitz П-subgroups. First we need an elementary fact about groups
in an avoidance class.
(4.25) Lemma. Let f) he a Schunck class, let A e a(f>), and let H e Proj 6(Л). If V is
an H-composition factor of Soc(/1) and if К = Ker(H on V), then the semidirect product
[F](H/K) belongs to />(f>).
Proof. If V = R/S, then H e Projy,(WK) by persistence, and H n R = 1 by definition
of u(f>). Hence, if T denotes the semidirect product [R/S]H, we have HR/S = T and
H e Projs(T), and consequently H/K e Proj6(T/K). But T/K is isomorphic with the
4. Examples
313
lor H/K. lhe group [ I] IH/Kf e. pnrmlive and so telougs И lhe boundary of & □
The following simple observation is also important.
ЦЙти’Let Q~pt'and let s be ,he Schunck class of a^s-Th™
Proof. Let G g b(£>). The class of primitive epimorphic images of G is generated by
G itself, together with the primitive epimorphic images of G/Soc(G), and the latter
belong to s43 n f> £ 'Vo- If G g then G e p*pn = §, contrary to the definition of
b(5f). Therefore G g and so 8G e fl'. □
(4.27) Proposition (Gaschiitz [17], Hawkes [13]). Let Q £ P*, let § = p<pn, and set
A = <?b(f>). Assume that at least one of the following three conditions is satisfied:
(i) 8a(§) = A;
(ii) Cl' contains all products of (not necessarily distinct) integers in A;
(iii) Cl' contains A and is multiplicatively closed.
Then 8a(S>) S fl', and consequently Statements (a), (b), and (c) of Theorem 4.23 hold.
Proof. It is obvious from (4.26) that Condition (i) yields the desired conclusion, and
also that Condition (iii) implies Condition (ii). Let A e a(f>), and let H e Projs(/4). If
V is an H-composition factor of Soc(zl), then [P](H/Ker(H on L)) 6 b(§) by (4.25),
and so | V\ g A. Since | Soc(X) | is the product of such integers | V[, evidently Condition
(ii) implies that 8 A = |Soc(/l)| g fT, and hence that <!a(Sf) £ fl'. □
Let fl be a set of powers of a fixed prime p, subject only to the proviso that p e fl. It
is easy to see that the class of fl-groups is then Gp and that the Gaschiitz fl-subgroups
of a group coincide with its Sylow p-subgroups. Thus a range of different subsets fl
of P* can give rise to the same family of Gaschiitz subgroups. The next result
quantifies this observation.
(4.28) Proposition (Hawkes [13]). Let Г, fl £ P*. and let § = p^V Assume that
Gaschiitz Г- and Cl-subgroups exist universally. Then Gaschr(G) = Gaschn(G) for all
G g G if and only if
(4.s)
acp n §) s г s p*Va(£>)-
Proof. If we set X = фг. Condition 4.e may be restated equivalently thus:
(4.C)
'РпУ>£Х£'Р\'Ш
As we observed in the discussion preceding (4.23), set Gasch (GL^cid« «nA
the set rf (G) described in the statement of Theorem 4.1; a nd so by that theorem
Gasch (G) = Proir(G) if and only if (4.f/) holds. Since Projg(G) - hn
Theorem 4.23, both the necessity and sufficiency of Condition £ are now c ear.
314
III. Projectors and Schunck classes
We now break off the general discussion to describe two examples.
(4.29) Examples, (a) Let SI = P*\{2}. Then contains all primitive groups except
Z2. and therefore the Schunck class § of П-groups consists of all groups which have
no epimorphic images of order 2; in other words, § is the class Q|21 of 2-perfect
groups. It is straightforward to verify that o(fj) = b(§) = (Z2), and so Condition (i)
of Proposition (4.27) is satisfied. It follows from (4.27) and (4.4) that Gaschn(G) =
{O2(G)} for all G g G. If Г is a set of prime powers such that Gaschn(G) = {O2(G)}
for all G g G, evidently 2 £ Г. Since 02(Alt(4)) = Alt(4), we conclude that 22 g Г and
hence that Г is not multiplicatively closed. In fact, the above <2 is the unique set of
prime powers for which the associated Gaschiitz subgroups are always normal
subgroups, but this needs some justification (see Exercise 11 below).
(b) Let p be a fixed prime, and let M be a set of natural numbers all of which are
coprime with p. Then take
n = p*\{p";nG^\M},
and let By (4.25) and (4.26) groups in a(fj) have p-power degree, and so it
follows from (4.18) that the ^-projectors of a group always have p-power index. We
assert that fl n 8a(£>) = 0. If not, there is a group A in u(f>) such that BA e SI. Let
A = NS, with N = Soc(G) and N n S = 1. Then by (4.16) the stabilizer S contains an
^-projector of A; call it H. Since | N | divides | A : HI, which is a power of p, and because
j TV | = BA g SI by supposition, we have |N| = pm for some me M. Therefore by choice
of M the FpS-module N has p'-dimension, and since |S: H| is a power of p, it follows
from B, 7.18 that NH is irreducible. Therefore NH e b(§), and |N| g Bb(f>); hence
|N| g SI' by (4.26). This contradiction proves the assertion that Sir-Ba(£>) = 0, and
so by (4.23) every group has a unique conjugacy class of Gaschiitz fl-subgroups.
Remark. Let Г and <2 be sets of prime powers such that Gaschr(G) = Gaschn(G) # 0
for all G g G, and assume that there is a fixed prime p such that P*\<2 S {p": n e LI};
in other words, <2 must contain all powers of all primes other than p. (The sets 12 in the
two preceding examples have this property.) For n 6 N, set r = p” — 1, and let G„
denote the primitive group E(r/p), whose socle has order p" and whose stabilizer has
order r. Let § = рфп, and observe that Gp. S = P'Lr- Therefore G„/Soc(G„) g f>,
and consequently G„ g и If G„ g f>, then clearly p" g (2 n Г. On the other hand,
if G„ g b(fj), then p" g <2' n Г' by (4.26). Hence P*\<2 = Р*\Г, and the powers of p in
12 are uniquely determined. In fact, we have shown that the set 5(ф n f>) и Bb(f>)
contains all powers of p, and since by (4.23) we have
5($ n 5) e <2 s Р*\5о(§) s P*\db($),
it follows that 8a(S)) = Bh(S)) in this situation. In Example 4.29(b) there are obviously
many choices of the set M for which the set SI' is not multiplicatively closed, and it
follows from these remarks that, for such choices, the Gaschiitz 12-subgroups cannot
be defined by a set of prime powers with a multiplicatively-closed complement.
4. Examples
315
We shall now investigate another kind of subgroup defined by
indices, also introduced by Gaschiitz [14].
restrictions on
(4.30) Definition. Let Q S P*, and consider the following condition on a subgroup
H of a group G:
GS3: If H < S < T < G, then |T: S| 6 Q'.
A subgroup H satisfying GS1 and GS3 is called a generalized Sylow Sl-subgroup of G
(and we shall omit the word ‘generalized’ when there is no ambiguity). We shall denote
the set of Sylow П-subgroups of G by Syln(G).
(4.31) Remarks, (a) Since Condition GS3 obviously implies Condition GS2, for
all groups G we have Syln(G) s Gaschn(G). Therefore, if Sylow Q-subgroups exist
universally, they must coincide with the Gaschiitz Q-subgroups by (4.24)
(b) If SI' is multiplicatively closed. Condition GS2 implies Condition GS3; hence
in this case the Gaschiitz Q-subgroups are Sylow Q-subgroups, and again the two
sets coincide. In fact, as the next result shows, the property of being defined by a
set of prime powers whose complement is multiplicatively closed characterizes the
generalized Sylow subgroups among the Gaschiitz subgroups.
(4.32) Theorem (Meyer [1]). Let Q s P*,and assume that Syln(G) 0 for all Ge G.
Then there exists a set Г s P* such that
(i) Г is multiplicatively closed, and
(ii) Sylr(G) = Syln(G) for all GeG.
Proof Let M = hi \ P*; let f> denote the Schunck class of Q-groups; and let A* denote
the set of all finite products of (not necessarily distinct) integers in 5b(§). Then define
Г = P*\A*,
and note that Г' = M и A*, which is clearly multiplicatively closed. By (4.25) we have
8a(f>) s A*, and therefore
Г £ P*\5u(£>).
Let n e ts*. By definition of A’
f]!=1 BBj = n. By Theorem 6.3, a
a projector of a direct product is
it follows that a Sylow Q-subf
Condition GS3 we must have n e
Г £ Q'; consequently
<X<Pnf>)£flsr.
Therefore Г fulfills Condition 4.8, and we deduce from (4.28) and from the rema
*, there exist groups B,, ..., B,e b(§) such that
result proved later which shows quite generally that
the product of projectors in the direct components,
’roup of Bt X X B, has index n. Therefore by
Q', and so A* £ O'. Since M s Q', we conclude that
316
111. Projectors and Schunck classes
of (4.31) (bearing in mind that Г' is multiplicatively closed), that
SylJG) = Gaschn(G) = Gaschr(G) = Sylr(G)
for all Ge S. □
(4.33) Remarks and Examples, (a) As we have pointed out, the Gaschiitz fl-subgroups
in (4.29) (a) and, for suitable choices of M, in (4.29) (b) cannot be defined by a set Г of
prime powers for which Г' is multiplicatively closed. Therefore by (4.32) they provide
examples of Gaschiitz Q-subgroups which are not generalized Sylow subgroups.
(b) Since IP' is multiplicatively closed, the supersoluble projectors are generalized
Sylow P-subgroups (see Example 4.20(b)).
(c) Another striking illustration of generalized Sylow subgroups, due to Gaschiitz
[14], is the following: Let n e M, let fl(n) = {q e IP* : q < n}, and let f> = р^П(п). Then
Gaschiitz proves that db(F>) S (Q(n))', and since (Q(n))' is multiplicatively closed,
every group has a unique conjugacy class of Sylow <2(n)-subgroups. In other words,
each finite soluble group G has a subgroup H such that
(i) if U < H, then |H : G| < n, and
(ii) if H < S < T < G, the [T: S| > n.
Moreover, the set of such subgroups forms a conjugacy class of G.
(d) Hawkes and Parker [1] have classified the Gaschiitz Q-subgroups with the
so-called D-property (namely, the property that every Q-subgroup of a group is
contained in some Gaschiitz Q-subgroup). These turn out to be just the Hall sub-
groups and the S2-residual (see Example 4.29(a)).
A worked example
The study of maximal links also yields a practical technique for finding projectors in
a specific group. The key to this is Proposition 4.18. If one is looking for an
^-projector of a group G and if G fj, then by (4.14)(e) our G has a b(Vj)-normal
maximal subgroup, Mt say. If M, £ Sy, choose a b(Vj)-normal maximal subgroup M2
of Mly and so on, until a subgroup Mr is reached that belongs to fj. Then by (4.18)
M, is an ^-projector of G. We now carry out this procedure for various Schunck
classes on a specific group G.
(4.34) Example. The group which we have chosen to illustrate this method is the
wreath product
G = Sym(4),
formed with respect to the natural representation of Sym(4) of degree 4. This example
has a sufficiently rich structure to throw up a diversity of projectors and yet is small
enough to ensure easy calculations.
We write S for Sym(4) and record the following elementary facts: S has a unique
chief series: I < V < A < S, where A is the alternating subgroup and V= {/, (12) (34),
(13)(24), (14)(23)}. The chief factor S/A is central, A/V is eccentric of order 3, and К
is a minimal normal subgroup on which S induces a group of automorphisms
4. Examples
isomorphic with Sym(3). We put
3t7
P = ^<(12)>,
Q = <(123)>, and
T = e<(12)>.
Then P is a Sylow 2-subgroup of S isomorphic with Dih(8), clearly Q e Syl (S) and
T is a complement to V in S isomorphic with Sym(3) 3 ’
gro^ -°- ^ -formation about the action of S on the base
g™.,P , G r'tC'G ' BS’v,ewed as a" internal semidirect product, and regard В
additively as an (F5S-module of f5-dimension 4. Let {.x,, x2, x:. x,1 be a basis of В
on which S acts naturally by permuting the suffices. Then it is clekr that the following
F5-subspaces are S-submodules: 6
Z = {Л(х, + x2 + x3 + x4): 2 e ff5};
N = 2,x;: 2, e Fs, £ 2, = di.
Because N is the kernel of the linear map £21.x1-»£2, from В onto F5, we
have Dim(/V) = Dim(B) — Dim((F5) = 3, and so the linearly independant set
{*i — ~ x3, *3 — хч} is an (F5-basis for N. Since NnZ = O, we have
В = N © Z. If X £ B, let (X) denote the (F5-subspace of В generated by X. For
i = 2,3,4 set
N; = <X, + Xi - Xj - Х„У,
where {i, j, k} = [2, 3,4}. It is straightforward to check that /V, is a submodule of the
restriction and that Kcr(F on NJ = {/, (1 i)(/k)}. Furthe-more,
N = N2 ф N3 ф 7V„,
and therefore C„,(F) = 0. Since V is the unique minimal normal subgroup of S, it
follows that N is faithful for S.
Now we consider N under the action of different subgroups of S, showing first that
N is S-irreducible. Suppose, by way of contradiction, that N is reducible. Then
by Maschke’s theorem we can write N = X ф К where without loss of generality
Dim,ДХ) = 1. Then S' < Ker(S on X) by B, 9.3(a). Since V < A = S’, it follows that
X < CN(V) = 0, a contradiction. Hence N is irreducible. Next, let H denote any
non-abelian subgroup of S such that NH is reducible. Then we assert that N„ is the
sum of two irreducible submodules. Certainly by Maschke s theorem we may write
4/ = L, © L2, where L< is an H-submodule of dimension i. If L2 were reducible, then
N would be the direct sum of three 1-dimensional F5H-submodules, and again by В
9.3(a) we should have 1 # H’ < Ker(H on N), contradicting the fact the X is faithful
318
III. Projectors and Schunck classes
for G. We now set H equal to P and T in turn. By inspection N2 and N* = N3 ф
are P-submodules, and both
= <*i + v2 + x3 — 3x4> and
L2 = <(X) - x2), (x2 - x3)>
are T-submodules of N. Therefore, by the preceding argument, it follows that
Nf = N, © N* and NT = L, ф L2 are completely reduced decompositions. It is a
simple matter to check that
Ker(P on N2) = {i, (12), (34), (12)(34)},
Ker(Ton L2) = T,
and that N* and L2 are faithful for P and T respectively.
We now have enough information to compute the projectors in G for various
Schunck classes, and we revert henceforth to multiplicative notation for B.
The Hall systems of G
It is clear from their orders that the subgroups BQ, BP, and PQ( = S) are respectively
Sylow 2-, 3-, and 5-complements of G. Thus
К = {BQ, BP, PQ, G}
is a complement basis of G. By intersecting these Sylow complements in pairs we
obtain the associated Sylow basis
В = {1, P, Q, B},
and together they make up a complete Hall system X = В и К of G. It is clear that
the subgroup D = Z<(12)> (sZ10) normalizes each subgroup in K. Furthermore, G
has the following chief series
I < Z < В < BV < BA < G,
in which Z/l and G/BA are evidently the only central chief factors. Their orders are
5 and 2 respectively, and therefore by I, 5.7 the subgroup D is the system normalizer
of X. Consequently G has |G: D| = 22 • 3 53 Hall systems.
For each of the Schunck classes 91, U, 9l2, S5G2, PQ(Dih(10)), and Q” in turn, we
hall describe the unique projector into which £ reduces. (Of course, the unique
с.-projector into which X reduces is the Hall л-subgroup belonging to X.) The
uniqueness of the projector is ensured by (3.22)(b) and I, 6.6.
4. Examples
319
The Carter subgroup
?ья?у CT‘er SubgrouP we construct a complete b(91)-normal maximal chain
such that 1 reduces into each term. Theorem 4.18 implies that the final term of this
?/BV h!”Ce the deSired Carter subgrouP of G First note that
G'™ = Jhat the SubgrouP M' = BP complements the socle
BA/B V of G/B V' Therefore M, is b(9l)-normal in C. Since the 2-group P acts faithfully
and irreducibly on Nf, we have Mt/ZN2 s [N*]P e b(91); therefore the subgroup
M2 = ZN2P is b(9i)-normal in M,. Ut P* = <(12),(34)>, the kernel of P on N,
Then M2/ZP*s[N2](P/p*) = Dih(10)eb(9i). Thus the subgroup M3 = ZP is
b(91)-normal in M2. Evidently 1 reduces into each term of the b(9!)-normal maximal
chain
ZP = M2<-M2<Ml<-G.
Since ZP e 9?, we conclude that ZP is the desired Carter subgroup of G. It has order
40 and contains the system normalizer D = Z<(12)>. The number of Carter subgroups
of G is the number of conjugates of ZP, namely |G: NG(ZP)| = |G: ZP\ = 3 53.
Go
BA <>
BEo
bA
3 !-SsSym(4)
A chief series of G
The supersoluble projector .
Clearly G/BF S Sym(3) 6 U and G/B s Sym(4) * U. Since Sym(4) e $ 11! follows
that G/B 6 b(U). The subgroup M* = ВТ clearly complements the socle of G/B, and
so we conclude that Mf is a b(U)-normal maximal subgroup of G Since L2 is a
2-dimensional irreducible (F5T-module, the quotient M /ZLt ~BP/ZP^ wb^b
isomorphic with [L2]T is not supersoluble; indeed, T acts faithfully wtaxx
Mf/ZL, belongs to and therefore to b(«). Consequently th^ubgL°uP^ “ f ' f
is b(U)-normal in In additive language, we showed thaiZ1
dimensional F5T-submodules of B. Hence Z and L are eye he“al "°™a
subgroups of and t^<-" ^^b^X.
а1^о^ТьГсХХ"оГcontain ai Rector, as is clear
from order considerations.
320
III. Projectors and Schunck classes
The 9i2-projector
It is easy to verify that 'Ji2 is a saturated formation and therefore a Schunck class by
(4.1). It is also straightforward to verify that the fe(U)-normal maximal chain M* <
< G constructed above is also fe(9i2)-normal at each link. Since MJ ells
we conclude that MJ is the unique 912-projector of G into which E reduces. The
projectors for 11 and 9i2 do not always coincide, of course; for example, the alternating
subgroup A of S has U-projectors of order 3 and is its own 9i2-projector. Note that
G has a self-normalizing 912-subgroup, namely BP, of bigger order that MJ, and that
therefore the combined property of being self-normalizing and belonging to 912 does
not characterize the 9i2-projectors. However, as we shall see next, BP is actually a
projector of G for a smaller Schunck class than 9i2.
The G5G2-projector
Again G5S2 >s a saturated formation and therefore a Schunck class. The Hall {2, 5}-
subgroup BP of G complements the socle of the primitive group G/BV s Sym(3),
which clearly belongs to b(S5S2). Therefore BP is a b(S5G2)-normal maximal
subgroup G, and since it belongs to G5G2 and to E, it is therefore the unique
G5G2-projector of G into which E reduces.
The projector for PQ(Dih(10))
The Schunck class = PQ(Dih(10)) is readily seen to consist of those groups G e
G5S2 all of whose complemented 5-chief factors H/K satisfy |AutG(H/K)| = 2. We
then obtain the following h(§)-normal maximal chain of G into which E reduces:
N2P < ZN2P < BP < G.
Since N2 is a 1-dimensional F5P-module such that |P/Ker(P on N2)| = 2, we have
N2P e S), and therefore N2P is the sought ^-projector of G.
The ^’-projectors
Since N is the unique minimal normal subgroup of NS and since S has the unique
chief series described at the outset, NS also has a unique chief series as follows:
1 < N < NV < NA < NS.
The terms of this series are the only normal subgroups of NS. This, combined with
the fact that G = Z x NS, makes it very easy to locate the ©„-residuals, and hence
the ©"-projectors of G. For example, the C|2,3'-projector is B, the Q|2,5'-projector
is NA, and the C|3,5'-projector is NS. Although a given Hall system reduces into a
unique projector of a group, a given projector may have many Hall systems reducing
into it, as these examples show.
Exercises
1. Give an example of a formation which is not a Schunck class and an example of
a Schunck class which is not a formation.
5. Locally-defined Schunck classes and other constructions
321
2. Let X be a class of finite groups such that X = X2 = zs сЛТ and bt <n a
.he cl... a „wle w ,tal ’
composition factors are in 4). groups wnose
3. Let C, be a Carter subgroup of G, for i = 1, 2. Prove from first principles that
<-! X C2 is a Carter subgroup of Gl x g2. P
4. Let C be a Carter subgroup of G. If |G| < 3 |C|, show that C = G.
5' at$ ,Lac^Ch'^kf.ClaSS’then °bviously S' r, fe(§) = 0. Can it ever happen that
<) vj о(у>) — (See Definition 4.10 for ‘dagger’ notation)
Let § be a Schunck class, and set Я = and ® = b(§). Show that if |Я| is finite
then |93| is infinite. If |®| is finite, does it follow that |Я| is infinite?
In the notation of the previous question, does the condition:
5.
6.
7.
sft n ф <= Я
imply that Sj is s-closed?
8. Let p, q e P, p 1 q, and let n be the order of q (modulo p). Let §(p/q) denote the
class of all soluble groups G with the property that if M < G, then either M is
normal of index p or M is non-normal of index < q”. Show that f>(p/q) is a
Schunck class. Find an §(2/3)-projector of the group G in Example 4.34.
9. Let X, 9) £ ф, and let 3"(G) denote the set of subgroups H of G described in the
statement of (4.19), but modified by substituting “4)-normal" for “.'/'-normal” in
Condition (ii). If 3*'(G) is a conjugacy class of subgroups for all GeG, prove that
:T(G) = Projj,(G) for some Schunck class Sj.
10. (Gaschiitz [16]) Let S £ hl, and set fl = S P*; let G be a soluble group, and
set Mo = M* = G. If M, is not an fl-group, choose an Mi+1 < M, with | M,: M(+11
as small as possible in S', and let M, be the fl-group (i.e. the final term) of this
descending chain. Let M* be the fl-group in a second, similarly-defined, starred
chain. Then prove that M, is conjugate to M*.
11. Let fl £ P*. Prove that any two of the following statements are equivalent:
(a) Gaschn(G) = {O2(G)} for all GeG;
(b) fl = P*\{2};
(c) |Gaschn(G)| = 1 for all GeG.
12 Let fl be the sei of prime powers not greater than n (cf. Example 4.33(c)), and
let denote the Schunck class of ^-groups. Prove that there are values of n for
which is not s-closed. .
13. Show that in Theorem 4.2 the assumption that the universe ® is Enclosed is
superfluous.
5. Locally-defined Schunck classes and other constructions
To begin with we discuss a versatile and practical method of “ctmg d^of
group!, including Schunck classes, by specifying the groups of
of forming new Schunck classes out of give
is (£.
322
III. Projectors and Schunck classes
Local Schunck classes
(5.1) Proposition. Let h : P -»{group classes} be a function which associates with each
prime p a (possibly empty) class of groups h(p). Then let X denote the class of all finite
groups G which satisfy the following condition:
(5.a) For all non-Frattini chief factors H/K of G and for all
primes p dividing |Я/К|, we have AutG(H/K) e Ji( p).
Then X is an G-Schunck class, and if h(p) £ S for all p e P, then X £ S.
Remarks, (a) Condition 5.a states that the set of automorphism groups induced by
G on the set of non-Frattini chief factors whose orders are divisible by p is contained
in h(p). Therefore, if h(p) = 0, the set of such chief factors must be empty, and by A,
11.8(b) this means that G e 6f.. Thus the interpretation of (5.a) when h(p) = 0 is that
(b) We also remark that by A, 9.13 it is only necessary in (5.a) to consider the chief
factors of a given chief series.
Proof of (5.1). Let В e h(X). It is obvious from the definition of the class X that it is
Q-closed, and from Remark (b) above it is also clear that I = hence Ф(В) = 1.
Suppose (for a contradiction) that В has distinct minimal normal subgroups M and
N. Then MN/N is a non-Frattini chief factor of B/N by A, 9.11. Since AutB(M) s
AutB(MN/N) and B/N e X, it follows that AutB(M) e h(p) for all primes p dividing
|Af|. Since B/M e X, by appealing as before to the Jordan-Holder theorem, we infer
that В e X, which contradicts the definition of b(X). Therefore В has a unique minimal
normal subgroup. Consequently В is primitive, and from (2.7), (c) => (a) we conclude
that X is a Schunck class.
If H/K is a non-abelian chief factor of a group G, then H/K = lnn(H/K) <
AutG(H/K), and so AutG(H/K) is insoluble; moreover, H/K is not Frattini. Hence, if
each h(p) consists of soluble groups, the chief factors of a group in X are abelian, and
in this case I £ S. □
We now furnish the situation described in this proposition with some suitable
terminology.
(5.2) Definitions. Let h be a function which associates with each prime p a class h(p)
of finite groups.
(a) The class £ of all finite groups which satisfy Condition 5.a will be denoted by
LC(h). It is called the class 'locally defined by h’.
(b) We call h a local function if h(p) is a homomorph for all p e P. (Recall that the
empty class is a homomorph.)
(c) A class X is called a local class if X = LC(h) for some local function h. (Although
the requirement that the classes h(p) be homomorphs limits the range of Schunck
classes that can arise as local classes (see Exercises 1 and 2), it helps the machinery
to run more smoothly and eases the transition from Schunck classes to formations
where the local theory plays such a crucial role.)
323
5. Locally-defined Schunck classes and other constructions
(d) The support of the function h is defined thus:
Supp(h) = {p e p: h(p) / 0}.
(5.3) Remarks, (a) If X = LC(h) and n = Supp(A), then X £ (£ .
(b) In a soluble group a non-Frattini chief factor is complemented by A, 9.10(a).
Hence, tn a soluble universe, we can substitute the word ‘complemented’ for ‘non-
Frattini’ in (5.a).
(c) If the condition of (5.a) is required to apply to all chief factors of G instead of
just to the non-Frattini ones, then the class X is a formation (see Chapter IV, 1.3);
but in this case we cannot conclude, even in a soluble universe, that X is a Schunck
class or a local class in the sense of (5.2) (c) (see Exercise 3).
(d) As an alternative approach to the construction of a ‘locally-defined’ class, we
mention the following: Letdenote a set containing exactly one representative of
each isomorphism class of finite simple groups. (Thus the map p —* Zp gives a natural
embedding of P into.'/'.) Let h:-»{group classes], and let X denote the ‘local’ class
of all finite groups G which satisfy Autc(///K) e A(S) when S is isomorphic with a
composition factor of the (non-Frattini) chief factor H/K of G. In this case, the element
S is uniquely determined by H/K because H/K is a direct product of isomorphic
simple groups. A greater variety of classes falls within the compass of this definition
of a ‘local’ class X. but it does not appear to lead to such a satisfactory generalization
of the soluble theory as the definition that we have adopted.
(5.4) Proposition. Let X = LC(A), where h: P -> {classes of groups] is a function
satisfying (1) e h(p) whenever h(p) / 0, and set
h*(p) = X n SpA(p)
for all primes p. Then X = LC(h*). Moreover, if h is a local function, then
(a) h* is a local function,
(b) Sh*(p) = A*(p) £ X, and
(с) X is contained in ep.epR0(A(p) n Л*(Р» if P e Supp(A) and in 6p. otherwise.
Proof. Let X* = LC(h*) and let G e X*. Let H/K be a non-Frattini chief factor of G
and put A = Autc(H/K). If p is a prime divisor of |H/K|,
that О (A) = 1. By assumption A e h*(p) £ <S„h(p). Hence A ~ A/O„( ) e (p),
consequently G e X. Thus X* S X. If H/K is a chief factor of a gro up X, then
Aut JH/K) I X/CAH/K) e q(X); hence, if X e X and H/K is non-Frattini, we have
Aut (H/K) e qX n hip) <= h*(p) for all primes p dividing |H/K|, and therefore X e X .
SZes that X = LC(h*). From the definit.on of а !
h(p) = 0 if and only if A*(p) = 0- ВУ (5 ') we have X = QX moreove h(p)
Q-closed, then so is 6pA(p) by II, 1.9. Therefore h* is a local Q we
blow let Hence,7nord'er toshow that G e X, by A, 9.13
A, 13.8(a) such a chief factor is centralized by R, we ave
324
III. Projectors and Schunck classes
Autc(H/K) S G/Cg(H/K) e q(G/K) <= Qft(p) = A(p),
and so H/K satisfies (5.a). Hence G e X. We have therefore shown that G e
X n Gph(p) = h*(p), and Assertion (b) is proved.
Let p e Supp(/i), and let G e X. Let C = C„} denote the set of centralizers
of those non-Frattini chief factors of G whose orders are divisible by p. If C = 0,
then G e by A, 11.8(b). Otherwise we have Q?=I C, = O„. p(G) by A, 13.8(a). Since
G С, e A(p) r> h*(p) by Part (a), we conclude that G/Op. p(G) e R0(A(p) r> A*(p)). Asser-
tion (c) is now clear. □
(5.5) Definitions. Let h be a local function, and let = LC(h). The h is called
(a) integrated if li(p) £ Sy for all p e P, and
(b) full if h(p) = Gph(p) for all p e P.
Thus Conclusions (a) and (b) of Proposition 5.4 mean that every local Schunck
class can be defined by a full and integrated local function h*. In the case of a local
formation g, it is possible to require without loss of generality that the defining classes
are themselves formations, and in Chapter IV, 3.7 we shall see that, among the local
formation functions defining g, there is then a unique one which is both integrated
and full. Without that requirement uniqueness is not guaranteed (see Exercise 4
below). For the continuation of this discussion of locally-defined classes we refer the
reader to Section 3 of Chapter IV.
Meanwhile we stipulate that for the rest of this section the universe is G.
The join of Schunck classes
As we remarked in the course of Example 4.34, an inclusion between two Schunck
classes does not imply a corresponding inclusion between their projectors. Perhaps
it is therefore surprising that, as the next construction shows, for arbitrary (soluble)
Schunck classes and Я it is possible to find a Schunck class £ such that an
£'-projector in every group contains both an ^-projector and a Я-projector; and that
furthermore there is a unique smallest such £. The construction that now follows is
due to Blessenohl [1].
(5.6) Definition. Let {У)А: A e A} be a set of Schunck classes. Their Join, denoted by
: A e A>. is by definition the class of soluble groups G such that G = (Et: A e A>
for every choice of a set {£;: £; is an f>;-projector of G for each A e A}.
We now proceed to show that the join is again a Schunck class.
(5.7) Lemma. The following statements are equivalent:
(a) G e : A e A>;
(b) If L is a Hall system of G and if F} is the unique Sy^-proJector into which L
reduces, then G = (JJ : A e A>.
Proof. It is clear from the definition of the join that (a) implies (b). Suppose then that
Assertion (b) holds, and for each A e A let E, be an f>;-projector of G. Then EA = Ff
5. Locally-defined Schunck classes
and other constructions
325
Theorem 69 S ChaMerT^T °f prOJeCtOrs are P^normal,
1 neorem 6.9of Chapter I applies, and we conclude that <Ед:2еЛ> contains a
conjugate of <£д: 2 e A> = G. Hence Assertion (a) holds. q
(5.8) Proposition Let {£>д : 2 e A} be a set of Schunck classes, and let § denote their
jotrr Let Ebe a Hall system of a group G. and let E, be the unique ^-projector of G
into which Z reduces. Then
(а) <£д : 2 e A> is an ^-projector of G into which X reduces, and
(b) § is a Schunck class.
Proof. If we can prove Assertion (a), then (b) will follow at once by (3.10) for the
universe S. Let E = <£д: 2 e A>. By (3.22)(a) the subgroup £; is an f>;-projector of
£, and by I, 6.9 the set L n £ is a Hall system of £ reducing into each £д. Therefore
£ e § by (5.7). If £ < £ < G with F e Sj, then £д is an §z-projector of F, and by
definition of the join § we have £ = <£д; 2 e A) = £. Hence £ is ^-maximal in G.
Let G -» G* be an epimorphism, and let H* denote the image of a subgroup H of G.
Then the set X* = {S* : S e X) is a Hall system of G* reducing into £*, which is an
^-projector of G*. Furthermore, £* = <£J: 2 e A>, and therefore by the above
argument £* is ^-maximal in G*. Hence E is an ^-projector of G. □
Let 6 and Я be Schunck classes, and let £ = <§, Я). Then it is clear from the
preceding result that in every group an £-projector contains both an ^-projector and
a Я-projector and that £ is the smallest Schunck class with this property. Further-
more, Proposition 5.8, together with I, 6.9, yields the fact that a join <H, К:
H e Projs(G), К e Proj1((G)> of minimal order gives an (Sj, Я>-ркуес1ог of G. In
Chapter VI we shall show how this ‘join’ construction gives rise to a lattice structure
on the family of Schunck classes, and how it is important in the study of a partial
order on Schunck classes called ‘strong containment’, which arises from the inclusion
relation between projectors.
The normalizer of a Schnnck class
The next construction is also due to Blessenohl and can again be found in Gaschiitz’s
Canberra notes.
(5.9) Definition. The normalizer IV(&) of a Schunck class § is defined to be the class
of all soluble groups which have a normal ^-projector. Clearly £ N(§).
(5.10) Proposition. Let & be a Schunck class, let G be a soluble group, and let H be an
f)-projector of G. Then
(a) Ng(H) is an A(5)-projector of G, and
(b) A(§) is a Schunck class.
в c i . м - n tn\ and let 6- G -»G* be an epimorphism. Denote the image
Г°?’Ус К Nc H Y bv Y* Because H* is an Я-projector of G*, by persistence
under 0 of a subgroup X by X . Because - SuDD0Se that A* <
it is a normal ^-projector of N*, and y h < L* Because H is
L* < G* with L* e A(&). Since H* e Proj6(£ ), we nave n _
326
III. Projectors and Schunck classes
pronormal in G, it follows from I, 6.3(c) that NC.(H*) = N*, and therefore L* < N*.
Hence N* is N(f>)-maximal in G*, and N is consequently an N(§)-projector of G.
Assertion (b) now follows from Assertion (a) by (3.10). □
The following result is an immediate consequence of(4.9) and the above description
of N(5)-projectors.
(5.11) Corollary. A(f>) = § if and only if 91 c §.
Prodncts of Schunck classes
Finally in this section, we touch briefly on the question of when the class product Я§
of two Schunck classes Я and § is again a Schunck class. From the little available
evidence it would seem that this rarely happens. At the time of writing no general
criteria are known, but in two special cases, namely Я = C and Я = 91, complete
solutions are available, and these are dealt with below.
(5.12) Proposition. Let nbe a set of primes, and let Sy be a Schunck class. Then
(а) £?f) is a Schunck class, and
(b) if U is an Sy-projector of G, then O”(G)U is a D” f>-projector of G.
Proof. By (5.8) the group On(G)U is a <£}“, §>-projector of G. Obviously <Q", fj> =
□
Thus ЕГ is among the Schunck classes Я with the property that ЯУ) is a Schunck
class for all Schunck classes f>. Another Schunck class with this property is described
in Exercise 10 below, and it suggests that the task of characterizing such Schunck
classes could be difficult.
(5.13) Theorem. Let Sy be a Schunck class. Then 9lSy is a Schunck class if and only if
Sy is a saturated formation.
Proof. First suppose that Sy is not a formation; we shall show that 9i§ is not
a Schunck class. Let X be a group of minimal order in Rof>\$>, a choice which
ensures that (q — 1)(Х)пꄧ £ f>. Thus Corollary 3.16 applies and X has distinct
minimal normal p-subgroups M and N such that X/M, X/N e §. Let q be a prime
unequal to p, and let V and V be F,X-modules such that Ker(X on 17) = M and
Ker (A on F) = N (for example, we could take for V the regular F, (X/M (-module
inflated to X). Let W denote the F,X-module U ф V, and form the semi-direct
product
G = [1V]X.
Since W is clearly faithful for X, the Fitting subgroup F(G) is a q-group, and therefore
F(G)nX < Op (X). Hence by Assertion (h) of (3.16) we have F(G)nX = 1, and it
follows that F(G) = kF; consequently G $ 91Sj. Let R/S be a chief factor of G in a chief
series passing through W. If R/S is above W, it corresponds to a chief factor of X via
5. Locally-defined Schunck classes and other constructions
327
the obvious isomorphism from G/W to X, and then G/Cc(K/S) e Q(X/F(X)) s ft On
H'V’11’6! hand"fX,S bel°W W'then itS centralizer in 6 contains either WM or
t W’and agjiln G/Cc(«/S) e ft. It follows that ф r, q(G) c 9lft. Therefore 91ft is not
a Schunck class.
Thus we have shown that if 9lft is a Schunck class, then ft is a saturated formation.
In the other direction, it is straightforward to verify directly that if ft is a saturated
formation, then so too is 9lft. However, for a formal proof the reader is referred to
Theorem IV, 1.9, from which it can be deduced that if ft is an s.-closed Schunck class
(as 91 certainly is) and if ft is a saturated formation, the ftft = ft о ft is a Schunck class.
□
Remark. The preceding result supports the view-point that a class product of two
Schunck classes is ‘rarely’ again a Schunck class. Further justification is provided by
a result of Forster [8], which states that if ftft is a Schunck class for all Schunck
classes ft, then the Schunck class ft is either (1) or S.
Exercises
1. Let h(2) = (1), h(3) = (1, Z2), h(5) = (1, Sym(3)), and h(p) = 0 if p > 5. Let ft be
the Schunck class locally defined by h. Prove that ft is not a local Schunck class
in the sense of Definition 5.2(c).
2. Let ft be a local Schunck class. Prove that b(ft) £ и ф2.
3. Let h(2) = Q(Sym(3)), h(3) = q(Z2), and h(p) = 0 for p > 3. Let ft denote the
class of all finite groups G such that AutG(H/K) e h(p) for all chief factors H/K
of G and for all primes p dividing | H/K |. Then show that ft is not a Schunck class.
4. Let G denote the class of cyclic groups. For all peP let h,(p) = SpG and
/,2(p) = e„9I. Show that /1, and h2 are both integrated and full local functions
defining the local Schunck class 9191.
5. Let ft be a Schunck class in a soluble universe. For each G e ftf, the basis of ft,
let p be the prime dividing |Soc(G)|, and let 9R(G) be a class of FpG-modules V
such that
(5,/j) Soc(G) < Ker(G on F).
Suppose that 9R(G) always contains Soc(G). Prove that the following class:
ft = ([F](G/Ker(G on F)): G e ft*. FeSUl(G))
is a Schunck basis- let ft(9K) denote the associated Schunck class pft. Prove that
local Schunck class containing ft. -. ., . r, proj (G),
6. (Gaschutz[15])LetftandftbeSchunckclasses,letGe S, and let HeProjs(
К e Proj „(G). Prove that the following statements are equivalent.
(a) G E (ft, ft\ r r G then either HSe, T S or
(b) If R/S is a complemented chief factor
KS n T / S.
328
III. Projectors and Schunck classes
7. Let 53 and Я be soluble Schunck classes, and consider the statements:
(а) Я £ N(§); (b) /?(§)п Я £ 81. Prove that (a)=>(b), and that if Я = 5„Я.
then (b) => (a).
8. Let § be a soluble Schunck class of characteristic n. Prove that fe(N(&)) =
(A e a(£): O"'(A/Soc(A)) e §\( 1)).
9. (Blessenohl—see Gaschiitz [15]). If § is a Schunck class, let Eroc(§) denote the
class (G: G = <ЯС> for some H e Proj,,(G)). (Note: ‘Eroc’ dualizes ‘Core’.) Re-
strict to a soluble universe, and let Я be an ^-projector of a group G. Prove that
<HG> is an Eroc(fj)-projector of G, and deduce that Eroc(.f>) is a Schunck class.
Prove further that if Char(fj) = n, then Froc(f>) = Q". How far do these ideas
extend to the universe G?
10. (Forster [8]) Let p be a fixed prime, and let Я be a Schunck class with the property
that 1 A OP(B) < B' for all В e Ь(Я), (e.g. p = 2 and Я = /i(Sym(3))). Prove that
Я§ is a Schunck class for all Schunck classes 5.
11. Find a Schunck class §, and a set n of primes, such that is not a Schunck
class.
6. Projectors in subgroups
An ^-projector of a group may be regarded as a measure of how close that group
comes in its quotient structure to membership of the class f>. On the other hand,
knowledge of the ^-projectors usually reveals little about the ^-subgroup structure
of a group, and, in particular, there is no connection in general between the projectors
of a group and those of a proper subgroup. (For example, in Sym(4) and Alt(4) the
Carter subgroups are respectively the Sylow 2- and Sylow 3-subgroups.) Two special
types of subgroup are exceptions to this general rule; these are the central factors and
the so-called well-placed subgroups, and they form the main concern of this section.
Projectors in central products
During this discussion we work in a general universe 81 satisfying Hypothesis 3.4.
(6.1) Proposition. Let G be a central product of subgroups Gt,.... G„. Thus
G = G1G2...G„,
where [Gj, Gj] = 1 for 1 < i A j < n. In addition, assume that if M <1 Gf for some i,
then G> Gt...Gi_lMGi+l...G„.LetX = Q(G) n ф and = Q(G,.) r> ф. Then -C £ £.
Furthermore, if X contains no primitive groups of type 3, then I = (J"=i X/.
Proof. By A, 19.8(a) we have G' = GJ. Therefore, if p divides |G: G’|, it also
divides | Gf: G-1 for some i. Assume now that p divides |G,: GJ for some i. Then Gf has
a maximal normal subgroup M such that | G,: M | = p. Because G(... Gt tMGitl ... G„
is a proper subgroup of G, it has index p, and therefore p also divides |G: G'|. It
therefore follows that
6. Projectors in subgroups
329
(6.a)
P||G : G'| if and only if p||Gj: GJ for some i.
Let Gf denote the subgroup Let H, e 3q; then H, G/T for some T <з G
‘if i z,,c?“* c“ " ““M'*
Is desfoed gr°UP G haS a qUOt'ent gr0UP ,somorPhic with and again H; e £
Now let X e X. Then № G/K, where G/K is primitive of type 1 or 2; in particular,
G K has a unique minimal normal subgroup, N/K say. Since К is a proper subgroup
of G ‘here exists а к e {1,such that Gt K, and it follows that N/K < G„K/K.
If G* < K, we have G/K £ o(G/Gf ) = q(Gj/(Gj n Gf)) £ Q(Gt), and then G/K e b.
Otherwise we have Gf £ K, and in this case
1 # N/K < (GkK/K) n (Gf K/K) < Z((GjK/K)(Gf K/K)) = Z(G/K).
Since G/K e ф, as before it follows that G/K s Zp, and again from (6.a) we conclude
that Zp e q(G,) n ф = Xi for some i. Thus in any case we have shown that X e (J?=1 .L,.
□
(6.2) Corollary. Let fj be a i/Schunck class, and assume that 'll is DD-closed and
contains no primitive groups of type 3. Then
& = Do5-
Proof LetG = Ff x -•-x witheachH,eThenGed(1'I1 ='ll.By(6.1)wehave
q(G) r> ф S |Jf=I (Q(H,) <£ Ф) £ §, and therefore G e p§ = fry □
In particular, this corollary tells us that a Schunck class of soluble groups is always
closed under forming direct products. We shall return to the question of closure
properties for Schunck classes in Section 2 of the next chapter.
If a group can be expressed as a central product, we can now show how its
projectors are related to the projectors of its centra] factors.
(6.3) Theorem. Assume that Ф and § satisfy the hypotheses of Corollary 6.2. Let
G = Gj G2 ... G„, where G, e Ф and [G„ Gj = 1 for 1 < i < n. If Hi e Projf(G,),
then И H, H is an ^-projector of G. Furthermore, if G has a unique conjugacy
class of ^-projectors (in particular, if 53 = S). then every ^projector of G has this form.
Proof. Bv A 194 there is an epimorphism from the external direci product
G, x • x G onto G. and consequently G 6 qd0S = moreover, since projectors
we consider the case G = G, x Gz.
330
III. Projectors and Schunck classes
Let N be a normal subgroup of a group X, and let N < V < X. Recall from (3.7)
that if 17 6 Proj^jL) and V/N e Proj&(A//V), then U e Projj,(X). First apply this
result with X = x G2, N = /7, x 1, and U = V = Ht x H2. Since x H, eft
by (6.2), it is clear that the hypotheses are fulfilled, and therefore Hi x H, e
Projj,(Ht x G2). Now apply the result again, this time with A' = G, N = 1 x G2,
V = Hi x G2, and U = Hi x H2; once more the hypotheses are obviously satisfied,
and we conclude that x H2 e Proj S(G), as desired. The final assertion is obvious.
□
Well-placed subgroups
First we specify that for the rest of this section the universe is S. We have just seen
in (6.3) that if L is a central factor of a group G, then an f>-projector of L is contained
in some ^-projector of G. We now describe another type of subgroup with this
property.
(6.4) Definitions, (a) A subgroup S of a group G is said to be critical (in G) if G =
SF(G); in other words, the critical subgroups of a group are the supplements to its
Fitting subgroup. If S is critical in G and S < T < G, then it is clear that S is critical
in T, and that T is critical in G.
(b) A subgroup W of G is said to be well-placed in G if there exists a chain of
subgroups
(6./?) W= Wo < Wi < < W„ = G
such that Wi-i is critical in Wt for i = 1,2,..., n. We shall denote the set of well-placed
subgroups of G by W(G\ By the final remark in Part (a) above, it is clear that
We'H "(G) if and only if there exists a chain of the form (6./?) in which each term is
both critical and maximal in the next. By A, 10.6(c) we know that O(G) < F(G) when
G # 1, and therefore soluble groups always possess critical maximal subgroups; in
particular, their identity subgroups are well-placed.
First we prove a result which shows a strong connection between the chief series
of a well-placed subgroup W and its parent group G. Roughly speaking, it says that
the factors of a chief series of W may be viewed as a subset of the factors of a chief
series of G. Before stating it, we recall from A, 10.5(b) that if M is a maximal subgroup
of a soluble group G, then M avoids a unique chief factor in a given chief series of G
and covers the rest.
(6.5) Proposition. Let M he a critical maximal subgroup of a soluble group G, and let
C: 1 =L0^Li < < Lr = L < U = Uo < U2 < < 17S = G
be an arbitrary chief series of G in which U/L denotes the unique chief factor avoided
by M. Let 17* = M л If for i = 0, 1,..., s. Then
C*. 1 = Lo< Li< - - < Lr = US < U* < •• <17* = M
is a chief series of M. If 0 is the injection from the chief factors of C* to those of C
6. Projectors in subgroups
331
defined by
g. “* Li/Lii-x (i — 1,2,..., r)
ЧЦ’/Ц^Ц/Ц,, (j = l,2,...,s)
and if V is a chief factor of C*, then
[П(М/См(П) S [6(ni(G/Cc(0(l/))).
Proof If И denotes any one of the factors LJL.^ or Ц/Ц_г in C, then CHI') > F(G)
by A, 13.8(b), and therefore V and G/CjfV) are covered by M since M is critical The
desired result now follows from A, 13.9. q
(6.6) Corollary, (a) If W is a well-placed subgroup of G, there exists an injection, 0
<,ay, from the chief factors of a given chief series of W into the chieffactors of a given
chief series of G such that the isomorphism of (6.y) holds with W in place of M.
(b) Well-placed subgroups are CfyP-subgroups.
Proof, (a) This follows from the Jordan-Holder theorem and repeated application
of (6.5) to the terms of a critical maximal chain from W up to G.
(b) Let W e 'U (G), and let H/K be a chief factor of G. Let M be a critical maximal
subgroup of G in the critical chain from W to G. If M avoids H/K, then so does W.
If M covers H/K, then (M n H)/(M n K) is a chief factor of M by (6.5). Since
W e by induction W either covers or avoids (M n H)/(M n K) and accord-
ingly covers or avoids H/K. □
As promised, we finally describe the connection between the projectors of a group
and those of its well-placed subgroups. In order to formulate part of the result, we
need to consider the following property of a class § of groups.
(6.Й) If Ge f) and S is a critical subgroup of G, then S e f).
This property is obviously equivalent to the closure of 5 under ‘taking well-placed
subgroups’, and we shall see in Section 2 of Chapter IV that it exactly characterizes
the saturated formations among soluble Schunck classes.
(6.7) Theorem. Let $>bea Schunck class, let W be a well-placed subgroup of a group
G, and let U 6 Proj&( W). Then there exists an H e Proj&(G) such that U < H. If £>
satisfies Condition 6.6, then H may be chosen so that U=Wr.H.
Proof. We prove both conclusions simultaneously by induction on |G|. Since We
WfG), there exists a chain of subgroups
W = M0<.Ml < - <M„ = G
with 4_, a critical maximal subgroup of И for i = L , » Since We ), by
332
III. Projectors and Schunck classes
induction we have U < L for some Le Projs(M„. j), with U = IT n L if (6.5) obtains.
Therefore we may suppose without loss of generality that W = M„_2, a critical
maximal subgroup of G.
Let F denote F(G), and consider the standard isomorphism ф : kV/(kV n F) ->
kVF/F. Since kVF = G and t/(kVn F)/(kVn F) 6 Proj s(kV/(kVn F)), it follows that
i//(U(B/r> F)/(kVn F)) = UF/F is an ^-projector of G/F. Thus by (3.25)(c) we have
UF = HF for some H e Proj&(G). Since H e Proj&(GF), by (3.23)(b) (with the sub-
groups E and L of that proposition equal to U) we may choose H so that U < H.
Because H = H n UF = U(H n F) < (Wn H)(H n F), and because H r^F < F(H),
the subgroup kVn H is well-placed in H. Therefore, if § satisfies (6.5), it follows that
И'п H e f>, and then U = kkn H by the f,-maximality of U in W. □
Exercises
1. Show that the final assertion of Proposition 6.1 and the conclusion of Corollary
6.2 may each fail to hold if primitive groups of type 3 are not excluded.
2. Show that for an G-Schunck class § the ^-projectors of a direct product G, x G2
need not have the form Я, x H2 with Я, e Projy,(G,) for i = 1, 2 (cf. Theorem 6.3).
3. If and are Schunck classes, show that fij ufi2isa Schunck class if and only
if vj or f>2-
4. Show that a Schunck class of soluble groups, while closed under forming central
products, need not be closed under forming normal products.
5. Let К sn G. Show that К e 'K'(G) if and only if К ZX(G) = G.
6. (a) Let N <! G e S. Show that Hr(G/N) = {WN/N: W e 7L~(G){.
(b) Let W e ~# (G) and W < L < G. Decide whether the following assertions are
true:
(i) #'(kk) e lT(L);
(ii) r (L) £ tT(G).
7. Let I be a Hall system of G, and let ^(G) denote the set of all well-placed
subgroups W of G such that £ reduces into each term of some critical maximal
chain from W up to G. Show that in general *L(G) is not closed under forming
intersections or joins.
8. In the notation and context of Proposition 6.5 show that if the chief factor 0(V)
of G is complemented, then the chief factor V of M is complemented, but that the
reverse implication is false.
9. Let § = pq(A.lt(4)), and let G = SL(2, 3). Show that G e Si and that G has a critical
maximal subgroup M ф (Thus Condition 6.5 is not satisfied by a Schunck class
in general.)
Chapter IV
The theory of formations
1. Examples and basic results
A formation is a (possibly empty) class g of groups with the following two properties:
(a) If G e g and N < G, then G/N e g;
(b) If Nj, N2 < G with Nt n N2 = 1 and G/Nf e g for i = 1, 2, then G e g.
In this section we gather together facts of a general nature about formations, we
describe a number of specific examples, and we develop several general methods of
constructing whole families of formations.
In Chapter II, 1.16 and 1.18(b), we saw that for any class 3E of groups the class
qr0I( = <q, r0>J) is the smallest formation containing 3E. Since r0<sd0 by II,
1.18(c), it follows that a <Q, s, D0)-closed class is a formation. The following examples
arise in this way:
(a) The class of nilpotent groups of class at most c is a formation. Its <q, s, d0)-
closure follows easily from elementary properties of the lower central series of a group
described in Chapter A, Section 8.
(b) The class Sl‘” of soluble groups of derived length at most d is a formation. (See
A, 10.2(a) for the q- and s-closure; the d0-closure follows from A, 19.8(a).)
(c) The class G(n) of groups of exponent at most n is a formation. Its closure under
each of the operations Q, s, and D„ is obvious.
We now present a construction method which exploits the invariance of projectors
under epimorphisms.
(1.1) Definition. Let § be a Schunck class in the universe 93 = <s, o, e„>93, and let
3E be any class of groups. Define a class (f> j i) as follows:
(5 J 3E) = (G e 93: Projs(G) e -I).
(We recall from III, 3.10 that the set Projs(G) is non-empty for all G e ®.)
This construction gives rise to a family of formations
where § is a Schunck class and g is a formation;
9! = <s, o, e„>® which is also a formation.
(1.2) Proposition. Let §bea ^-Schunck class, and let g be a formation. Then (S| g)
is a formation and has the following properties.
(a) § n g = 5 r> (& j gk
(b) (5jg) = (5H&nS)>;
parametrized by pairs (§, 5),
it works for any universe
334
IV. The lheory of formalions
Let and ft2 be formations.
(c) If$t £ g2, then (5 j ft,)<=(&I 82);
(d) if ft, =s^ft,then Si (51Й2)e (518182)-
Proof. Since projectors are preserved by epimorphisms, the Q-closure of (§ j ft)
follows at once from the Q-closure of ft. Let i 6 {1, 2} and let G/N, e (f, j ft) with
/V,r>N2 = l. If HeProj£,(G), then H/(H n NJ S HNJNt e Projs(G/Nf). Hence
H/(H n NJ 6 ft, and H 6 Roft = ft. Consequently (§ j ft) is R„-closed and is therefore
a formation.
Assertions (a), (b), and (c) are obvious from the definitions. To prove Assertion (d),
consider a group G in ft,(&|ft2): it has a normal subgroup К such that
and G/Ke(5|ft2). If NeProj^G), then HK/K 6 Proj6(G/N) e ft2. Hence
H/(H n K) e ft2, and since H n К e sft, = ft, by assumption, we have H e ft, ft2.
Therefore G e (5 j ft, ft2). □
From this proposition it follows, for example, that the classes consisting of all finite
groups whose Sylow p-subgroups have a fixed upper bound on their nilpotency class
(alternatively, on their derived length, or on their exponent) are formations, and it is
clear that by varying § and ft (and the universe S3) in this result one can obtain a
rich variety of formations.
Another versatile method of construction formations is to specify the groups in a
class by their action on certain chief factors, a technique which we have already
encountered in the development of local (Schunck) classes in Section 5 of Chapter III.
(1.3) Proposition. Let i and 9) be given classes of groups, and let f'-f)-* {classes of
groups} be a mapping which satisfies f(G) = /(Я) whenever G = H e '!). Define classes
^(respectively ft®) to consist of all finite groups G satisfying the following two
conditions-.
(a) All chief factors of G belong to 3E;
(b) Autc(S) e f(S) for all chief factors (respectively all Frattini chief factors) S of G
which belong to 9).
Then ftv and ft® are formations.
Proof Let § denote one of the two classes ft¥ and ft®. It is obvious that I efi and
& = Q& Let N, and N2 be distinct minimal normal subgroups of a group G such that
G/N, and G/N2 are in £>. By II, 2.6 it will be enough to prove that G 6 and by the
strong version of the Jordan-Holder theorem (see A, 9.13) it will then be sufficient to
show that Conditions (a) and (b) are satisfied by the chief factors in a given chief series
of G. Therefore consider a chief series passing through N,. Since G/N, e Sj, we have
then only to verify Conditions (a) and (b) for the minimal normal subgroup N,. We
use the well-known G-isomorphism
N, = N,N2/N2
and the consequent fact that Autc(N,) s Autc(N,N2/N2). Since G/N2 6 fi. it follows
ihat N, N2/N2 e I, and therefore that N, e 3E. Moreover, if N, e 'I), then N, N2/N2 e T),
1. Examples and basic results
335
and in this case Autc(/V,) e Ж /V2//V2) = f(Nl). Hence, in the case f, = 8v we have
shown that G e F>, as required. On the other hand, when Aj is Frattini then so is
At TV2//V2, and therefore in the case 5 = Дф we can also conclude that Ge f>. □
(1.4) Illustrations. Retaining the notation of Proposition 1.3, we now describe some
special cases.
(a) Let 3E = 9) = G and /(G) = 0 for all Ge?). Then = (1), and So =
(G: G has no Frattini chief factors).
(b) Let 3E = 9) = 6 and /(G) = (1) for all Ge?). Then gv = 91, and ЙФ =
(G: the Frattini chief factors of G are central).
(c) Let X = <&p. \j (G: G is an elementary abelian p-group generated by at most n
elements), and let ?) = 0. Then Sv and So, coincide with the class of p-soluble groups
of p-chief rank at most n (the p-chief rank of a p-soluble group is the maximum of
the ranks of its p-chief factors as elementary abelian p-groups—see A, 4.19.)
(d) Let 2 be a class of non-abelian simple groups. Put 3E = 9) = Z and f(T) = (T)
for all T e I. Then Sv = d03:, and
So = (G: the chief factors of G belong to T).
(The truth of the Schreier conjecture—see Huppert [5], 1,18.5—implies that, in fact
Sv = 8® here.)
(e) Let g: P -> {classes of groups}, let 3E = ?) = 6, and let /(G) = fj Jp(p): p||G|}.
Then consists of all finite groups G such that for all chief factors S of G and for
all p 11 S|, we have Autc(S) 6 p(p). As we pointed out in III, 5.3(c), the class Sv need
not be a local class in the sense of III, Definition 5.2(c). However, if g(p) is a formation
for all p e P, then we shall see in Theorem 3.2 of this chapter that Sv = LC(p).
In the preceding results we have obtained formations by specifying the permitted
actions of a group either on all its chief factors of a given isomorphism type, or just
on its Frattini chief factors of a given type. In contrast, restriction of the actions of a
group on its non-Frattini chief factors does not in general g've rise to a formation,
although it does lead to Schunck classes (see HI, 5.1). The reason for this is made
clear by the next result, which shows that if a group with a prescribed action appears
as a Frattini chief factor of a group in a given formation, then it will also appear
as a complemented chief factor of a group in the same formation. This result will
be particularly useful in the next section, where we study the interface between
formations and Schunck classes.
(1.5) Proposition (Barnes and Kegel [1]). Let R/S be a normal section of a group G
in a formation g, and let К be a normal subgroup of G contained in CG(R/S). With
respect to the following action of G/K on R/S:
(rSf* = g~‘rgS, reR.geG,
form the semidirect product H = [R/S] (G/K). Then H 6 8-
IV The theory of formations
J JO
Proof. Consider the following subgroups of the direct product D = (G/S) x G
G* = {(gS,g) .g eG},
K* = {(kS,k):keK},
Rt = {(rS, l):reR),
and observe that Rt < D.
The map 0:g->(gS,g) is obviously an isomorphism from G to G* such that
0(K) = K*. Therefore 6 induces an isomorphism 0*\ G/K -> G*/K*. The action (by
conjugation) of G* an Rt is given by
(gS, g) l(rS, l)(gS, g) = (g~lrgS, 1)
for all g e G, r e R. By defining 0* on R thus:
0*(r) = (rS, 1) e Rt,
it is clear that 0* extends to an isomorphism from the semidirect product [R](G/K)
to the semidirect product (R1](G*/K*). Since G/S eQg = g, and since G*R, is
evidently subdirect in D, it follows that G*Rt e Rog = g. Because К centralizes R/S,
the subgroup K* centralizes Rt, and, in particular, K* < G*Rt, Furthermore, it is
clear that the group G*Rt/K*, which can be factorized as the (internal) semidirect
product (R1K*/K*)(G*/K*), is isomorphic with [Rj](G*/K*) = 0*([R](G/K)).
Hence [R] (G/K) s G*R, /К* e Qg = g. □
The supply of formations produced by the recipes of (1.2) and (1.3) can be further
increased by exploiting the elementary observations that if 3- is any family of forma-
tions, then fj {g: g e 3-} is again a formation, and furthermore that if 3 is a directed
set with respect to the partial order of inclusion, then (J { g: g e 3} is also a
formation (see II, 1.7(b)). However, another familiar device for building new classes
from old, the class product, which was shown in III, 5.13 not to respect Schunck
classes, fails to preserve formations also. The example showing this which now follows
is an application of the special case of (1.3) outlined in (1.4)(e).
(1.6) Example. Let p e P, p > 7, let E = E(3/7), a non-abelian group of order 21, and
let g. P -> {classes of groups} be a mapping defined as follows:
1. Examples and basic results
337
S(7) = (Z2,Z3),
9(p) = (Z2, E), and
g(q) = 1 for all q*7,p.
Let ® = (G e S: for all r e P and for all r-chief factors S of G the group Autc(S) is in
g(r)). By (1.4) (e) the class ® is a formation, and, of course, so is the class Ш of abelian
groups. We now show that the class product is not a formation. Let H = E(6/7),
the primitive extension of a cyclic group of order 7 by a cyclic group of order 6 By
B, 10.7 the group H possesses a faithful irreducible module over Fp; denote this by
Ni, and let N2 be the 1-dimensional F„H-module such that the kernel of И is O2(H).
Finally, set G = [Л/1 © Л/2]/7, and let and M2 denote respectively the normal
3- and 2-complements of G.
Then evidently = Dih(14p) e g and M2/N2 = [N3]E e g; consequently
G/Nj e gill for i = 1,2. But G has four normal subgroups К such that G/K e Й, and
it is easy to check that none of these belongs to g. Hence G e R0(g9I)\g9l, and so
gSI is not a formation.
The failure of class products to preserve formations can fortunately be rectified by
a simple modification of the definition of a product. Recall from II, 2.3 that if g is a
formation, then each finite group G has a smallest normal subgroup with quotient
in g; it is called the ^-residual of G and is denoted by Gs. In (2.4) it was shown that
0(G) я = 0(GR) for all epimorphisms 0 of G, and we shall henceforth use this important
property without further comment.
(1.7) Definition (Gaschiitz [10]). Let ® be a class of groups and g a formation. We
define
®og = (G:GRe®),
and call ® ° g the formation product of ® with g. Clearly ® ° g S g £ ® ° Й
338
IV. The theory of formations
whenever 1 e ®, and if ® = s„®, then ® о ft = Sft. We are about to show that when
ft and ® are both formations, then ® ° ft is also a formation; hence Example 1.6
shows that in general ® ° ft ®8-
(1.8) Theorem. Let ft and ® be formations and § a class of groups. Then
(a) ® о ft is a formation,
(b) Ga°8 = (G 8)lfi for all GeG, and
(c) (§°®)° 8 = 5°(® ° Й)-
Proof, (a) Let .1 = ® о ft.
(1) 3E = q3E: Let G e 3E. If 0 is an epimorphism of G, then 0(G)8 = 0(G8) 6 q® = ffi.
Therefore 0(G) e 3E.
(2) 3E = Rn.U Lett e {1,2},andlet G/Nt e IwithA, n N2 = I.ThenG8/(G8nA’,) s
G^NJNt = (G/Nt)9 e ®. Hence G8 e r0® = ®, and therefore Gel.
(b) First observe that, since G/(G®°8) e ® о ft, we have G8/(Gffio®) =
(G/(G®°8))8 e ®. Therefore (G8)® < G®°8 On the other hand, we have (G/(G8)®)8 =
G8/(G8)®6®. Consequently G/(G8)® e ® о ft, and therefore G®°8 < (G8)®. This
proves Assertion (b).
(c) Using Assertion (b) and the definition of a formation product, we obtain the
following sequence of equivalent statements:
G e (§ ° ®) о ft о G"e(f)o<5)
« (G8)®6£>
о G®°86£>
•» G e § о (® о ft). □
Although we know of no way of modifying the definition to ensure that the class
product of two Schunck classes is again a Schunck class, when the top class is a
formation, then the formation product again comes to the rescue in favourable
circumstances.
(1.9) Theorem. In the universe G, let be a Schunck class of characteristic n, and let
8 be a formation. Assume that either (i) ft = Eoft, or (ii) ft = Gpft for all p e n. Then
fl ° ft is a Schunck class.
Proof. Let G e p(f> о ft). By III, 2.7 it will suffice to show that Gefi» ft. Let R = G8,
let H e ProjjJR), and put N = NG(H). Because of the conjugacy of projectors in a
soluble universe, the Frattini argument applies, and we have G = RN. Suppose, for
a contradiction, that N < G, and let W < M <• G. Set К = Corec(M). By hypothesis
G/K e 5 о ft, and therefore R/(R n K) RK/K e ft. Consequently H(R n K) = R,
and we have G = NR = /VH(R n K) — N(R г. К) < M < G, the desired contradic-
tion. Hence we conclude that N = G and H < G. If we now assume that G/H has a
maximal subgroup M/H supplementing R/H in G/H, the argument that we have just
used leads to a similar contradiction. Therefore R/H <, 0>(G/H) and G/H e E«,g.
Consequently, if ft is saturated, we have G/H e ft; but then R = H e ft, and so
G e § о ft, as desired. If ft = Gpft, then clearly R = 0₽(R), and in this case R/H is a
1. Examples and basic results
339
p -group because R/H e 91. Therefore Assumption (ii) implies that R/H is a n-group
On the other hand, from III, 4.8 it follows that R/H is a тг'-groupbecause R < N (m
Hence R/H = 1, and again we have G e § о 8- G
Remark. Using deeper techniques, we shall show in (4.8) that the restriction to a
soluble universe in this theorem (in the case of saturated formations) is unnecessary.
Another useful construction for formations comes out of the following result.
(1.10) Proposition. Let ft he a formation. Assume that with each group G in 8 there
is associated a subgroup T(G) such that 6T(G) < T(0G) for all epimorphisms 0 of G.
Define
8(T) = Q(G/T(G): G e 8)-
Then
(a) 5(T) is a formation, and
(b) if T(G/T(G)) = 1 for all Ge®, then 8(T) = q{G e 8: T(G) = 1}.
Remarks. The condition on T implies in particular that T(G) is always characteristic
in G. Clearly the subgroup functions O„, Ф, Soc, Z, and the radical of a Q-closed
Fitting class can all play the part of T in this proposition, and for Or and Ф even the
hypothesis of Part (b) is fulfilled.
Proof, (a) We must show that R08(T) = 8(T)- Let GetpfpT). Then there exist
groups Gj, G2 e 8(T) and, for i = 1, 2, epimorphisms apG-^G,- such that
Ker(aj)n Ker(a2) = 1, and the map ocG-vG, x G2 defined by ag = (atg, a2g)
is a monomorphism. Since G2, G2 e R(T), for i = 1, 2 there exists a group Ht e 8
and an epimorphism ftp. H, -»G, such that Кег(Д) > T(Hf Then the map
/?: Hi x H2 -* Gi x G2 defined by P(ht, h2) = (ft/ij, p2h2) is clearly an epi-
morphism. Let H = p~'(aG), the full inverse image of aG under p, and let л,- denote
the projection of H, x H2 onto H, for i = 1, 2. Since H is obviously subdirect
in Hi x H2, we have H e r„8 = 8 Moreover, by hypothesis we have n,T(H) <
Т(л,Н) = Т(Н,) < Ker(ft) for i = 1, 2, and hence 7(H) < Кег(0,) x Ker(ft) =
Ker(/?). We can therefore conclude that
GeaG^ H/Ker(/l)eQ(H/T(H)) with He 8-
Thus G e 8(T) _
(b) This follows at once from (a). LJ
With an eye to later applications, we consider the special case T = O, of the preceding
proposition in more detail.
(1.11) Lemma (D’Arcy [3]). Let 0 # л £ P.tet % be a formation, and let (h = 8(<W
Then'.
340
IV. The theory of formations
(a) © is a formation, and © £ 8 S ©„©;
(b) If X is a formation which satisfies X — d„X £ 8 £ ©nX>
then X = ©.
Proof, (a) This is clear from (1.10).
(b) Set ?) = (G e 8:O„(G) = ') Then Q'i) = © by (1.10)(b).
Since 8 s ©„£, we have ?) £ i, and hence © = Q'i) £ i Now let GeX and pen.
Then the wreath product
W = Z„ %reg G
is in ©„Jt = X £ 8- But evidently О„.(И/) = 1; therefore W e ?) and G e q?) = ©. □
(1.12) Definition. The closure operation s,: Recall that a subgroup H of a group G is
well-placed in G if there is a chain of subgroups
H = Hn < H, < < H„ = G,
such that = H, for i = 1,..., n. If I is a class of groups, define
s„X = (H: G e it, H is a well-placed subgroup of G).
It is obvious that I £ swX £ sjl) whenever jE £ ?) Since the relation of being
well-placed is transitive, it is also clear that sw is idempotent. Therefore s„ is a closure
operation.
In general a formation is not subgroup-closed; for example, Z3 ф QR0(Sym(3)).
However, we shall shortly prove that s„ < qr0 and hence that a formation is always
s.-closed. This surprising fact has important consequences for several aspects of the
theory of formations. First we need to prove the following technical lemma.
(1.13) Lemma (Bryant, Bryce, and Hartley [1]). Let W he a subgroup of a nilpotent
group G of nilpotency class c > 1, and let T = (WGf Then the following statements
are equivalent:
(a) The class of W is less than c;
(b) The class of T is less than c.
Proof. We shall first prove the following statement by induction on n:
(’•«) K„(T) < K„+1(G)K„(H') for all n e N.
(Recall that K,( ) denotes the ith term of the lower central series.) Let n = 1, w e W,
and geG. Since w’ = fg, w_| ]w, we have
Ki(T) = T = <[g, w“‘]w: we W, g e G)
<K2(G)H' = K2(G)K1(1F).
1. Examples and basic results
341
Now suppose that the inclusion in Statement l.a holds fora given value of и > l.Then
= [K„(T), T] < [K„+1(G)K„(fy), T]
= [K„+I (G), T] [K„( 1У), T] (by a, 7.4(f))
< K„+2(G)[K„(fV), K2(G)IV]
= ^„+2(G) [K„(W), K2(G)] [A„(fy), fy] (by A, 7.4(a) and (f))
= A„t2(G)K„u(fy),
because [K„(fy), K2(G)] < K„+2(G) by A, 7.8(b). This completes the induction step
and with it the proof of (l.a).
Ifc(fV) < c, then КД1У) = 1. Since KcH(G) = 1 by hypothesis, we conclude from
(l.a) with n = c that KC(T) = 1; in other words, c(T) < c. Hence Assertion (a) implies
(b), and the reverse implication is obvious. □
(1.14) Theorem (Bryant, Bryce, and Hartley [1]). Let К be a nilpotent normal sub-
group of a finite group G, and let G = WK. Then W e QR0(G). (For a more general
version of this result, the reader is referred to Theorem VII, 1.1.)
Proof. The proof will be by induction on c = c(K). If c = 0, then К = 1 and W = G.
Therefore assume that c > 1, and consider the following subgroups of the direct
product G x G x G:
1У* = {(w, w, w): w e W] ( = W),
D = {(k,k, l):keK]{ = K), and
E = {(1, к, к) : к e K] ( = K).
Let H = < И7*, D, E). Because W* obviously normalizes D and E, it also normalizes
the nilpotent subgroup F = (D, Ef, moreover H = W*F. Since WK = G, evidently
H is subdirect in G x G x G and therefore belongs to Ro(G). Let Z = Z(E) =
{(1, z, z): z e Z(K)}. Then Z centralizes D and is normalized by W*, and so Z < H.
Since DnZ=l, we have DZ/Z = D = K. Therefore the group F/Z contains a
subgroup DZjZ of class c and hence has class c itself. But E/Z{~ K/Z(K)) has class
c — 1; consequently, by (1.13) the group <Ef>/Z( = <(E/Z),f/z’>) has class c — 1. Since
W*DZ/Z supplements <EF>/Z in H/Z, by induction it follows that W*DZ/Ze
Q^(H/Z) s qrcqr„(G) = qr0(G). Let x e W*D n Z; then л = (wk, wk, w) = (1, z z)
for suitable weW, ke K, zeZ. Then 1 = wk = z, and consequently x = (1,1,1).
Hence W*D nZ = 1, and it follows that W*D s W*DZ/Z e qr0(G). Thus finally we
can conclude the W = W* ~ W*D/D e qr0(G). □
(1.15) Corollary, (a) s„ < qr0; in particular, formations are always sw-closed.
(b) QR„Eps, < QRoS.E, for any prime p.
342
IV. The theory of formations
Proof. Part (a) follows immediately from (1.14) and the definition of s„. To prove
Assertion (b), let G e t:rs.T for some class X of groups. Then G has a normal
p-subgroup К such that G/K s H, where H is a subnormal subgroup of some
Л-group X. Let В denote the base group of W = К Qj„g X; then BH sn
fVeEpX, and so BH es.E„X By A, 18.8(a) the group BH contains a subgroup L
with
(1-/0
LsKrL,„H,
such that BH = BL and BryL = N, where N (<L) maps to the base group
of KrLire(,H under the isomorphism (1.0). Moreover, from A, 18.9, using the
same isomorphism, we see that L contains a subgroup G* s G such that L - NG*.
Thus BH = BNG* = BG*, and since В is nilpotent, (1.14) yields the conclusion that
G* e QRo(BH). Consequently G e QR„snErX, and therefore
QROEPS„ < (QR0)2S„Ep = QRn\Ep
since by II, 1.18(b) the class map qr0 is idempotent.
In contrast to formations, Schunck classes need not be s„-closed; in fact, we shall
prove in the next section that a Schunck class is s„-closed if and only if it is a saturated
formation.
Our second application of (1.14) exploits the fact that every subgroup of a nilpotent
group is well-placed.
(1.16) Theorem (P.M. Neumann [2J, Vaughan-Lee (unpublished)). A formation
consisting entirely of nilpotent groups is subgroup-closed.
The following lemma, which also depends in part on (1.14), will be used frequently
in the sequel.
(1.17) Lemma. Let ft be a formation, and let G = UN with U < G and N < G. Then
(a) U~'N = G®N,
(b) if N e 91, then U* < G®, and
(c) if N e <S„ for some p e P and if G is the formation ft(Op)
described in (1.10), then [N, I/®] < G®.
Proof, (a) Let 6 denote the epimorphism from U onto G/N defined by 6(u) = uN.
Then U^N/N — 6(17®) = (G/N)® = G^N/N, and so Assertion (a) is true.
(b) Let R = G®. Then (UR/R)(NR/R) = G/K e ft, and therefore UR/R e ft by
(1.14). Consequently U/(U cyR)eft, and 17® < R.
(c) In view of Part (a) we may pass to the quotient group G/G ® and may suppose
without loss of generality that G® = 1. Then by (1.11 )(a) we have G e G„.®, and
therefore G® is a normal p'-subgroup of G. By Part (b) we have UK’ < G® and
therefore [N, 17®] < [N, G®] < N n G® = 1. □
1- Examples and basic results
343
As a further application of (1.14) we shall show that for a formation » of soluble
groups the g-residual respects the operation of forming direct products.
(1.18) Theorem (Doerk and Hawkes [2]). Let % be a formation of soluble groups, and
let D = Gr x G2 x x G„. Then
P8 = (G1)8x(G2)8x-x(G„)8.
Proof. By induction on the number of direct factors it is clearly sufficient to handle
thecasen = 2. Therefore let D = G1 x G2,let T = D", and let Tn(Gt x 1) = Tt x 1,
Infix G2) = 1 x T2. Let v denote the natural homomorphism from D onto
D/(Ti x T2), let a denote the canonical isomorphism from D/(T, x T2) to
(G1/T1) x (G2/T2), and let G — a о v. Put G* — G(/7] for i - 1,2, and denote the image
of T under 6 by R. Since residuals are preserved under epimorphisms, we have
К = (G* x GJ)8. Also, it is clear that Rn(G* x 1) = R n (1 x GJ) = 1, and hence
that R < Z(GJ x GJ). Since by hypothesis g-groups are soluble and since R is
abelian, the group GJ x GJ is soluble. If GJ = 1, then 7 = 7, x T2. If GJ / 1,
then G J e <S has a proper subgroup U such that GF(GJ) = GJ. Obviously V x GJ <
GJ x GJ with (U x G J)F(GJ x GJ) = GJ x GJ. By induction on the order of D we
may assume that
Vs x (GJ)8 = (U x GJ)8.
Moreover, by (1.17)(b) we have
(U x GJ)8 < (GJ x GJ)8 = R.
Therefore 1 x (GJ)8 <Rn(l x GJ) = 1, and so GJ e g. Similarly GJ e g. Hence
R = 1, and again we have T = 7, x T2. Consequently,
(G/(l x G2))8 = (T, x G2)/(l x G2).
However, G, x 1 complements 1 x G2 in G, and so
(G/( 1 x G2))8 = (G, x 1 )8(1 x G2)/(1 x G2) = (G8 x G2)/( 1 x G2).
It follows that 1\ = Gp and similarly that T2 = Gf. □
The corresponding statement to (1.18) for Fitting classes is not true. We shall see
in Chapter IX (Example 2.14 (b)) that there exists a Fitting class g and a soluble group
G such that the g-radical(G x G)a is strictly greater than GB x Gs. This phenomenon
leads to the so-called theory of Lockett sections for Fitting classes. Although the
analogous theory for soluble formations is reduced to a triviality by Theorem 1.18, in
Exercise 12 ofChapterX, Section 1 we describe an example which shows that (1. 18) is
false in the larger universe G. Because of this, it is indeed possible to develop a
non-trivial theory of‘Lockett sections’ for formations of arbitrary finite groups and
how this can be done is indicated in Exercises 10 and 11 of Chapter X, Section 1. n
the exercises at the end of Chapter X, Section 4 we also discuss a dualized ‘Lausch
group’ for a formation.
344
IV. The theory of formations
Exercises
1. Decide whether the following classes of groups are formations:
(a) The class of all groups that possess a Sylow tower;
(b) The class of groups which are direct products of simple groups;
(c) The class C” of л-perfect groups (in a soluble universe);
(d) The class of all finite groups G such that Z(G) has exponent 2;
(e) The class of groups G, described in (d), satisfying the further condition that
condition that G/Z(G) is a direct product of non-abelian simple groups.
2. Find a Schunck class § and a saturated formation g such that the formation
§ | g defined in (1.2) is not saturated.
3. Describe the formations generated by Sym(3) and Sym(4). (See Blessenohl and
Brewster [1] for a more general construction.)
4. Let g and © be formations. Show that in general © £ G о g.
5. Let § be a Schunck class and g a formation. Show that § о g is not in general
a Schunck class.
6. Let g and © be formations such that © о g = (S. Prove that either g = <S or
© = S.
7. Let g be a formation, and let H be an g-projector of a soluble group G. Prove
that H is an Bag-projector of G.
8. (D’Arcy [3]) Let g be a formation, and let © denote the formation g(O„)
described in (1.10). Show that <S„© = © if and only if G„(G e g : O„.(G) = 1) e g.
9. Let D and Q be non-isomorphic, non-abelian groups of order 23. Prove that
QRr,(O) = qr0(2). Is the analogue for an odd prime true?
10. Let G and H be nilpotent groups of class 2 with the same exponent. Is
qr„(G) = QR„(H)?
11. (Pense) If g is a formation, define
Mg) = (G e G: G ф g and (q - 1 )(G) e g).
Show that
(a) g2 = g if and only if s„(bQ(g) и (1)) = bQ(F) и (1), and
(b) g ° g = g if and only if bQ(g) contains Soc(G) whenever it contains G.
2. Connections between Schunck classes and formations
Closure properties of Schunck classes and formations, those that they have in
common and others that set them apart, are the main theme of this section (here our
main source is Forster [5] and Hawkes [7]). We also investigate certain forma-
tions that may be naturally associated with a given Schunck class (using ideas of
Kattwinkel [1]). For simplicity’s sake we work in a soluble universe throughout, and
leave the reader to decide which results can be extended to a more general setting,
and how.
By III, Definition 2.6 the <S-Schunck classes are the non-empty PQ-closed classes
of soluble groups (by III, 2.7 they may also be characterized as the p-closed classes,
2. Connections between Schunck classes and formations
345
but p is not a closure operation). We now look for descriptions of Schunck classes
by other closure properties. In Chapter III we showed that a Schunck class is closed
under the following operations: q (III, 2.7); Еф (III, 2.10); d0 (III, 6.2) However these
three together are not enough to characterize Schunck classes, as the followine
example shows. 6
(2.1) Example, Let ?) = (1, Z2, Sym(3)), and let I = e.d0?). We aim to show that
the class £ is (q, еф, Do)-closed but is not a Schunck class. The first step is to prove
that d09) is Q-closed. A group D in d(,4) clearly has the form
D = E x Sj x • • x S„
where E is an elementary abelian 2-group and s Sym(3) for i = 1, t. Let
R ~ O3(D), and let R,- — O3(5j for i — 1,..., t. Then R is clearly the direct product
of the pairwise non-D-isomorphic minimal normal subgroups R,,.... R,. If /V is a
normal subgroup of D, by B, 3.5 we may therefore write
N n R = R, x • • x Ru (u < t)
after suitable renumbering of the components S2,..., S,. If 7; e Syl2(Sj), we evidently
have
D/(N riR) = E x Tt x x T„ x Su+1 x • x S„
and therefore D/(N r>R) = E* x S, where E* is an elementary abelian 2-group and
S S Su+l x x S,. Since N n R = O3(N), it follows that N/(N n R) is a 2-group and
is therefore contained in 02(D/(N n R)) = E*. Hence
D/N s (D/(N r> R))/(N/(N r> R)) S (E*/(N/(N n R))) x S,
which clearly belongs to d0'2). Therefore d09) is Q-closed. Since оеф<ефо by II,
1.17(iii), it follows that
q£ = QE»D0?) e E»QD0?) = ЕфО0?) = i
and hence that £ is Q-closed. It was also shown in II, 1.17 that e»d0 is a closure
operation, and therefore X = ЕФо0Х = <e», d0 )£
To see that X is not a Schunck class, consider a group G which is the semidirect
product of an elementary abelian group of order 9 with an inverting automorphism
(of order 2). Clearly G 4 '!) Since G is directly indecomposable and has trivial Frattini
subgroup, it follows that G 4 e»d0?) = Jt But the primitive epimorphic images of G
evidently belong to ?) and hence to Jt. Consequently G e pi'\X, and therefore I is
not a Schunck class.
Of the ‘standard’ closure operations that appear in the list in II, 1.5, the three
Q, Еф, and d0 are the only ones under which a Schunck class is invana у c os
(see Exercise 1 below). We wish to characterize Schunck classes among saturated
346
IV. The theory of formations
homomorphs by the imposition of an additional closure property. We observe
that R<,-closure is obviously too strong a requirement (identifying only the satu-
rated formations), whereas D„-closure is too weak (as the above example shows).
However, there is a new operation Ro, which lies between d0 and r0, and which
precisely achieves this objective.
(2.2) Definitions, (a) (Gaschiitz [7]) Let H/K be a complemented chief factor of a
group G. Let C — CG(H/K), and let
R = p {T: T < С, T < G, and C/T g H/K}.
Then the normal section C/R is called the crown of H/K. A crown of G is by definition
the crown of some complemented chief factor of G.
(b) (Hawkes [7]) The closure operation R„: If £ is a class of (soluble) groups, we
define Rtl£ to be the class of all groups G which possess a set .4' of normal subgroups
Ni,...,Nt such that
(i) G/N, e £ for i = 1,2,..., t,
(ii) QUi Ni — •> and
(iii) for each crown C/R of G there is at least one N, such that C f NtR.
(2.3) Remarks, (a) If p e IP and G/G'G11 1, then G/G'G11 is a crown of G.
(b) Let C/R be the crown of H/K, and let p||C/R|. Then by B, 3.6 the section C/R
is a completely reducible, homogeneous FpG-module. Since C/R is a self-centralizing
normal subgroup of G/R, and since Op(G/C) = 1 by A, 13.6, it follows that C/R =
F(G/R). If C/T = H/K, then C/T is a self-centralizing minimal normal subgroup of
G/T. Hence C/T is complemented in G/T by A, 15.8(b) and A, 15.7. Therefore
®(G/R) = 1, and C/R is complemented in G/R by A, 9.2(f). Furthermore, each chief
factor of G between R and C is complemented and has C as its centralizer.
(c) Let C/R be the crown of a complemented chief factor H/K of G, and let T be
a maximal G-subgroup of C containing R. Then G/T = [H/K}(G/Cp, in particular,
the group G/T is primitive, and its isomorphism class depends only on C/R and not
upon T.
(d) We shall subsequently use the obvious fact that ф r> гфк„£ £ £.
First we justify our claims for r0.
(2.4) Proposition (Hawkes [7]). The class map Ro defined in (2.2)(b) is a closure
operation and satisfies Do < Ro < rc.
For the proof of this proposition it will be helpful to have the following criteria
for a normal subgroup not to cover a crown.
(2.5) Lemma. Let C/R be a crown of G, and let N <3 G. Then any two of the following
statements are equivalent.
(a) The image of C/R under the natural homomorphism from G to G/N is a crown
of G/N -,
2. Connections between Schunck classes and formations
347
(b) N does not cover C/R;
(c) RN < C.
PJ°°f' . S1I1re crowns are by definition non-trivial, Statement (a) implies that
RN CN and therefore that C RN.
(b) => (c): Let L = RN n C (= R(N г-. C)). If Statement (b) holds, then L is a normal
subgroup of G properly contained in C. Since [IV, C] < N n C < L, we have N <
CG(C/L}, which equals C because C is the centralizer of every non-trivial normal
section of C/R. Therefore RN = R(N C) = L < C.
(c)=>(a). Since C/RN is completely reducible and homogeneous as an
FpG-module, and since C is the centralizer of each chief factor of G between RN and
C, it is clear that (C/N)/(RN/N) is part of a crown of G/N. However, if K/N is a normal
subgroup of G/N such that (C/N)/(K/N) is isomorphic with a chief factor of G/N
between RN/N and C/N, then C/K is G-isomorphic with a chief factor of G between
R and C, and hence RN < K. Thus (C/N)/(RN/N) is evidently a full crown of G/N.
□
The proof of Proposition 2.4. Obviously r0 is expanding and monotonic; to prove
that it is a closure operation, it remains to show that it is idempotent. Let G e =
r0‘(r„X). Then G has normal subgroups N,,..., N, such that
(i) G/Ni e R„X for i = 1,..., t,
(ii) Q!=1 /V,- = 1, and
(iii) each crown of G is not covered by at least one of the N\.
It follows that if i e {1,..., t}, the group G/N, has normal subgroups {N^/N,: j =
1,..., t,} such that
(i)' G/N^cX,
(ii)' Ni} = Nh and
(iii)' each crown of G/Nt is not covered by at least one of the Ny/N,. The normal
subgroups of G in the set N = {/Vy: i = 1,..., t; j = 1.t,} obviously have trivial
intersection. If C/R is a crown of G, by Condition (iii) above there is an i e {1,..., t}
such that C RN.. But then by (2.5), (b) => (a), the factor (C/N^RNJN^ is a crown
of G/N,, and in this case Condition (iii)' implies the existence of а уб{1,
such that C f RNir Hence the set N satisfies the three requirements of Definition
2.2(b), and therefore G e R0Jf. It follows that Ro2 = Ro and hence that r0 is a closure
operation.
It is obvious that r0 < r„. To see that d0 <. r0, let G e d„I. If G = 1, then
Geis R„Jt'. Otherwise, we may suppose that G = G, x x G, with 1 / G, e X
for i = 1,.... t. For ie {1,..., t] set (Vf = Пл.»6; <if' = we obtain K is
easy to see that each complemented chief factor of G is G-isomorphic with a comple-
mented chief factor above A, for some i e {1....t}, and it is then obvious that its
crown cannot be covered by this Nr Therefore the set Л' = {Nf}!=i clearly satisfies
the three requirements of Definition 2.2(b), and so G e R^X.
Although it is not difficult to construct explicit examples to show that d0 / r0 /
we remark that these inequalities also follow from (2.7) below, in the light of Example
2.2 and of the fact that not all Schunck classes are formations. □
348
IV. The theory of formations
(2.6) Theorem (Forster [5]). Let X be a non-empty class of groups. Then ефкоч£ is
the smallest Schunck class containing X.
Proof. Since the Schunck class generated by £ is the same as that generated by q£,
we may suppose without loss of generality that £ = q£. Let § = (G: q(G) e £),
which is the Schunck class generated by £, and suppose that §\ефк„£ is non-empty;
it therefore contains a group G of minimal order. Clearly 0>(G) = 1, and G is not
primitive. Let
.Ж = {W 3G: G/N c ^}.
Since G ф ф, each N e .Ж is non-trivial, and so because of the choice of G we have
G/N e ф n ефк„£ = £ by (2.3)(d). Let К < G. Since O(G) = 1, there is a maximal
subgroup M of G complementing K, and therefore К f Corec(M) e Л'. Hence
Q [JV: N e .4' } = 1. If C/R is a crown of G, and if C/T is a chief factor of G above R,
then G/T is primitive; thus T is a subgroup in which does not cover C/R.
Consequently .Л" satisfies the requirements of Definition 2.2(b) with respect to G, and
so G e r„£ c eor0£. This contradiction proves that § e ефк0£.
Now let G e ефе0£. Then G has a normal subgroup К contained in ®(G), and
normal subgroups Nt......N, such that (i) G/Nt e £, (ii) Nt = K, and (iii) each
crown of G is not covered by some Nt (because all crowns of G obviously lie above
K). Let G/T be a primitive quotient of G, let C/T = Soc(G/T), and let C/R be the
crown of C/T. Then RNt < C for some i e {1,..., t}. Let C/L be a chief factor of G
above RNj. Then by (2.3)(c) we have G/T s G/L e <j(G/Nj) c £. Therefore G e p£ —
f>, and consequently гфк0£ £ □
(27) Corollaries (Forster [5], Hawkes [7]).
(a) Any two of the following statements about a non-empty class X of soluble groups
are equivalent:
(i) £ is a Schunck class;
(ii) £ = e4,r„q£;
(iii) £ is <Q, R„, ^/-closed.
(b) The class map Ефкос; is a closure operation and coincides with <Q, Ro, Еф>.
Proof, (a) The equivalence of Assertions (i) and (ii) is an immediate consequence of
(2.6), and the equivalence of (ii) and (iii) follows from (b).
(b) Let £ be a class of groups. By (2.6) the class e4,ri,q£ is a Schunck class, and
hence by the implication: (i) =>(ii) of Part (a) we have (eor„q)2£ = EeR0Q£. Thus the
class map Ефк0<} is idempotent, and since it is obviously also monotonic and expand-
ing, it is therefore a closure operation and coincides with <q, r0, e*> by II, 1.15.
□
Although <q, R0>-closed classes do not play a significant part in the theory of
soluble groups, we nevertheless give them a name to emphasize their position as a
common ancestor of Schunck classes and formations.
(2.8) Definition. A (q, R0)-closed class of (soluble) groups is called a preformation.
2. Connections between Schunck classes and formations
349
The next result follows directly from (2.6)
(2.9) Corollary If % * 0 is a preformation (in particular, if g is a formation), then
ЕфЦ is the smallest Schunck class containing g.
The upshot of (2.7)(a) is that an <S-Schunck class is a saturated preformation. We
shall shortly characterize formations of soluble groups as s„-closed preformations.
But first we need a new concept.
(2.10) Definition. Let R/S be an abelian normal section of a group G. Then we call
the semidirect product [R/S](G/Cc(R/S)) the split image of G derived from R/S. We
denote the set of all primitive split images of G by Psi(G); obviously this consists of
just those split images which are derived from chief factors of G, and evidently, when
G is soluble, we have ф n q(G) £ (Psi(G)).
(2.11) Lemma. Let X = <q, s„> £ <= <S. Then Psi(G) e £ for all G e £.
Proof. We use induction on |G|. If G = 1, then Psi(G) = 0. Suppose that G / 1, and
let M be a critical maximal subgroup of G. Let L = Corec(M), let U/L = Soc(G/L),
and note that G/L is isomorphic with the split image of G derived from U/L.
Then from III, 6.5, and in particular from Equation 6.y, it follows that (Psi(G)) =
(G/L) и (Psi(M)). But G/L e q£, and furthermore, since M e s„£ = £, by induc-
tion we have Psi(M) c £. Therefore Psi(G) £ £ □
(2.12) Theorem (Forster [5]). Let g be a class of soluble groups. If one of the
following statements is true, then they all are.
(a) g is a formation;
(b) g is an s„-closed preformation;
(c) g is a preformation, and Psi(G) e g for all G e g.
Proof. The implication: (a) =>(b) follows from (1.15)(a) and the fact that Ro < Ro. The
implication: (b) => (c) is obvious from (2.11). It remains to show that Statement (c)
implies (a). Let G be a group with distinct minimal normal subgroups Nt and N2 such
that G/Nt e g for i = 1, 2. To prove that g is R0-closed, by II, 2.6 it will suffice to
show that G e g, and this is certainly the case if no crown of G is covered by both
Nt and N2 because of the R„-closure of g. Therefore suppose that G has a crown
C/R which is covered by Nt and N2. Then for {1, j] = {1,2} we have
NiNj/NjSN^ NjR/R = C/R,
G G
and, in particular, C = Cc(N,). Consequently C/R is the unique crown of G covered
by N, and N2. It follows that G/R = ^,N2/N2](G/C) e Psi(G/N2), and so G/R eg
by the hypothesis of Statement (c). It is then clear that the normal subgroups in the
set Ж = {Nj, N2, R} satisfy Requirements (i)-(iii) of Definition 2.2(b), and therefore
G e R„g = g. Hence g is a formation.
350
IV. The theory of formations
From (2.7) (a) and (2.12) we can now readily deduce the following promised result.
(2.13) Corollary. A Schunck class is a saturated formation if and only if it is s. -closed.
Next we make two elementary observations about the primitive split images of a
group.
(2.14) Lemma. Let G be a group.
(a) If N <G, then Psi(G//V) £ (Psi(G)).
(b) If N2,N2<G and N2 n N2 = 1, then (PsifC/Nf) и Psi(G/N2)) = (Psi(G)).
Proof. The class of primitive groups generated by Psi(G) obviously depends only on
the isomorphism classes of the G-modules which appear as chief factors of G, and so
Assertion (a) is clear. Furthermore, because N, S each chief factor of G is
G
G-isomorphic either with one above Nt or with one above N2, and then Assertion
(b) is also clear. □
Let X be a class of groups and c a unary closure operation. We recall from II, 1.19
that Xе = (G: c(G) <= X).
(2.15) Lemma. If X is a class of soluble groups, let /(X) denote the following class:
f(X) = (G 6 6: Psi(G) <= X).
Then
(a) f(X) is a formation,
(b) if ?) = <q, s„>?) <= X, then ?) с /(X), and
(с) /(X) = X<°-s~> if and only if f(X) £ X.
Proof. Part (a) follows at once from (2.14) and Part (b) from (2.11). In Part (c) the
necessity of the condition is obvious. To prove the sufficiency, suppose that f(X) £ X.
Since /(X) is a formation, it follows from (2.12) that /(X) is <Q, su>-closed, and then
Part (b) implies that /(X) is the largest <q, s„)-closed class contained in X. □
The following theorem was proved independently by Kattwinkel [1] and Schaller
[5], and was subsequently improved by Forster [5].
(2.16) Theorem. Let § be an <5-Schunck class. Then § contains a unique maximal
formation, denoted by S)°\ and
= =
Proof. If J is a formation contained in §, then by (2.12) we have g = <Q,
consequently g £ /(§) by (2.15)(b). By definition the class /(§) is contained in
p£> — & and so by (2.15)(a) it is the largest formation contained in §. The last
equation is now clear from (2.15)(c). □
2. Connections between Schunck classes and formations
351
A further contribution to the conclusion of this theorem can be derived from the
following property of the set #“(G) of well-placed subgroups of a group G.
(2.17) Lemma. Let К be a normal subgroup of a soluble group G. If H/K e W/G/K),
then there exists a W e it(G) such that H = WK.
Proof. We use induction on |G|. First suppose that К •< G.
Case (a): К is complemented in G, by L say. Then there is an obvious isomorphism
from G/K onto L, under which H/K maps to L<~» H, which is therefore a well-placed
subgroup of L. As L is obviously well-placed in G, it follows thatLnHe #(G), and
since H = (L n H)K, we obtain the desired conclusion by taking W = L n H.
Case (b): К < <D(G). Let M/K be the maximal subgroup of G/K in a critical maximal
chain running from H/K up to G/K. Then (M/K)F(G/K) = G/K, and from A, 9.3(c)
it follows that MF(G) = G. Since H/K is obviously well-placed in M/К, and since
I AL | < |G|, by induction there is a subgroup W in #'(Af) such that H = WK, and
because 0(M) £ 11(G), this W also satisfies the requirements of the lemma.
Finally, if К is not a minimal normal subgroup of G, then either К = I, in which
case the result is clear, or there exists a minimal normal subgroup N of G contained
in K. Since (H/N)/(K/N) is obviously well-placed in (G/N)/(K/N), by induction there
exists a subgroup W*/N in KfG/N) such that (W*/N)(K/N) = H/N. By setting
К = N and H = W* in the case К •< G which we have already dealt with, we deduce
that G has a well-placed subgroup W such that W* = WN. Then we have H =
W*K = WNK = WK. □
(2.18) Corollaries, (a) For all GeG and all epimorphisms в: G-> 0(G) we have
e(1C(G)) = W(G))-
(b) S„Q < QS„.
(c) X<0-s-> = Xs- for all I = Ql c
Proof, (a) Let 0 be an epimorphism of G. If S is a critical subgroup of G, then
0(G) = 0(SF(G)) < 0(S)F(0(G)); therefore 0(5) is a critical subgroup of 0(G). It follows
that critical chains of G are mapped by 0 to critical chains of 0(G), and consequently
0(# (G)) e tT(0(G)). On the other hand, if 0(H) e #'(0(G)), by (2.17) there is a IF in
V/ (G) such that 0(W) = 6(H), and Assertion (a) is now clear.
(b) Let H e s_q(G). Then H/K e U'(G/K) for some К < G, and by (2.I7) we have
H = WK for some WeII(G). Consequently H/K s W/(W с. K) e OS.(G), and As-
sertion (b) follows. . . .
(c) Since s„ is unary, the class Xs» is certainly defined, and from its definition we
have X' °-s-> e Xs". But in view of Part (b), it follows from II, 1.20 that X - is Q-closed;
therefore Xs- e X<o s">, and the two classes are equal. □
(2.19) Theorem. Let c be a function which assigns to each soluble group G a set c(G)
of subgroups satisfying the following properties:
(a) #(G) s c(G) for all GeS:
352
IV. The theory of formations
(b) 0(c(G)) = c(0(G)) for all G e Z and all epimorphisms 6 of G;
(c) If H 6 c(G) and L e с(Я), then L e c(G).
For a class X of groups, let Xе denote the following class:
Xе = (G e to: c(G) <= I).
Then, if Xis a Schunck class or a formation, the class X' is a formation.
Remarks. Given a function c with the properties described in the statement of this
theorem, we can define an associated class map by setting
cl = (c(G): G 61).
Since G 6 T^(G), it follows from Property (a) that X cX; moreover, from Property
(c) we have <2X = cl. Since c is obviously monotonic, we therefore conclude that
c is a unary closure operation (see Definitions II, 1.19). In the notation of closure
operations Properties (a) and (b) then imply that (i) s„ < c < s, and (ii) cq < qc.
Proof. Since X = qX and cq < qc, by II, 1.20 the class Iе is Q-closed.
Next we assert that R()X' X. Since Iе I by Property (a), this assertion .is
certainly true if X is a formation. Therefore suppose that X is a Schunck class. If
G e X', then c(G) Xе by Property (c), and therefore by Property (a) we have
W'(G) Xе. Consequently Xr is sw-closed, and it follows that Xе Xs- = by
(2.18)(c) and (2.16). Hence Rf,Xr r0Xo,<° = X0R° X, as asserted.
To complete the proof we must show that Xе is Ro-closed. Let N2 be normal
subgroups of a group G such that G/Nt and G/N2 are in Xе and n N2 = 1.
Let H 6 c(G), and let i e {1,2}. Then HNJNj e (fG/Nj) by Property (b), and there-
fore НЦН n NJ s HNJNj e ci' = Xе. It follows that H e R0Xr X, and hence that
G 6 Xе, which proves that X' = R(1Xr. □
Evidently the hypotheses of this theorem are fulfilled when c( ) = ifl ), but in this
case the theorem yields nothing new (see (2.16) and (2.18)(c)). However, the hypo-
theses are also fulfilled when c(G) is defined to be the set of all subgroups of G, in
which case we derive the following result.
(2.20) Corollary (Kattwinkel [1]). If § is a Schunck class or a formation, then §s
is a formation.
We have therefore shown that for a Schunck class § we have
(2.«)
§s <= §OR” <= 5.
We now discuss possible patterns of equality in this sequence of classes. First we
make the obvious remark that all three classes coincide if and only if § is subgroup-
closed.
2. Connections between Schunck classes and formations 353
(2.21) Proposition (Kattwinkel [1], Schaller [5]). Let § be a Schunck class of soluble
groups. Then ft"»» = & if and only if ft""» is saturated.
Proof. The necessity of the condition is obvious. To prove the suficiency, assume that
S0R° is saturated, and let G be a group of minimal order in the class ft\ft"“°,
supposing by way of contradiction that this class is non-empty. By III, 2.11 the
group G is primitive, and therefore in the notation of (2.10) we have (Psi(G)) =
(G) u(Psi(G/Soc(G))). From the minimal choice of |G| it follows that G/Soc(G)e
S0"" and hence that Psi(G/Soc(G)) s ft""» by (2.16). Consequently Psi(G) s ft, and
therefore G 6 ftagain by (2.16). This contradiction confirms that ft = ft""». □
It is not hard to find saturated formations which are not subgroup-closed, and which
therefore provide examples of a Schunck class ft for which
fts c £<;«.. = §
In this case, however, the formation fts is also saturated by a theorem of Carter,
Fischer, and Hawkes (see VII, 6.13). On the other hand, the saturation of fts does
not appear to influence the other two classes. For example, if ft = Q", then ft0"» is
the class of all groups which have no central л-chief factors (this is a proper subclass
of ft); and fts = <5„ which is indeed saturated. Finally, we remark that it is possible
to have fts = ft°"« <= ft; such an example is suggested in Exercise 4 below.
The next result sets precise limits on the range of Schunck classes ft for which
ft0"» is a given fixed formation. We should mention, however, that not every forma-
tion can be expressed in the form ft0*» for some Schunck class ft; those that can be so
expressed have been characterized by Schaller [5] (see Exercise 5).
(2.22) Proposition (Kattwinkel [1], Schaller [5]). Let ft be a Schunck class, and let
8 denote the formation ft0"». Let 8 = (В: В e b(f>), B/Soc(B) e 8)- Let ft0 and ft0
denote the Schunck classes еф8 and /1(8) respectively. Then the following statements
about a Schunck class Я are equivalent:
(a) 8 = ft0R«;
(b) ft0 £ Я £ &°.
Remark. Schaller [5] characterizes the class ft0 differently as follows: Let 6 -
ft i g; then ft0 = (G: G« < Ge).
Proof. First note that the class 8, as a subclass of b(ft), is a Schunck boundary.
Therefore 8 = 6(ft°) by III, 2.9. Assume that Assertion (a) holds. Since 8 Я we
have ft0 = Еф8 S е®Я = Я. We suppose that Я £ ft0 and derive a contradiction.
Let G be a group of minimal order in Я\§°. By III, 2.2(a) we have G e fe(ft )
8 £ ф, and consequently G/Soc(G) e Я°"» by definition of 8. Therefore applying
(2 16) twice, in the notation of (2.10) we conclude that Psi(G/Soc(G)) £ Я, hence that
Psi(G) £ Я. and finally that G e Я""» = 8 S & = S’- 11118 contradiction therefore
proves that Я £ ft0, and Assertion (b) follows.
354
IV. The theory of formations
Now assume that E.g£ Я S ft0 Since Я is by hypothesis a Schunck class, by
(2.16) it contains a unique maximal formation Я°\ which therefore contains g. If
there exists a group, G say, of minimal order in B0R«\g, and then by
П, 2.5(a) this G has a unique minimal normal subgroup N, where G/N eg. Let
B = [N](G/Cc(N)), and observe that (Psi(G)) = (B) u (Psi(G/N)). By (2.16) we have
Psi(G/N) £ ft and Psi(G) £ Я, whereas Psi(G) £ ft because G £ g = ft0’».
Therefore the primitive group В is in ft\ft. Since B/N S G/Cc(N) e OtG/N} £ Qg =
g £ ft, it follows that В e b(ft) and hence that В 6 ® = 6 (ft0). But Be Я £ ft0,
and we have a contradiction. Consequently g = Я0'-, and we have shown that
Assertion (b) implies Assertion (a). □
This completes our discussion of the various formations that are associated in a
natural way with a given Schunck class; some further related ideas are explored in
the exercises that follow. If we turn this situation on its head and look for Schunck
classes that may be naturally associated with a given formation, no comparable
pattern emerges. As we have seen, the smallest Schunck class containing a formation
g admits the elegant description r^g, but the Schunck classes that are contained
in g do not seem amenable to easy analysis. To show that there certainly need not
be a unique maximal one is the purpose of the following example, with which we
close this section.
(2.23) Example. Define a function g: P -+ {classes of groups} by setting g(7) =
(1, Z2, Z3) and g(p) = <5 for all p 7, and denote by 3E the formation constructed
from g according to the procedure of (1.4)(e). (Thus I consists of all soluble groups
which induce on their 7-chief factors groups of automorphisms in ry(7).) Then set
g = e79i|2.3)ni,
and note that g is a formation.
Let D = Dih(14) and E = E(3/7), a non-abelian group of order 21. Let ft, = pq(£>),
the Schunck class generated by D, and let ft2 = pq(E). We claim that ft, e g. The
formation T generated by D is easily seen to consist of all supersoluble groups which
are extensions of elementary abelian 7-groups by elementary abelian 2-groups (see
Example 3.4(f) in the next section). If G 6 гфТ, then G is still supersoluble; therefore
GeG7S2 and the Frattini chief factors of G are either central 2-chief factors or 7-chief
factors on which G induces a group of automorphisms in (1, Z2). Thus ефТ> £ g. By
(2.9) the class ефТ> is a Schunck class, and so ft, = pq(O) s роефТ = г.фТ £ g, as
claimed. Similarly ft2 £ g.
Let C be a cyclic group of order 6, and let Ц be an irreducible (F-,C-module such
that |Ker(C on I'd = i for i — 2,3. The modules are both 1-dimensional and V2 ® V3
is faithful for C. Denote by H the Hartley group Я(И2, F3) (see B, 12.11) and by G
the semidirect product Then H'/l is a Frattini 7-chief factor of G on which G
induces a group of automorphisms of order 6, and consequently G ф g. However,
since q(G) n ф = (Z2, Z3, В, E), it follows that G is in p(ft, uft2), which is the
Schunck class generated by the homomorph ft, и ft2. Thus we see that there is no
Schunck class contained in g that contains both ft, and ft2.
2. Connections between Schunck classes and formations 355
Exercises
1. Find a Schunck class which is not closed under any of the following operations:
$n> Ro, No, E, Ez.
2. Prove that qr0 = <q, r0> roq.
3. If S is a class of groups, show that ecqr0I is a Schunck class containing 3E but
that it does not necessarily coincide with eor0qX, the smallest Schunck class
containing X.
4. Let Gbethe unique non-abelian extension of Z3 by Z4. Show that G has a faithful
irreducible module, V say, of dimension 2 over (F5. Let В = [L]G, and let § be
the Schunck class pqs(B). Prove that = §0,<> J
5. (Schaller [5]) If g is a formation, set A4g = (G: G e E4g, Psi(G) £ g). Prove
that g arises as the largest formation in some Schunck class if and only if
g = A®g-
6. Justify Schaller’s characterization of §° described in the remark following the
statement of Proposition 2.22.
7. (Schaller [5]) Let § and Я be Schunck classes, and let g =
(i) Show that Statement (a) of Proposition 2.22 is equivalent to each of the
following conditions:
(с) §° = Я0;
(d) §0 = Яо.
(ii) If g = Я0"», then g = (§ n Я)о1<« = <§, Я)0'».
8. (Schaller [5]) Let § be a Schunck class, let g = and assume that § =
E«g ( = §o 'n the notation of Proposition 2.22). If c = s, or n0, show that § is
c-closed if and only if g is c-closed.
9. (a) Let § be a Schunck class whose boundary consists of single-headed groups.
Prove that § = Nof>.
(b) Let В denote a primitive group of degree 25 such that B/Soc(B) s Sym(3),
and let § = h(B). Prove that §s is not n0-closed.
10. (Kattwinkel [1]) If § is an ©-Schunck class, put
bs(f>)=(G 6 b(§): (s - 1)(G) £ f>) and 5* = h(bs(f>)).
Prove the following assertions.
(a) If § and Я are Schunck classes, then §s = if and only if bs(S) = МЯ)
if and only if = Я*.
(b) MS) = b(S*) = MS*)- л s
(c) S* is the largest Schunck class such that (S ) = &
(d) (&*)*=&* &
(e) In each soluble group an S* -projector contains an ^-projector.
11. (Kattwinkel [1]) If § is a Schunck class, set S„ = П {C: £ IS a Schunck class
and C* = £>*}. Prove the following assertions about Schunck classes S an
Я.
(a) (&,)s = Ss
(b) (&#)»=&#•
(c) = Я5 if and only if S # £ Я £ &
356
IV. The theory of formations
(d) S»# =r«f)s.
Deduce that the formation §s is saturated if and only if is s-closed.
12. Lei § be a Schunck class. Prove that an § ““"-projector of a group, if it exists,
is already an ^-projector.
13. (Kattwinkel [1]) Let G be a primitive group such that G/Soc(G) is cyclic of prime
power order. Prove that h(Gf is a saturated formation.
14. Show that a formation g of finite soluble groups has characteristic n if and only
if
u 9I(P) f= g f= (Q” )0R«.
pen
3. Local formations
In his foundation work on formations Gaschiitz [8] shows that in a soluble universe
every group has a unique conjugacy class of subgroups associated with each saturated
formation g, namely the g-projectors. At the same time he introduces the concept
of a local formation with which to construct a rich variety of examples of saturated
formations. Our study of local formations in this section is confined to an investiga-
tion of their properties as classes of groups; in particular, we examine some of the
different local functions f which define a given local formation g, and we study the
interplay between properties of f and g The relation of local formations to their
projectors will be the theme of Section 5, but until then we work as generally as
possible in the universe (5.
(3.1) Definitions, (a) A local function f: P -» {homomorphs} is called a formation
function if /(p) is a formation for all p 6 IP.
(b) A class g of finite groups is called a local formation if there exists a formation
function f such that g = LC(f) in the sense of Definition 5.2(a) of Chapter Ш. But
in this case we adopt the notation
g = LF(f),
which will henceforth always carry the implicit meaning that f is a formation function,
(c) Iff: IP -> {classes of groups}, a chief factor H/K ofa group G is called/-centra/if
AutG(H/K) e/(p) for all primes p dividing \H/K\.
Otherwise it is called f-eccentric.
Thus, in the above terminology, a group belongs to the local class LC(f) if and
only if its non-Frattini chief factors are /-central. The next result shows that for local
formations the restriction to non-Frattini chief factors is unnecessary.
(3.2) Theorem. Let f be a formation function, and let r. be the support of f. Any two
о/ the following statements are equivalent:
3. Local formations
357
(a) G 6 LF(f);
(b) G 6 (%, n Gp.Sp/(p) for all pen-,
(c) All chief factors of G are f-central.
Proof (a) => (b): Let G e LF(f), and let p e IP. If H/K is a non-Frattini chief factor of
G such that p\\H/K\, then G/Cc(tf/K) =; Autc(H/K) e/(p) by III, Equation 5.«
(on page 322). By A, 13.8(a) the subgroup Op-P(G) is the intersection of the nor-
mal subgroups Cc(H/K) as H/K runs through such chief factors, and therefore
G/OP-.P(G) e R0/(p) = f(p). Consequently G 6 Gp.e„/(p), and in view of III, 5.3(a) we
see that Statement (b) holds.
(b) => (c): Assume that Statement (b) holds, and let p be a prime dividing the order
of a chief factor H/K of G; then clearly pen. Since 0p. p(G) < CG(H/K) by A, 13.8(a),
it follows that Aut6(H/K) = G/Cc(H/K) 6 q(G/O„.,p(G)) £ Qftp) = /(p), and State-
ment (c) holds.
Since Statement (a) is weaker than (c), the circle of implications is complete. □
Remark. If H/K is non-abelian, the condition:
Autc(G/K)e/(p) for all p||H/K|
is equivalent to the condition:
AutG(H/K)6g
provided that /(p) £ g for all such p.
We can now deduce the following theorem, which was originally proved for <5 by
Gaschiitz [8].
(3.3) Theorem. LF(f) is a saturated formation.
Proof. By III, 5.1 the class LF(f) is an G-Schunck class and therefore certainly
saturated. Since Gp.<5p is obviously an s„-closed formation, we have GpGp/(p) =
Gp. £p о /(p), which is a formation by (1.8)(a). Therefore by the implication: (a) => (b)
of (3.2) the class LF(f), as the intersection of formations, is itself a formation. □
Remark. The basis of the G-Schunck class LF(f) clearly consists of all primitive
groups G such that if N < G and p| |N|, then G/Cc(N) e f(p) о LF(f).
(3.4) Examples, (a) Letrr £ IP, and let/be the formation function defined as follows.
(G if pen,
/(p) — if рфп.
Then by (3.2) we have LF(f) = A₽6,,Gp.<SpG n G„ = G„. By replacing G by S in the
definition of /, we obtain the local class
358
IV. The lheory of formations
(b) Let g be a formation, and set /(p) = 8 for all pe P. Then by (3.2) we have
LF(/) = n₽epC₽e,>S = 9lS.
(the last equality follows from II, 2.7(a)).
(c) If/6 N, let
(I copies),
the class of soluble groups of nilpotent length at most I. If I > 1, take /(p) = 9i<,_ 1 ’
for all p e P. Then by Example (b) above we have LF(f) = 911, which is therefore a
local formation. (The symbol 91° is to be interpreted as the class of groups of order
1, and this class itself is a local formation, defined by setting /(p) = 0 for all p e IP.)
(d) Let f be the formation function defined by
ifq = p,
ifq^p.
By (3.2) we have LF(f) = Gp. Sp, and if G is replaced by <5, we obtain the class Gp.6p
of soluble p-nilpotent groups.
(e) Let к e M, and define
f(G.£ )k~*(E •
/(<?) = V ₽ '
if <7 = p,
ifq * p.
Then by (3.2) we obtain LF(/) = Gp.<Sp(Gp.<Sp)l-1Gp. = (<Ep <5p)k(Ep-, the class of
p-soluble groups of p-length at most k.
(f) Let f be the formation function defined by setting
/(p) = 9I(p — 1) for all p 6 P.
(We recall that 9l(p — 1) = 9lr\G(p — 1), the class of abelian groups of exponent
dividing p — 1, which is clearly a formation.) The saturated formation LF(f) so
defined is called the class of supersoluble groups and is denoted by the symbol U in the
sequel. Since /(p) £ <5 for all p 6 IP, clearly U s <5.
If G e U and H/K is a chief factor of G, then G/Cc(H/K) e 9I(p — 1), and therefore
I H/K | = p by B, 9.8(d). Conversely, if H/K is a chief factor of a group G and
\H/K\ = p, then G/Cg(H/K) e 9l(p — 1) by A, 21.1(b) and (c). It follows that a super-
soluble group is characterized by the property of all chief factors having prime order
(or, equivalently, being cyclic). We will discuss further properties of the class U and
will go into generalizations of it in Section 2 of Chapter VII.
(g) Let -< be an arbitrary linear ordering on the set IP of all primes, and for each
finite subset Г of P, let
8r = e₽eP2...GPn,
3. Local formations
359
whereT {pj,...,p„} and p, -<p2<- - -<p„. It follows easily from (1.8) and (1 9)
Sr a saturated formation, and since 5ru gr. s Rr, jr, the set of all such Rr
is a directed set with respect to the partial order of inclusion. Therefore the class
= U !8r: Г cp, infinite}
is also a saturated formation. We call <X< the class of Sylow tower groups of type (or
complexion) -<.
We now describe a local definition for J If p e P, let n(p) = {<? e P: p < q}
and put /(p) = SBlp|. Then by III, 5.1 we have Lf(/)s£. Clearly
A₽6 ®n(p) = LF(f)- If the class Lf(/)\I< is non-empty, it contains a
group of minimal order, G say, which by III, 2.11 is a primitive soluble group
(since 14 is an ©-Schunck class by III, 4.1). Let N = Soc(G), and suppose that
N is a p-group. Since G/N e I < n ©,(p) c e„., it follows that N e Sylp(G). But
then G obviously has a Sylow tower of type < by definition of n(p). This con-
tradiction shows that T < — LF(f).
(3.5) Remarks. Let f, g, and f2 be formation functions for all 2 6 A.
(a) If f(p) S„s(p) for all p 6 P, then LF(f) Q LF(g);
(b) A {LF(Л): ze A) = LF(h), where h(p) = А !Л(р): 2 e A} for all p e IP;
(c) Let N < G and G/N e LF(f). If G/Cg(N) e f(p) for all p| |A| (in particular, if
N is an f-central minimal normal subgroup), then G e LF(f).
Proofs, (a) If H/K is a chief factor of a group G and if p is a prime divisor of |Я/К|,
then Op(Autc(H/K)) = 1 by A, 13.6(b). Therefore, if Aut6(///K) e Gpg(p), it follows
that Autc(W/K) 6 g(p). Let G 6 LF(f). Appealing twice to (3.2), we see that, as all chief
factors of G are /-central, they are therefore «/-central, and consequently G 6 LF(g).
(b) This follows at once from the definition of a local formation.
(c) The chief factors of G above N are /-central by (3.2). Let H/K be a chief factor
of G below N. If p| | H/K |, then G/Cc(H/K) 6 Q(G/Q( A)) 5= qftp) = /(p), and so H/K
is /-central. Therefore by the Jordan-Holder theorem all chief factors of G are
/-central, and so Ge LF(f) by (3.2). □
In general a local formation possesses many local definitions, as the following
example shows.
(3.6) Example. Define formation functions / and/ as follows:
/(p) = (l), and
f(p) = (G 6 <5: all central chief factors of G are p-groups)
for all p e P. It is easy to check that /(p) is a formation. Now let / be any formation
function such that
f(p) <= f(p) S /(P)
360
IV. The theory of formations
F(P) =
for all p e P. Then we assert that LF(f) = 91. Certainly 91 = LF(f) LF(f).
Suppose, by way of contradiction, that LF(f )\91 is non-empty and therefore con-
tains a group G of minimal order. Then G is primitive, and Soc(G) is a minimal normal
p-subgroup for some prime p. Then G/Soc(G) = Aut6(Soc(G)) e 9ln/(p)e9ln/(p) =
S . Therefore G is a p-group, and we have G e 91 contrary to supposition. Hence
LF(/) = 9».
By III, 5.4 (see III, 5.5 and the subsequent remarks) each local formation g = LF(f)
can be defined by a full and integrated local function g given by
(3.a) g(p)= 5^sr/(p)
for all p e P, and it is clear that because f is a formation function, then so is g. Our
next result shows that, in contrast to the situation for arbitrary local function (see
Chapter III, Section 5, Exercise 4), a formation function defining a local class is
uniquely determined by the requirements of being full and integrated.
(3.7) Theorem. Let g = LF(f), and define a function F: IP -»{group classes} as
follows:
C(G e g: Op.(G) = 1) forp e Char(g),
0 for p £ Char(g).
Then F is the unique full and integrated formation function such that g = LF(F).
Proof. Let g = LF(g) with g integrated and full; we know that at least one such
function exists, namely the g given by Equation 3.a. If p £ Char(g), clearly g(p) = 0.
Let p e Char(g). Then by definition and by (3.2) we have
g(p) = <5pg(p) cgc Gp.Spfl(p) = Gp.p(p).
Consequently g(p) = g(Op.) by (and in the notation of) Lemma 1.11(b), and it follows
from (1.10)(b) that g(p) = F(p). Therefore g = F. □
(3.8) Proposition. Let g = LF(F) with F integrated and full, and let f he a formation
(a) If g = LF(f), then F(p) = &„f(p) n g = <Sp(/(p) n g) for all p e P.
(b) If f(p) £ <3 and F(p) = <5p(f(p) n g) for all p e P, then g = LF(f).
Proof, (a) Because the formation function defined by Equation 3.a is integrated and
full, we deduce from the uniqueness of F in (3.7) that F(p) = Sr/(p)ng. It then
follows that F(p) s Gp(/(p) n g). But it is obvious that f(p) n g s F(p), and there-
fore Sp(/(p) n g) c SpF(p) = F(p). Consequently F(p) = Sp(f(p) n g).
(b) Let g* = LF(f), and note that g* c S by 111, 5.1 Since F(p) E Gp/(p), it
follows from (3.5)(a) that g s g*. If possible, choose a group of minimal order in
g* \g- From II, 2.5 we know that G is a primitive soluble group, and if N = Soc(G),
3. Local formations
361
then G/N e g. Let q be the prime divisor of |N|. Then G/N s AutJN) e f(q\ and
therefore G e S,(/(q) n g) = F(q) c g. This contradicts the choice of G, and so we
conclude that g = g*. (£
Remark. In Exercise 5 below we describe an example to show that the hypothesis of
solubility is necessary in (3.8)(b).
(3.9) Definitions, (a) The uniquely determined full and integrated formation func-
tion defining a local formation g is called the canonical local definition of g. It will
be identified by the use of an upper case Roman letter. Thus use of the notation LF(F)
will carry with it the tacit assumption that F is the canonical local definition of LF(F).
(b) By (3.5)(b) a local formation g always has a smallest local definition, namely
the formation function f defined thus:
/(p) = AW8 = LF(0)}
for all p e P. It will always be denoted by the use of a ‘lower bar’.
(c) Using (3.5)(b), one can easily show that if Ft is the canonical local definition
of g7 for all 2 e A, then Q д Ft is the canonical local definition for Qд gA.
(3.10) Proposition. If g is a local formation, and if pe Char(g), then
f(p) = O(G/Op..p(G):Geg).
Furthermore, for g £ S we also have
f(p) = qr0(G/Soc(G): Ge goф”).
Proof. Define a function h as follows:
Jq(G/O (G): G e g)
ад = 10
if p e Char(g),
ifp</Char(g).
From (1.10)(a) (setting T = Op .p) we deduce that ft is a formation function. Moreover,
if G e g, we have G/Op. p(G) e ft(p), and hence g S (£p Spft(r) for all p e Char(g).
Therefore g £ LF(ft) by (3.2). If p e Char(g) and G e ft(p). then G e OtH/().. (H))
for some H e g. By (3.2) we have H/OP ,P(H) 6 f (p), and hence G e af(p) - Ж Соп'
sequently ft(p) e ftp), and then (3.5)(a) implies that LF(ft) £ LF([) - g. Hence
g = LF(ft), and ft = f by definition off. f
Now assume that g E 6, and let ?,„(G) denote the set of p-chief factors of a group
G e g. If H/K 6 tf„(G), by the implication: (a) - (c) of (2.12) the sem.d.rect product
[H/K]AutG(H/K) belongs to gn*P”, and since Op.p(G) is the °
centralizers of the p-chief factors of G. we conclude from the above description f
that
362
IV. The theory of formations
f(p) = qr<,(G/Cc(H/K): H/K e %(G), G e 8)
= qr„(AutG(H/K): H/K e %(G), G e 8)
= qr„(G/Soc(G): G e 8 <3 ф”). □
We prove next that an inclusion between two local formations is equivalent to
a corresponding inclusion between either their canonical or their smallest local
definitions.
(3.11) Proposition. Let 8 = CF(F) = LF(f) and ® = LF(G) = LF(g). Then any two
of the following statements are equivalent.
(a) 8 S ®;
(b) f(p)£g(p)forallpeP:
(c) F(p) £G(p) for all pe P.
Proof. The implication: (a) => (b) follows directly from the characterization in (3.9) of
the smallest local definition. Since f(p) S 8 and g(p) £ ® for all p e P, the implica-
tion: (b) => (c) follows from (3.8) (a). Finally, the implication: (c) => (a) is clear from
(3.5)(a). □
Next we describe a local definition of the smallest local formation containing a
given formation.
(3.12) Proposition (D'Arcy [3]). Let 8 be a formation, let it = {p: 3G e 8 such that
p| |G|}, and define a class mapf thus:
(q(G: Ge Band0„.(G) = 1) ifpen,
/<P)=l0 if рфп.
Then f is a formation function, and LF(f) is the smallest local formation containing 8-
Proof. That f(p) is a formation follows at once from (1.10)(a) and (b) (with T = 0p-).
By definition of it we have 8 E and by definition of f(p) we have 8 £ Gp/(p) E
Gp Sp/fp). Therefore jt<;Lf(/) by (3.2). On the other hand, the description
of the canonical local definition given in (3.7) shows that if 3 is contained in
some local formation, LF(H) say, then f(p) H(p) for all pen, and hence that
LF(f)zLF(H) by (3.5)(a). □
Remark. This is not our last word on local definitions. Eventually we shall character-
ize all local definitions of a soluble local formation by showing the existence of a
unique largest one (in the obvious sense). But since this requires the concept of an
B-normalizer, it must wait until Chapter V, Theorem 3.18.
The next theorem states (i.a.) that the formation product 3 ° ® of two local
formations 3 and ® is again a local formation, and is proved by finding an explicit
local definition (in fact, the canonical one) for 8 ° ®-
3. Local formations
363
riS The°rem’ Lef S = LF(F’’ le‘ ® Ье ° П0П-етР,У fOm°liOn- Assu™
(i) ® = LF(G), or
(ii) Sr® = ® for all p ф Char(g).
Then ff о (5 = LF(H), where
F(p) о ® ifp e Char(g),
H(p) = - G(P) if P i Char(g) in Case (i),
® ifp Ф Char(g)inCase(ii).
Proof. By (1.8)(a) the class g о (5 is a formation and H is a formation function. We
set § = LF(H), and first prove
Step 1: We will obtain a contradiction by supposing that §\go ®
contains a group X of minimal order. Such an X has a unique minimal normal
subgroup N by II, 2.5, and X/N e 5 ° ®. Let R = X® < G. If N R, then R = 1,
and X e ® £ g о ®, contrary to supposition. Therefore N < R. Let p be a prime
divisor of |N I, and suppose first that p ф Char(g). Then X/Cx(N} e H(p) £ ® (since
the function G is integrated in Case (i)), and consequently N < Z(R); in particular, N
is a p-group. If N < ®(R), we have R e Eog = g by (3.3); but then X e g ° ®, contrary
to supposition. Therefore N is supplemented in R by some maximal subgroup, U say,
of R. Since N is central, U is normal in R, and R/U is a p-group. Thus R3" is a normal
subgroup of G, which is contained in U and which therefore does not contain N.
Because N is the unique minimal normal subgroup of G, it follows that R3p = 1, in
other words, that R is a p-group. Since R/N e g and p ф Char(g), we conclude that
R = N. Then in Case (ii) we have X e Sp® = ® s g ° ®, which contradicts the
choice of X. For Case (i) we observe that, as X e the p-chief factor N/l of X is
H-central and therefore G-central by definition of H. But then by (3.5)(c) we obtain
the same contradiction that X e ®. Thus we can suppose that each prime dividing
| N| belongs to Char(g). Hence, if p| |N|, we have X/Cx(N)e H(p)=F(p)°®. There-
fore R/Cr(N)S RCx(N)/Cx(N) = (X/Cx(N))® e F(p). Since R/N e g, it follows from
(3.5) (c) that Reg and hence that X e g о ®. This final contradiction proves that
S g ° ®. Next we prove
Step 2: g ° © £ £>. Suppose that this is not the case, and choose a group Y of
minimal order in g о 6\f>. Then Y has a unique minimal normal subgroup N, and
G/N e f>. Let R = Y ®, and first suppose that N f R. Then R = 1 and Y e ®. In Case
(i) it is clear from the definition of H that G(p) £ H(p) for all p e IP since by hypothesis
G is integrated, and so by (3.5) (a) we have ® £ f>. In Case (ii) w'e even have 6 £ H(p)
for all peP, and then Tef> by (3.5)(c). In either case we conclude that Te$>,
contrary to supposition. Hence N < R. and since R e g, it follows that all prime
divisors of | /V| are in Char(g). Because R < G, the subgroup N is a direct product
of minimal normal subgroups of R by A, 4.13(c), and therefore, if p| I N|, it follows t at
(T/CJN))® = RCr(N)/Cr(N) R/Cr(N) e n,F(p} = F(p).
364
IV. The theory of formations
Consequently Y/Ct(N) e F(p) ° ft, and so N is H-central in Y. Since Y/N e f>, we
conclude from (3.5)(c) that Y e This contradiction completes Step 2 and hence
shows that 5 = g ° ®. It remains to show
Step 3: The formation function H is integrated and full. If p e Char(g), we have
SpH(p) = Qp(F(p) о (5) = (S,F(p)) о (5 = F(p) о (5 = H(p)
because F is full, and we also have H(p) = F(p) °®ego® = §.
If p ф Char(g) and H(p) / 0, in Case (i) we have SpH(p) = SpG(p) = G(p) be-
cause G is full, and in Case (ii) by hypothesis <apH(p) = Sp® = ® = H(p). Hence H
is full. Finally, in either case we have H(p) £ ffi £ §, and therefore H is integrated.
□
If c is a closure operation, one can ask two obvious questions about its effect on
a local formation g = LF(f).
(1) If/(p) = c/(p) for all pep, does it follow that g = eg?
(2) If g = eg, can one infer that /(p) = c/(p) for all p e P?
We do not attempt to answer these questions in general, but instead consider some
special cases relevant to later needs.
(3.14) Proposition. Let g = LF(f), and let c be one of the closure operations s, s„, or
N„. If ftp} = cflp) for all p e P, then g = eg.
Proof. First let c = s (respectively s„), let G e g, and let U be a subgroup (normal
subgroup) of G. Let p e Char(g). Then U/(U rs 0p. r(G)) s UOp-p(G)/Op.p(G) and
U0p. p(G)/0p p(G) is a subgroup (normal subgroup) of G/Op. p(G)ef(p). Since
G n 0p. p(G) < 0p. p(U), it follows that V/0p- P(U) e Qf(p) = f(p). Therefore Leg,
and g = eg.
Now suppose that c = n0, and let G = NtN2 with JV( e g for i = 1, 2. Let R =
0p. p(G), and observe that for 1=1,2 we have RrtNt = 0р -р(^) by A, 13.4(e). Let
I e {1,2}. Then NtR/R s e f(p), and consequently
G/R = (N, R/R)(N2R/R) e к<Др} = /(p).
Hence G e g, and from 11, 2.11 we therefore conclude that g is n0-closed. □
(3-15) Examples, (a) If f is the formation function defined in Example 3.6, we have
LF(f) = 91 = s9l, but obviously f(p) / sf(p) for all p e P.
(b) It is straightforward to verify that the class 9191 is a local formation whose
canonical local definition is given by F(p) = Sr?I for all p e P. Then 9191 = e„9WI,
but F(p) E®F(p) for all p e P.
Thus the answer to the second question about closure operations may be negative,
even when the canonical local definition is used. However, in this case the answer
may also be positive as the next result shows.
3. Local formations
365
(3.16) Proposition. Let c be one of the closure operations s, s„, or N„ let g = LFIF)
and assume that g = eg. Then F(p) = cF(p) for all p e P.
Proof. First we handle the case when c = s (respectively s„). Let p e Char(g), let
G e F(p), and let U be a subgroup (normal subgroup) of G. Let W = Z 4>„ G, and
let В be the base group of W. Then W e &„F(p) = F(p) c 5, and UB e cg’= g.
Since CH.(B) = B, we have Op.(UB) = 1, and therefore UB e SpF(p) = F(p) by (3.2).
Consequently U = UB/Be QF(p) = F(p), and hence F(p) = cF(p).
Now suppose that c = No. By П, 2.11 it will suffice to show that if G = NtN2 with
Nt < G and Nt e F(p) for i = 1, 2, then G e F(p). Let W = Zj\s,^ G with В as the
base group, and observe that W is the product of normal subgroups А, В and N2B,
which both belong to <=„F(p) = F(p) e g. Therefore W e Nog = g. As before we
have Op.(W) = 1, therefore IV e &pF(p) = F(p}, and hence G s W/B e QF(p) = F(p).
□
The following lemma will be useful, both for our immediate purposes and for later
applications.
(3.17) Lemma. Let I be a positive integer, and assume that g = LF(f) £ 91'. Then
g = LF(g), where g is the formation function defined by setting g(p) = /(p)n 9l'-1
for all p e P.
Proof. Let f be the smallest local definition of g. If G e 91', then G/0p. P(G) e
q(G/F(G)) s 9l1-1, and therefore from the description of f given in (3.10) we deduce
that /(p) £ g(p) for all p e P. Hence by (3.5) (a) we have
g = LF(f) c LF(0) c LF(f) = g. □
The fact that a formation of nilpotent groups is subgroup-closed carries over to
the local definition of a metanilpotent local formation and yields the following
result.
(3.18) Theorem. If g = LF(/) £ 9l2. then g = sg.
Proof. By (3.17) we have g = LF(g), where g(p) = f(p) n 91 for all p e P. By (1.16)
the formation g(p} is s-closed for all p e P, and hence g = sg by (3.14). □
Exercises
1. Let g = LF(F) in the universe SB = <s, Q, E., e>SB. Prove that any two of the
following statements are equivalent:
(a) g =
(b) F(p) = F(q) for all p, q e P;
(c) Ftp) = g for all p e P. x . ,.
2. In each of the following cases find the smallest local f°™atl°" c°" ®‘"mg
group G by describing its local definition: G = Z2, Sym(3), Sym( ),
366
IV. The theory of formations
3. (a) Let 5 be a local formation, and let G = AB, where A, В < G. Assume that
A n В e 91 and A e g. Prove that G e g if and only if В e g.
(b) Show that the corresponding statement for Schunck classes is false.
4. Let g = LF( f). (a) Show that the formation function g defined by setting g(p) =
if P e Char(g) and 0 otherwise is a full local definition of g, and is
canonical if / is integrated, (b) Show that the formation function h defined by
setting h(p) = f(p) n g for all p e P is an integrated local definition of g, and is
canonical if f is full.
5. Let G be a non-abelian simple group, and for all pe P let f(p) = qr<>(G) and
F(p) = Sp. Prove that
(a) F(p) = Sr(/(/>) n 91) for all p e P, and
(b) LF(F) = 91 / 91(qr0(G)) = LF(f).
(Thus the solubility hypothesis of (3.8) (b) cannot be dispensed with.)
4. The theorem of Lubeseder and the theorem of Baer
This important and celebrated theorem states that saturated formations of finite
groups are local formations. It was proved originally for a soluble universe by
Lubeseder in 1963 in her Kiel dissertation, which was written under the supervision
of Gaschiitz. Her proof uses some elementary ideas from the theory of modular
representations, which were later dispensed with when the first widely-available
published account of the theorem appeared in Huppert’s book [5]. In 1978 Schmid
[3] showed that the restriction to soluble groups is unnecessary, although his proof
reinstates the facts about blocks used by Lubeseder and also makes essential use of
a theorem of Gaschiitz about the existence of non-splitting extensions. Most of this
section is devoted to a proof of this theorem in its full generality, but including within
the development a treatment for soluble groups that avoids the more sophisticated
machinery. The universe throughout is G.
We begin with a sequence of preparatory lemmas.
(4.1) Lemma. Let X be a finite group, and let M be a faithful X-module over Fp. If N
is an irreducible FpX-module, then
[N]X 6 QRoEoR0([lW]X).
In particular, [W]X belongs to every saturated formation that contains [M]X.
Proof By the theorem of Steinberg (B, 10.13) there exists an refJ such that N is
isomorphic with a quotient of the module М|г| = M ® • • • ® M (r copies). Let . // =
{M,,..., Mr} be a set of r copies of the Fp X-module M, and let II denote the Hartley
group H(.//) defined in B, 12.11. There we showed that II admits X as a group of
operators in such a way that
4. The theorem of Lubeseder and the theorem of Baer 367
(1) Н/Ф(Н) s Mi ® • • ® M„ and
(2) Z(H) contains an X-invariant elementary abelian p-subgroup T which is iso-
morphic with when viewed as an FpX-module.
With respect to this action of X on H form the semidirect product G = [HIX
and observe that from (1) we have G/«D(H) = [M, ® • •• ® MJX e R„([M]X) Since
Ф(Н) < Ф(С), it follows that G e eor0([M]X). Then by (1.5) we have [MW]X a
[T](G/H) e qr0(G), the formation generated by G. Finally, as N is a factor module
of M'r\ we conclude that [H]X e o([M|r,]X) <= qr0(G) <= qroe.ro([M]X). □
(4.2) Lemma. Let g be a saturated formation, let G e g, and let pbea prime divisor
of |G|. Then g contains a cyclic group of order p.
Proof. By (4.1) it will suffice to find a group X which has a faithful module M over
Fp such that [M]X e g. For then, with N as the trivial ffpX-module, we obtain
N x X = [AQX e g, and hence N e Qg = g. First we give a proof for
(a) The soluble case: Assume that G is p-soluble. Let R = 0p- p(G), and let S denote
the residual of R for the formation of elementary abelian p-groups. Since p||G|, we
have R/S / 1, and by A, 9.6(a) and A, 10.6(c)(ii) we can regard R/S as a faithful
G/R-module over Fp. Since [R/S](G/R) e g by (1.5), we can therefore take X = G/R
and M = R/S.
We now give another proof which handles
(b) The general case: Here we have to appeal to Gaschiitz’s theorem on the
existence of Frattini extensions, but this is the only point at which it is used in the
proof of the Lubeseder theorem. Since p| | G|, it follows from В, 11.8 that there exists
a group H and a minimal normal subgroup M of H such that (i) G = H/M, and (ii)
M < Ф(Н). Therefore H e E*g = g, and setting X = H/C„(M), we then conclude
from (1.5) that [M]X eg. □
(4.3) Corollary. Let g be a saturated formation of characteristic n. Then
n = {p e P: 3G e g such that p||G|}.
Thus a saturated formation g has characteristic n if and only if 9i„ S g S
The next lemma involves elementary properties of blocks and is not required for
the proof of the soluble case of the Lubeseder theorem.
(4.4) Lemma (Forster [2]). Let g be a saturated formation, and let G e g. Let p e P,
and let В be a block in FpG. Finally, let T be an irreducible module in B, and let N be
an <FpG-module all of whose composition factors belong to B. If the semidirect product
[T]G belongs to g, then [H]G belongs to g.
Proof. By B, 4.6 there exists a directly indecomposable FpG-module P such that
P/Rad(P) S T, and by B, 3.14 we have Rad(P) < Ф([Р]С). Therefore [P]Ge Еф'’“
g. Let IV be a composition factor of P. and let P* be a directly indecomposable
368
IV. The theory of formations
projective Fp G-module which has a composition factor isomorphic with W. First note
that by (1.5) we have [W]G e g. Among the submodules of P* which have W as an
epimorphic image, choose a minimal one. К say. Then К has a maximal submodule
L such that K/L s W, and obviously L = Rad(K) by the minimality of K. Con-
sequently, [K]Ge еф([IVJG) S g. By B. 4.10 the module P* contains a unique
minimal submodule, S say, which is therefore contained in A, and so by (1.5)
again we have [S]Ge g. But by B, 4.10 we have S = P*/Rad(P*), and therefore
[P*]G e Eog = g. From B, 4.17 we can then conclude that [Q]G e g for all in-
decomposable projective modules Q in the block B. But by B, 4.19 the module
N is an epimorphic image of a direct sum of such modules Q, and it follows that
[/V]GeQRog = g. □
(4.5) Lemma. Let g be a saturated formation, let Ge g, and let p be a prime divisor
of \G\. If X = G/Opp(G), and if N is an irreducible FpX-module, then [A]X e g.
Proof, (a) The soluble case: If G is p-soluble, we repeat the argument of Case (a) in
the proof of (4.2) to find a faithful X-module M over Fp such that [M]X e g, and
then apply (4.1).
(b) The general case: Let M be the direct sum of a complete set of representatives
of the isomorphism classes of irreducible FpG-modules in the first block. By B, 4.23(b)
we have Ker(G on M) = Op-p(G), and we can therefore view M as a faithful X-module.
Let T be the trivial irreducible FpX-module. Since g contains Zp by (4.2), it follows
that [T]G = T x G e Dog = g, and an application of (4.4) to the first block of FpG
then yields [M]G e g. Consequently [M] X = [M]G/Ker(G on M) e Qg = g, and
from (4.1) we conclude that e g. □
We are now ready to prove the main theorem.
(4.6) Theorem (Gaschiitz [8], Lubeseder [1], Schmid [3]). A formation of finite
groups is saturated if and only if it is local.
Proof. The sufficiency is proved in (3.3). To prove the necessity, assume that g is a
saturated formation of characteristic n and set
. (Q(Geg:O„.(G)= 1) if pen,
J(P)=)^ .r ,
10 if p £ n.
By (1.10) this f is a formation function, and obviously J(p) E g for all p e P. Set
§ = LF(f). Let G e g. If p e P\n, by (4.2) we have G e Gp.; therefore G e G„. For
pen, by definition of f we have G/OP(G) e /(p), and so G e GpSp/(p). Hence Ge§
by (3.2), and consequently g £
In order to prove the opposite inclusion we first need to show that if p e n, then
14 «) f(p) = Gp/(p).
Suppose, by way of contradiction, that (4.a) is false, and let G be a group of minimal
4. The theorem of Lubeseder and the theorem of Baer
369
order m £рДр)\Др). Since SJfp) is Q-closed and f(p) is a formation, the group G
has a unique minimal normal subgroup, N say, and G/N e f(p) Furthermore N
is obviously a p-group, and therefore 0p.(G) = 1. If N < <b(G), then G e Ефftp) c
e®<5 - Я, and then evidently G e Др), contrary to supposition. Therefore by A
15 8(a) the group G is primitive of type 1; in particular, N has a complement in G
call it H, and H is faithfully represented on the ^-module N. Since H s G/N ef(p),
by definition of f there exists a group L in g and a normal subgroup К of L such
that 0p.(L) = 1 and L/K = H. If L is a p'-group, then L = 1, and G = N s Z But
Z„ belongs to g by (4.2) and hence to Др), and then we have the contradiction that
G 6 Др). Consequently we can suppose that p||L|. Since N is a faithful H-module,
we can consider N as an FpL-module with Ker(L on N) — K. Let X = L/0 (L)
Since 0p(H) < Hr, 0p(G) < H n N = 1, we have 0p. r(L) = 0p(L) < K, and ^so we
may regard N also as an F^X-module with Ker(X on N) = K/0 . (L). By (4.5) we
have [N]X e g, and it follows that G = NH [N](L/K) e q([/V]X) e og = g. But
then, since 0„.(G) = 1, we conclude that G e Др) by definition of f. This final contra-
diction proves Equation 4.a.
We can now summarily complete the proof. By (3.2) we have
5 = П (GP-eP/(p)nGn)
pen
= n (G„./(p)nG„)
pen
S П (G„.gnG,)
pen
eS,
where II, 2.7(b) is applied for the final inclusion. □
(4.7) Remarks, (a) For the proof of this theorem in a soluble universe we need only
Lemma 4.1, together with Lemmas 4.2 and 4.5 with their special proofs for the soluble
case, and none of these involves concepts from block theory or requires the cited
theorem of Gaschiitz.
(b) We shall henceforth make free and frequent use of the fact that the concepts
of‘saturated formation’ and ‘local formation’ are equivalent without explicitly citing
Theorem 4.6.
Return for the moment to Theorem 1.9, which was proved by elementary methods.
It is a consequence of that theorem that in a soluble universe the formation product
of two saturated formations is again a saturated formation. We can now show that
the restriction to S is again unnecessary. The following theorem is a generalization
of (1.9) to the universe G in the case where the Schunck class § is a saturated
formation. It can be deduced at once from (3.13) and (4.6), and may also be prove
directly without appeal to the Lubeseder theorem; however, we know of no proof
that avoids the representation-theoretic machinery of this section.
370
IV. The theory of formations
(4.8) Theorem. Let g and © be formations of finite groups. Then the formation
product 8 о 6 is a saturated formation if either of the following two conditions is
fulfilled.
(a) g and ('> are both saturated;
(b) g is saturated and 0> = Sp(fi for all p ф Char(g).
In an unpublished work Baer gives another generalization of the Lubeseder
theorem to the universe Й. His approach uses a different concept of a local formation,
and it leads to a family of formations containing all saturated formations of the
universe <f, and coinciding with the saturated formations when the universe is
restricted to S. The following presentation of Baer’s theorem includes an interesting
variation of the proof of Lubeseder’s theorem for the universe S.
We recall that 3 denotes the class of finite simple groups.
(4.9) Definitions, (a) A formation g of finite groups is said to be solubly saturated if
the condition:
С/Ф(А) 6 g for a soluble normal subgroup TV of G
always implies that G 6 g.
(b) A map f; 3 -> {classes of groups} is called a Baer function provided that f(J)
is either a formation or the empty class whenever the simple group J is cyclic. If
J = Zp, we may write ftp) instead of f(J).
(c) Let f be a Baer function, and let H/K be a chief factor of a group G. Then H/K
is a direct power of some J e 3. and we say that H/K then has composition type J;
furthermore, we say that H/K is f-central in G if Autc(H/K) 6 f(J).
(d) Let /be a Baer function. It follows from (1.3) that the class of all finite groups
whose chief factors are all /-central is a formation; we call this formation the
Baer-local formation defined by f and we denote it by BLF(f). A class ® of finite
groups is called a Baer-local formation if S3 = BLF(f) for some Baer function /.
Remarks, (a) If / is a formation function in the sense of (3.1)(a), and if we set
gtJ) = -
ftp) when J Zp, and
A f(P) when J e 3\SI,
.plui
then it is clear from (3.2), (a)«=-(c), that LF(f) = BLF(g). Thus local formations are
a special case of Baer-local formations.
(b) In the universe of finite soluble groups the concepts of local formation and
Baer-local formation evidently coincide.
(c) The concept of a Baer-local formation will be used for the construction of
several types of Fitting class in Section 2 of Chapter IX.
The rest of this section is devoted to proving that the solubly saturated formations
of finite groups are precisely the Baer-local formations.
4. The theorem of Lubeseder and the theorem of Baer 371
(4.10) Definition. The subgroup C-(G). Let p e P, and let G e G. The subgroup C”(G)
ofc6 wBh be‘he ‘"1ег5ес“оп of the centralizers of all the abelian p-chief factors
ot G, with C (G) = G if G has no abelian p-chief factors. Clearly 0(C”iG)} < Cp(0Gi
whenever 0 is an epimorphism of G, and therefore, if g is a formation, it follows from
(LIU) that the class
Q(G/C₽(G): G 6 g)
is also a formation.
In proving the next lemma we appeal to (6.7), a theorem in the final section of this
chapter. We would like to reassure the reader that the proof of that theorem is
completely independent of the result which we are about to prove.
(4.11) Lemma. Let N be a soluble normal subgroup of a finite group G Then
C₽(G/®(N)) = C₽(G)/®(TV).
Proof. Let 4/®(TV) = C₽(G/®(TV)). Clearly ®(TV) < C₽(G) < A. Let/be the formation
function defined as follows:
/w =
(1) for q = p,
£ for q ± p-
Since [Л, TV] < A n N, it is certainly true that A acts /-hypercentrally on N/(A n TV)
in the sense of Definition 6.2(b) below (which requires that A induces an /(g)-group
of automorphisms on every Л-composition factor whose order is divisible by q}. Since
TV e S, the G-chief factors between Ф(ЛГ) and A n TV whose orders are divisible by a
prime q are in fact abelian ц-chief factors, and therefore by definition A acts /-
hypercentrally on (Л n TV)/®(TV). Thus the group A acts /-hypercentrally on TV/W(TV),
and hence by (6.7) it acts /-hypercentrally on TV. Since A < G, and since every abelian
p-chief factor of G is therefore completely reducible as an 4-module, it follows that
A centralizes all p-chief factors of G below ®(TV), and consequently that A = C₽(G).
□
(4.12) Theorem. A Baer-local formation is solubly saturated.
Proof. Let /be a Baer function, and let g = BLF(/). Furthermore, let N be a soluble
normal subgroup of G such that G/O(TV) 6 g, and note that by definition all chief
factors of G above G/W(TV) are /-central. Let p be a prime divisor of |<D(TV)|. By
A, 11.8(a) the prime p divides /V/Ф (TV), and because N is soluble, G has an abelian
p-chief factor above <D(TV). Therefore by definition of g the class /(p) is a non-empty
formation, and consequently, because G/W(TV) e g, we have (G/®(TV))/C₽(G/W(TV)) e
R„/(p) = /(p). From (4.11) we then deduce that G/C₽(G) e f(p) and conclude that all
p-chief factors of G below ®(TV) are /-central. Since all chief factors of G below Ф(А)
are p-chief factors for some prime divisor p of |®(TV)|, it follows that G e g and
therefore that g is solubly saturated.
372
IV. The theory of formations
(4.13) Lemma. Let % be a solubly saturated formation, let X be a finite group, and let
M be a faithful X-module over Fp such that [M]X eg. If N is an irreducible FpX-
module, then [A/JX e g.
Proof. This follows at once from the proof of (4.1), granted the elementary observa-
tion that the Hartley group used there plays the role of a soluble normal subgroup.
□
(4.14) Lemma. Let $ be a solubly saturated formation, let p be a prime, and let G be
an ft-group which possesses an abelian p-chief factor. Then Zp e g.
Proof. Let H/K be an abelian p-chief factor of G, and put C = Cc(H/K). Then H/K
may be regarded as a faithful G/C-moduleover Fp, and by (1.5) we have [H/K] (G/C) e
g. If N denotes the trivial Fp(G/C)-module. by (4.13) we have [N](G/C)e g, and
therefore Zp s [IV] (G/C)/(G/C) e Qg = g. □
(4.15) Lemma. Let g be a solubly saturated formation, and let p be a prime. Let N be
an elementary abelian normal p-subgroup of a group G. Assume that [/V] (G/N) ё g
and that Zp e g. Then G e g.
Proof Let T be a trivial irreducible Fp(G/N)-module. Our assumptions imply that
G/N e Qg = g, and therefore that [T](G//V) s T x (G/N) e D„g = g. Regard N as
an Fp(G/N)-module, and form the Hartley group H = H(N, T) described in B, 12.11.
It follows from B, 12.17 that H admits G/N as a group of operators in such a manner
that
Н/Ф(Н) s N © T~
Ф(Н) s N
- as Fp(G/N)-modules.
Let X = [f/](G/N). Since [/V](G//V) and [T](G/N) both belong to g, it follows that
Х/Ф(Н) e Rog = g, and hence that X ё g because g is solubly saturated. Let Y =
G x X, and let D/(N x H) denote the diagonal subgroup of Y/(N x H) =
(G/N) x (G/N)(seetheHassediagramopposite.).ClearlyD/(N x Ф(Н)) s Х/Ф(Н)ея
Since Ф(Н) is G/Л/-isomorphic with N. the subgroups N x 1 and 1 x Ф(Н) of D are
isomorphic as D-modules, and we can form the diagonal subgroup, call it N*, of
N x Ф(Н). Then N* < D, and we have (N x H)/N* = N*H/N* £ H/(N* r,H)^H;
thus (N x Ф(Н)}/И* = Ф((Л/ x H)/N*), and as g is solubly saturated, we conclude
that D/N* 6 g. Furthermore, we have D/(N x 1) s D(G x 1 )/(G x 1) = Y/(G x 1) s
leg, and consequently D e R„g = g because N* n (N x 1) = 1. Finally, we deduce
that G s F/(l x X) = D(1 x X)/(l x X) s D/((l x X)nD)eQg = g. □
4. The theorem of Lubeseder and the theorem of Baer
373
(4.16) Lemma. Let % be a solubly saturated formation, and let pbea prime. Let H eft,
and let CP(H) < L <! H. If N is an irreducible Fp(H/L)-module, then [N](H/L) e g.
Proof. Let Fj, Vr be the p-chief factors of H below CP(H). Then by (1.5) we
have [V,] (H/CP(H)) e g for i = 1,..., r, and consequently, if M = (Jj-, f<, we have
[M] (H/CP(H)) e Rog = g. Since CP(H) centralizes all chief factors of G above CP(H),
it is clear that M is faithful as an (///C₽(H))-module. Hence, viewing N as an
(H/CP(H)(-module by inflation, it follows from (4.13) that [N](H/CP(H)) e g, and
therefore that [N] (f//L) e Qg = g. □
Our preparations are complete, and we can now prove Baer’s theorem.
(4.17) Theorem (Baer). The solubly saturated formations of finite groups are precisely
the Baer-local formations.
Proof. By (4.12) the Baer-local formations are solubly saturated. Now let g be a
solubly saturated formation. The candidate f for its Baer-local definition is defined
as follows:
(a) If J ё 3\9I, then put
f(J) = (Autc(H/K): G e g, H/K is a chief factor of G of composition type J);
(b) If p e P, and g contains a group with an abelian p-chief factor, then put
ftp) = Q(G/Cp(G): G e g);
(c) If no group in g has an abelian p-chief factor, put f(p) = 0.
374
IV. The theory of formations
Observe that/(J) E 8 for each J e 3. and that, as mentioned in (4.10), the class f(p)
in Case (b) is a formation; thus, in particular, f is a Baer function. Let ® = BLF(f).
It is clear from the above definition of f that g s ®, and so the burden of the proof
is to show that 8 s g. Suppose that this is not true, and let G be a group of minimal
order in ®\g. We shall show that this supposition leads to a contradiction. Since g
is a formation, it follows easily that G has a unique minimal normal subgroup, N say,
and that G/N e g. Suppose that N has composition type J for J e g\9I. Then
Cc(N)r> N = 1, and therefore CG(N) = 1. But G e ®, and we conclude that G s
Autc(N) e f(J) £ g, which is a contradiction.
Therefore N is a p-group for some p e P. Since G e ®, it follows that /(p) # 0;
therefore the class g contains a group possessing an abelian p-chief factor and hence
contains Zp by (4.14). By (1.5) the group В = [N](G/N) belongs to the formation ®.
First suppose that Cc(N) > N. Then the intersection M of CB(N) with its complement
G/N in В is a non-trivial normal subgroup of B, and therefore by definition of G we
have B/M e g. Hence В = B/(M n N)e Rog = g. Since g contains Zp, we can now
apply (4.15) and deduce that Ge g. But G ф g by supposition, and so we must have
Cg(N) = N.
Thus N is a faithful G/N-module, and consequently G/N e f(p) since Ge®. But
then by definition of f(p) there exists a group H in g and a normal subgroup L of H
containing CP(H) such that H/L s G/N. Since N is an irreducible G/N-module
(equivalently ///L-module), it follows from (4.16) that В = [(V](G/N) e g. Then once
more from (4.15) we conclude that G e g, and we have reached a final contradiction.
Therefore Beg and equality holds. □
In 1985 Forster [13] published a theorem which embraces both the Gaschiitz-
Lubeseder theorem and the above theorem of Baer’s as special cases. We will not
prove Forster’s theorem, but will end this section with a short description of its
content.
Forster’s point of departure is a class 3E of finite simple groups satisfying <r(.T) =
Char(X), = / say. Set 9) = еЭЕ, the class of groups whose composition factors belong
to X; evidently 9) is a Fitting class and so each group G has a largest normal 9)
subgroup, the 9)-radical Gs. A chief factor which belongs to 9) is called an -chief
factor. In order to formulate the “^-saturation” of a formation, Forster defines a
subgroup ФЖ(С) of a finite group G as follows;
(1) If p is a prime and Op.(G) = 1, set
фр ) = ГФ(О if Soc(G/O(G)) and <D(G) belong to ?), and
ж (Ф(С?1) otherwise.
(2) For a general group G, set
ФЖ(С)/О„.(С) = ФУС/О„.(С)).
(3) Then put ®x(G) = Gsn(Qp6Z®₽(G)).
A formation g is called X-saturated if whenever G/N e g for some normal
subgroup N <ФЖ(С), it follows that Ge g. An X-formation function is a map
5. Projectors and local formations
375
Z -* {formatlons}- If/is such an ^-formation function, the X-local forma-
tion defined by / is the class of all finite groups G which satisfy the following two
properties: Б
(a) If H/K is an X-chief factor of G, then G/Cg(H/K) e fip) for all p e a(H/Ky
(b) If L is a normal subgroup of G such that Soc(G/L) is (i) a minimal normal
subgroup of G/L and (ii) a direct power of some group J e 3\3E, then G/L e f(J). (If
f(J} = 0, no such L must exist.)
Finally, a formation $ is said to be ^.-locally definable if there exists an ^-formation
function f which -locally defines g. With this terminology we can now state the
promised result.
Theorem (Forster [13]). A formation g of finite groups is ^-saturated if and only if
it is X-locally definable.
If £ = g, the class of all finite simple groups, then we obtain as a special case
the Gaschiitz-Lubeseder theorem. At the other extreme, with X = (Zp: p e P) this
theorem yields the above theorem of Baer.
Exercises
1. Let G be a non-abelian simple group. Then qr0(G) is a solubly saturated but not
a saturated formation.
2. (Herzfeld [2]) If g is a formation, define
Ool(„(g) = (G e g: If § S g such that g = QR„(G, §), then g = QR,,(§)).
(a) If there exist maximal subformations in g, then <I>QRo(g) is the intersection of
all such maximal subformations. Otherwise <I>ORo(g) = g
(b) If G e g is monolithic, then (Q — 1)(G) s OQR.(g).
(c) IfX is a saturated subformation of g, then.t OORii(g). In particular, Ф,)Во(3) =
g if g is saturated.
5. Projectors and local formations
As a point of departure, consider the following observations: If g is a saturated
formation, then so is g n S. Therefore, since every section of a soluble group G is
soluble, it follows that an (g n 3)-projector of G is an g-projector, and vice-versa.
Then, because local formations are saturated and are therefore Schunck classes by
III, 4.1, from III, 3.21 we can deduce the following theorem.
(5.1) Theorem. Let g be a local formation of finite groups, and let Gbea soluble group.
Then G has a unique conjugacy class of ^-projectors, and these are at the same t,me
^-covering subgroups.
376
IV. The theory of formations
In this section we study projectors in the context of saturated or local formation,
concentrating on results which depend specifically on some aspect of the formation
property (e.g. the existence of a local definition or a residual), and which either cannot
be formulated or are not true for general Schunck classes. It is not a comprehensive
treatment of this theme, however, and further results of this nature will be presented
at the appropriate places (e.g. in Section 3 of Chapter V). Because we need the
universal existence of projectors, we make the blanket hypothesis that henceforth all
groups considered in this section are soluble.
Our first goal is a theorem about a property of projectors which applies exclusively
to saturated formations and therefore characterizes them among Schunck classes. In
order to formulate it, we first need some definitions.
(5.2) Definitions, (a) Let Л (G) denote the lattice of normal subgroups of a group G.
Thus the join of two normal subgroups N, and N2 is their product Nj N2, and their
meet is the set-theoretic intersection Nj n N2.
(b) If U < G, a map p:. 1(G) -<, i (If) is a lattice homomorphism if
p(Nj r, N2) = pNjr, pN2, and p(NtN2) = (pN2)(pN2)
for all normal subgroups Nj, N2 of G.
(c) If is a Schunck class and H an ^-projector of G, then we define a map
p(G, H): .4'(G) -> ЛУН) as follows:
p(G, H)N = Nr.H.
(5.3) Theorem (Necessity. Ti Yen—unpublished; Sufficiency. Huppert [7]). Let be
a Schunck class. A necessary and sufficient condition for the map p(G, H) defined in
(5.2) (c) to be a lattice homomorphism for each GeG and for each ^-projector H of G
is that is a saturated formation.
Proof, (a) Necessity. Let Nt, N2 be minimal normal subgroups of a group G such
that Л/, n N2 = 1 and G/Nt e Sj for i = 1, 2. Further, let H e Proj6(G), and assume
that the map p — p(G, II) is a lattice homomorphism. If ie{l, 2}, then clearly
HNt = G. Suppose that G ф then N:r,H = 1, and therefore
N, N2 n H = p(N, N2) = p(JV, )p(N2) = (IV, c, 1/)(N2 n //) = I.
It follows that |Л\||Л12||Н| = |N,N2HH| = |G| = which implies that N2 = 1.
This contradiction shows that G e But then by II, 2.6 the Schunck class § is
R„-closed and is therefore a saturated formation
(b) Sufficiency. Let § be a saturated formation, let H e Projs(G), and let NA and
N2 be normal subgroups of G. For obvious set-theoretical reasons the equation
(N, n N2) r> H = (/V, n //) n (N2 n H)
is correct. Tо prove that the other desired equation
5. Projectors and local formations
= (N,nH)(N2nH)
377
(5 а)
holds, we proceed by induction on |G|. Let M = N, n N2, and suppose that M * 1
Since HM/M e Proj j,( G/M), we can apply induction to G/M. Then we obtain
NtN2r>H = (N,N2Mr~HM)r,H
(NtMn HM)(N2M nHM) ГН (by induction)
= ((N, n H)M(N2 n H)M) n H
= c\ IffN^r, H)(M r<H)
= (Nt n H)(N2 ci H), as required.
Therefore we can suppose that NI ci N2 = 1. Since ^-projectors are persistent, we have
H e ProjgfNjH). Therefore, if NtH < G, the induction hypothesis yields:
Nt N2 ci H = (Nj N2 ci TV, Hj ci H = NfN2 ci N, H) ci H = (by induction)
(Nj n H)(N2 <xNlH<xH) = (N1<x H)(N2 n H),
and again Equation 5.a is verified. Similar reasoning leads to the same conclusion if
N2H < G. Hence we can also suppose that NJ1 = G = N2H. But then G e R„f> = f>,
and consequently H = G, in which case (5.a) is obviously true. □
Remark. If g is a Fitting class, and if F is an g-injector of G, one can similarly define
a map p: .T’(G) -> .1(F) by p(N) = N c.F. But in this case p is a lattice homomor-
phism for all soluble groups G only for the classes g = S„ for n с P (see Chapter
IX, Section 1, Exercise 6).
The following theorem comes directly from (5.3) and A, 1.2.
(5.4) Theorem (Rose). Let g be a saturated formation, and let F be an ^-projector of
G. If Nt, N2 < G, then
FNlnFN2=FlNinN1).
The next topic we broach concerns the traffic of information between chief factors
and maximal links, a theme already familiar from work in Section 4 of Chapter III,
where the concept of an X-normal maximal subgroup is defined. For a saturated
formation g we can refine the concept of g-normal by using the local definition.
(5.5) Definitions. Let f be a formation function, and let M be a maximal subgroup
of a group G. Since our universe is soluble, the index |G: M| is a power of some prime
p (in which case we recall that M is said to be p-maximal in G). If M/CoreG(M) e f(p),
we say that M is f-normal in G; otherwise we call M f-abnormal in G.
378
IV. The theory of formations
(5.6) Remarks. Let f be a formation function, and let M be a p-maximal subgroup
of a group G. Put К = CoreG(M), and let C/K denote the socle of the primitive group
G/K.
(a) If S/T is a chief factor of G complemented by M (there is exactly one such in
each chief series of G), then CG(S/T) = C by A, 15.5. Since M/К s G/C = Autc(S/T),
it follows directly from the definitions that
(i) M is f-normal if and only if S/T is f-central, and
(ii) M is f-abnormal if and only if S/T is f-eccentric.
(b) Let g be a formation function such that Gp</(p) = Sp/(p). Since S/T is an
irreducible (FpG-module, from A, 13.6(b) we have Op(G/C) = 1. Therefore
(i) S/T is f-central if and only if it is g-central, and
(ii) M is f-normal if and only if it is g-normal.
If g = LF(f) with f integrated, we call S/T also ^-central. By (3.8)(a) and (i) this
definition is independent of the choice of the integrated f.
(c) Let g = LF(f). Then the concept of ‘g-normality’ as defined in III, 4.13 is
related to the concept of ‘/-normality' defined above in the following way.
(i) If M is ^-normal, then M is f-normal in G;
(ii) If M is f-normal and f is integrated, then M is ^-normal in G.
In particular, if F is the canonical local definition of g, then g-normal (g-abnormal)
is the same as f-normal (f-abnormal).
Proof of Remark (c). (i) If M is g-normal, then G/K e g by definition. Since G/K is
primitive, we have C/K = Op- p(G/K). Therefore M/K = G/C (G/K)/Opp(G/K') e
/(p), and consequently M is /-normal in G.
(ii) If M is /-normal, then (G/K)/(C/K) = G/C e f(p) £ g because by hypothesis /
is integrated. Since C/K is /-central, it follows from (3.5) (c) that G/K e g, and hence M
is g-normal in G. □
(5.7) Theorem. Let g = LF(f). Any two of the following statements are equivalent.
(a) Geg;
(b) Every chief factor of G is f-central',
(c) Every maximal subgroup of G is f-normal.
Proof. The equivalence of Statements (a) and (b) has already been proved in (3.2),
and by Remark 5.6(a) it is clear that Statement (c) follows from (b). On the other
hand, by the same remark Statement (c) implies that every complemented chief factor
of G is /-central, and this is precisely the requirement for a soluble group to belong
to LF (/). □
This theorem can be improved when the local definition of g is integrated.
(5.8) Theorem. Let g = LF(f) with f integrated. Any two of the following statements
are equivalent.
(a) Geg;
(b ) Every chief factor of G between Ф(С) and F(G) is f-central',
(c ) Every critical maximal subgroup of G is f-normal.
5- Projectors and local formations
379
Proo/ Since a maximal subgroup of G is critical if and only if it complements some
chief factor between Ф(С) and F(G), it is clear from (5.6)(a) that Statements (b'( and
(с') are equivalent. Furthermore, the implication: (a)^(b') is obvious from the
corresponding implication of (5.7). Now assume that Statement (b') holds Bv A
10.6(c) we have ' ’ ’
F(G)/O(G) = A,/®(G) x x А,/ф(С),
where each А/Ф(С) is a complemented chief factor of G. Let i e {1,..., t), and put
ci = Сс(М/ф(6))- Then G/C, e f(p} s 8 by hypothesis, and since Q = F(G) by
A, 13.8(b), we conclude that G/F(G) e R„g = g. It follows that all the chief factors
in a chief series of G/®(G) are /-central, and therefore G/®(G) e g. Consequently
G e r.og = g, and so Assertion (b') implies (a). □
(5.9) Remark. In fact. Assertion (b') of the preceding theorem implies that G e g if
and only if / is integrated. For suppose that / is not integrated. Then we can find a
p e Char(g) such that the class /(p)\g is non-empty and therefore contains a group,
G say, of minimal order. Here, as usual, G is primitive and G/Soc(G) e g. If Soc(G)
is a p-group, then Soc(G) is /-central, and consequently G e g by (3.5)(c), which is a
contradiction. Hence Op(G) = 1. Let W = Zp rL„g G. and let В be the base group of
W. It follows that F(IV) = B, and as W/B s Ge f(p), it is easy to see that all the chief
factors of W between ®(ILj and FflL ) are /-central. However, because G ф g, then
certainly W ф g, and so Assertion (a) does not follow from Assertion (b') in this case.
The following elementary observation is helpful in studying the embedding of an
g-projector in a group; it follows easily from (4.3), 1, 3.3(c), and the conjugacy and
persistence of projectors.
(5.10) Lemma. Let ftbea saturated formation of characteristic n, and let G e S. Then
Proj8(G) = U {Projg(H): H e Hall„(G)}.
The characterization of projectors by properties of maximal chains described in
Section 4 of Chapter III can be reformulated for local formations in the following way.
(5.11) Proposition. Let g = LF(/), and let g(p} =f(p)^3 for all p e P. Let <(G)
denote the set of subgroups H of G which satisfy the following two conditions.
(i) If U < H, then U is f-normal in H;
(ii) If H < S < T < G, then S is g-abnormal in T.
Then :C(G) = ProjR(G).
Proof Condition (i) implies that H e g by (5.7), and so we can substitute the local
definition g for / without changing that condition. But g is obviously integrated, and
therefore by (5.6)(c) the term p-normal (y-abnormal) means the same as g-normtU
(g-abnormal). The desired conclusion now follows from III. 4.19 on setting g
and observing that Condition 4/ of that result is obviously satisfied. U
380
IV. The theory of formations
(5.12) Definitions. Let 5 = LF(f).
(a) A subgroup U of a group G is said to be /-subnormal (respectively ^-subnormal)
in G if there is a chain of subgroups
U = l/0 <•••<• = G
such that Ц_, is an /-normal (g-normal) maximal subgroup of U, for i = 1, .... n.
We shall write Uf-snG (U g-snG) in this case. By (5.6)(c) g-subnormality implies
f-subnormality, and if / is integrated, the two concepts coincide. If (1) e ftp) £
for all primes p. then /-subnormality and 'Ji-subnormality both coincide with the
usual concept of subnormality.
(b) Let U < G. A subgroup S of G with the following properties:
(i) U g-sn S, and
(ii) if U g-sn T, then T < S
is called an fi-subnormalizer of U; it is obviously unique if it exists. (In A, 14.12 an
example is described which shows that subnormalizers need not always exist.)
(5.13) Remark. IfN<G and U f-sn G, then UN/N f-snG/N (andlikewise for‘^-sn’).
Proof. This follows at once from the observation that if M < G and then
N < CoreG(M), and therefore (M/N)/Corec/N(M/N) s M/Corec(M) (and likewise
(G/N)/Corec/N(M/N) s G/CoreJM)). □
Although not all subgroups of an g-group need be /-subnormal (see Exercise 8 for
example), in a positive direction we have the following result.
(5.14) Lemma. Let g = LF(f).
(a) If G is in g and W is well-placed in G, then W is an f -subnormal ^-subgroup of G.
(b) If W is an ^-subgroup cf a group G, and if there exists a chain
W=W0 < <Wn = G
with Wj_y f-normal and critical in W/ for i = 1,..., n, then G e g.
Proof, (a) Let M be the maximal subgroup of G in a critical maximal chain from W
up to G. Then M e sH.g = g by (2.1'2), and since W is well-placed in M, an obvious
induction on the group order yields W e g and IV/sn M. Since M is /-normal in G
by (5.7), we conclude that IV/sn G, as required.
(b) By induction on |G : IV| it will be sufficient to deal with the case where И7 <• G.
Let
C: 1 =L0< Lj < <Lr = L<U = U0<U1< <U,= G
be a chief series of G in which U/L is the chief factor avoided by W. Since W is /-normal
by hypothesis, it follows from (5.6)(a) that U/L is /-central. But by III, 6.5 there is a
bijection from the chief factors of IV to the remaining chief factors of G in C preserving
5. Projectors and local formations
381
the groups of automorphisms induced by the parent group. Since W e R by (3 21 the
chieffactors of IF are all/-central. It follows that all the chief factors in tte chief series
C are /-central, and therefore G e R.
(5.15) Proposition. Let R = LF(F). Let N be a nilpotent normal subgroup of a group
G, and let H be an ^-subgroup of G such that G = HN. Then the unique ^-projector
of G containing H is the /y-subnormalizer of H in G.
Proof. By III, 3.23(b) there is a unique R-projcctor of G containing H; denote it by
E. Since Ec-.N is a nilpotent normal subgroup of E and E = H(E n N), it is clear
that H is well-placed (in fact, critical) in the R-subgroup E; therefore by (5.14) the
subgroup H is F-subnormal, and hence R-subnormal, in G. On the other hand,
suppose that H R-sn L < G. Then there is an R-normal maximal chain from H up to
L, and since L = H(L rt N), each link in this chain is critical. Hence L e R by (5.14) (b).
But by III, 3.23(a) an R-maximal subgroup of G containing L is an R-projcctor of G
(containing //), and therefore by the uniqueness of E, we have L < E. □
If U is a supplement to a nilpotent normal subgroup of a group G, the following
theorem shows (inter alia) how to construct an R-projector of G from an R-projector
of U. If I denotes the nilpotent length of G, it therefore enables one to construct an
R-projector of G in at most I steps.
(5.16) Theorem (D’Arcy [2]). Let R = LF(F) with Char(R) = n. Let N be a nilpotent
normal subgroup of a group G, let U < G = UN, and let H be an ^-projector of U.
For p e n set Cp = COp(/il(HFI'’1) and C = Cp. Finally set E = HC. Then
(a) E is an ^-projector of G, the unique such containing H,
(b) E is the 'ft-subnormalizer of H in G, and
(c) if G = HN, then Gs = O,.(G) x (XPe«[Op(W), Hf,f"J).
Proof, (a) First it should be remarked that Cp is //-invariant because H normalizes
Op(N) and HFW; hence H < NG(Q and, in particular, E is a subgroup of G. Let L be
an R-maximal subgroup of G containing H. Since HN/N < LN/N cqR = R, and
since HN/N is an R-projector of G/N by the invariance of projectors under epimor-
phisms, it follows that HN = LN. By III, 3.23(a) and (b) the subgroup L is the unique
R-projector of HN containing H and by III, 3.7 it is an R-projector of G. We conclude
that L is the unique R-projector of G which contains H. and hence it will suffice to
show that E = L.
First we show that E e R. Since E/C S H/(H n Gj e R, by (3.2) and the Jordan-
Holder theorem it will be enough to prove that the chief factors of E below C are
R-central, and, again by the Jordan-Holder theorem, we can as well consider a p-chief
factor S/T below C. for some pen (since obviously E e £„). Since C < F(HC). the
centralizer of S/T contains C; therefore E and H induce isomorphic groups о
automorphisms on S/T. By definition HFW centralizes Cp and hence S/T. and i
follows that AutJS/T) s H/C„(S/T} e q(H/Hf,p') <= F(p). Therefore S/Tis F-central
in E, and we have proved that E e 8- Thus E < L.
382
IV. The theory of formations
Since L is an ft-projector of G, by (5.10) it is contained in some Hall n-subgroup
G„ of G, and by (5.3) we have Ln N = G„ n X₽e P(En Op(N)) = Xpe„(Ln Op(N)).
We assert that if p e n, then
(5./?) LoOp(N)<.Cp.
For, if (5./1) is true, it follows that L = L n HN = H(L r,N)< Н(Хре„Ср) = E, as
desired. To prove this assertion, first observe that Hegc Qp.QpF(p) = Sp.F(p),
and that Hr(₽’ is therefore a p'-group. Moreover, since L = H(L n N), we have
//'<"> < LF(p} by (1.17) (b). Since L e ft, the chief factors of L below L n Op(N) are all
F-central; they are therefore centralized by LFIP> and, a fortiori, by the p'-group HF(P>.
Hence by A, 12.3 we have [LnOp(N|, HF(₽,J = 1, and therefore (5./J) holds. Thus
E = F, and Assertion (a) of the theorem is proved.
(b) Since F is the canonical local definition of ft, we need make no distinction
between the concepts ‘F-subnormal’ and ‘ft-subnormaf. Suppose that H F-sn J < G.
Then HN/NF-sn JN/N by (5.13). Since the subgroup HN/N(= EN/N) is an ft-
projector of G/N, by (5.11) it is F-abnormal in every subgroup of G/N in which it is
maximally contained; therefore HN = JN. Since E is the ft-subnormalizer of H in
HN by (5.15), it follows that H F-sn E and also that J < E. Thus E is the ft-subnor-
malizer of H in G.
(c) Since ft has characteristic n, obviously O„.(G) < G®, and so we may suppose
without loss of generality that O„.(G) = 1; in particular, N is a n-group.
First we deal with a special case by supposing that N is a p-group (for some p e n).
Set R = HFtp\ and recall from earlier in the proof that R is a p'-group. By A, 12.5 we
therefore have N = [N, R]CN(R) and by Part (a) also E = CN(R)H; consequently G =
NH = [A, R]E. Since NR < G and [N, R] = (NR)S- s- char NR, we have [N, R] <
G. It follows that G/[N, R] e q(E) £ ft, and hence G® < [N, R], On the other hand,
by the invariance of residuals under epimorphisms, RG^/G* is the F(p)-residual of
IIG^/G", and so, applying Part (a) to G/G® e ft, we obtain [N, R] G®, which yields
the desired conclusion when N is a p-group.
In the general case, let pen, and let P = Op(N) < G. Since N is nilpotent, we may
write N = P x Q, where Q = Op.(N) a G. Then, because N/Q is a normal p-subgroup
of G/Q, we deduce from the special case already proved that
G»e/e = [PQ/Q, RQIQ\ = [P, R~\Q/Q.
Clearly [P, R] is the unique Sylow p-subgroup of the nilpotent group [P, R]Q
( = G®Q), and its order divides |G®|. Therefore [P, R] is the Sylow p-subgroup of G®,
and the desired conclusion for the general case is now clear. □
(5.17) Theorem (D’Arcy [2]). Let ft = LF(F) with Char(ft) = n. Let N be a nilpotent
normal subgroup and H an '^-subgroup of a group G = HN. If p e Tt and Op{G^) is
abelian, then H c-. Op(G®) = 1.
Proof. Let К = Op(G®), and let R denote the p’-subgroup HF,P>. By (5.16)(c) we have
К = [OP(N), R], and so
5. Projectors and local formations jgj
[К, R] = [Op(A), R, R] = [Op(A), R] = к
by A, 12.4(b). Since К is abelian, by A, 12.6 we have [H R] = H r> К But an
application of (5T6)(c) to the group H = (H n K)H with H n К in therole of A yields
[HnK, R]<H»= 1; therefore HnК = 1. a
As a corollary we obtain the following useful splitting theorem.
(5.18) Theorem (G. Higman [1], Carter and Hawkes [1], Shult [1]). Let g be a
saturated formation, let R denote the ^-residual of a group G, and assume that R is
abelian. Then R is complemented in G, and its complements are precisely the g-
projectors of G.
Proof Since G/R e g, certainly each R-projcctor E of G is a supplement to R in G.
If E n R # 1, then E n Op(R) 1 for some p e Char(g), which contradicts the conclu-
sion of (5.17). Therefore R is complemented in G by E. On the other hand, each
complement to R in G belongs to g, is therefore contained in some R-projector E of
G by III, 3.23(a); and hence equals E by order considerations. Consequently Projs(G)
is precisely the set of complements to R in G. □
Next we describe another distinguished local definition of a saturated formation R;
it is formulated in terms of properties of R-projectors.
(5.19) Theorem (Doerk [3]). Let g = LF(F), and for all peP let f*(p) = (g jF(p)).
Then the following statements are true:
(a) The formation function f* is a full local definition of R;
(b) If R = LF(h) with h(p) = s/i(p) for all p e P, then h(p) £ f*(.p) for all p e P;
(c) The f*-central chief factors of a group G are precisely those chieffactors which
are covered by an ^-projector of G.
Proof (a) Let peP, and recall that (RjF(p)) is by definition the class (Grthe
R-projectors of G are in F(p)) and is a formation by (1.2); thus f* is a formation
function. Let G e Sp/*(p), and let E be an R-projector of G. Then E/(En OP(G)) S
EOp(G)/Op(G) e F(p), and therefore E e SpF(p) = F(p). Consequently G e/*(p), and
we have shown that Sp f *(p) = f *(p), i.e. that f* is full. Since /*(p) <y = F(p), it
follows that <3p(/*(p) n R) = SpF(p) = F(p), and therefore by (3.8)(b) we have R =
LF( f*).
(b) If G e h(p) and E e Proj R(G), then E e sA(p) n R = h(p) n g = F(p) by (3.8) (a).
Hence G e /*(p), and consequently/1(р) e/*(p).
(c) Clearly it is sufficient to prove the statement for a minimal normal subgroup
N of G. Let H be an g-projector of G, and first suppose that N is /‘-central.
Then H/C„(A) S HCc(N)/Cg(N) e F(p),and therefore by (5.16)(c) we have (HA)- -
[A, = 1. From the g-maximality of H it follows that H = HN, and so H
contains (covers) N. Conversely, if N is contained in H, then we have F- H<
[А, /Л1'”], again by (5.16)(c). Therefore HCc(A)/Cc(A) = H/C„(A) e <jF(p) = F(g
and we conclude that N is /*~central in G.
384
IV. The theory of formations
Finally, we prove some more results which will be needed in Section 5 of Chapter VII
(5.20) Lemma. Let F be the canonical local definition of a saturated formation 8 =
LF(F], and let л S Char(8) be a set of primes. Any two of the following statements
are equivalent :
(a) In each group G the ^-projectors have n'-index (or, equivalently, Zn « 8 in the
notation of VI, 1.1);
(b) F(q) F(p) for all pen and all q e P;
(c) F(p) = 8 for all pen:
(d) f*(P) = <= f°r a,l P^tt:
(e) S„8 = 8-
Proof, (a) => (b): We derive a contradiction by assuming that (a) holds and that (b)
fails. Let pen, and let G be a group of minimal order in F(q)\F(p). Then G has a
unique minimal normal subgroup N, the quotient group G/N is in F(p), and N is a
p'-group because F is full. Let W = 2prLirts, G, and let В denote the base group of W
(= BG). Since Geg and « 8, it follows that W e 8- But В = Op. p(IV), and
therefore G И7О • (H7) e F(p), which yields the desired contradiction.
(b) => (c): Let p = Char(8) and p e n. By (3.2) we have
8 = n (П,6„ s,.F(4)) s ep n s,.F(P)j = f(P).
Since 8 is integrated, we have F(p) e 8, and therefore F(p) = 8-
(c)=>(d): This follows at once from the definition of f* in Theorem 5.19.
(d) => (e): If f*(p) = S, then (5.19)(c) implies that all the л-chief factors of a group
are covered by its 8-projectors and hence, in particular, that S„8 = 8-
(e)=>(a): If S„8 = 8. then it follows easily that all л-chief factors of a group are
covered by its 8-projectors and hence that S„ « 8- □
(5.21) Lemma. Let 8 = LF(f), and assume that an ^-projector E of a group G
satisfies E e f(p) for all primes p. Then E = G.
Proof. It will suffice to show that G e 8- If G f 8, then G has a p-chief factor G"'/K
for some prime p. But G/G® = EG^/G7' s E/(E r\ G®) e Q(/(p)) = f(p}, and con-
sequently G/K e Qpf(p) £ 8- This gives the contradiction G® < K. Therefore G e 8-
□
Again using the notation of Theorem 5.19, we deduce the following.
(5.22) Corollary. Let 8 = LF(F), where F is the canonical local definition, and
assume there exist primes p and q such that F(q) n F(r) £ F(p) for all primes p. Then
f*(q)<~\f*(r) = F(q)c>F(r).
Proof. Let G e f*(q) ryf*(r), and let E e Proj„(G). Then E e F(q) n F(r) t= F(p) for
all primes p, and by the preceding Lemma E = G. Hence f* (q) n f*(r) £ F(q)r\ F(r),
and since the reverse inclusion is obvious, equality holds. □
5. Projectors and local formations
385
Exercises
(All exercises are for a soluble universe; f* will denote the local definition described
in Theorem 5.19.)
1. Let ft be a saturated formation and let F e Projn(G). Prove the following:
(a) (Beidleman [1]) If p(G, F) is a lattice epimorphism, then F is a CAP-subgroup
of G;
(b) The converse of (a) is false;
(c) If p(G, F) is injective, then G = F.
2. If p(G, F) is an epimorphism for all Ge S, prove that ft = (1) or g. (Hint: Use
the description of CAP saturated formations in Chapter VI, Section 5.)
3. (Huppert [7]) let ft be a saturated formation and let FeProjs(G). If
<K, Ly n F = (K r\F, Lr\ Fy for all K, L sn G, prove that F is a CAP-subgroup
of G. Show also that the converse is false.
4. Let & be a formation and let ft = 9i§. If Fe Projs(G) and if К is a normal
subgroup of F containing Fb, then show that F = /VC(K).
5. Let ft = LF(f), and let <J>P(G) denote the intersection of the p-maximal subgroups
of a group G. Prove that G e ft if and only if for all p e Char(ft) all chief factors
of G between <!>P(G) and Op. p(G) are /-central.
6. Let ft = LF(f), and assume that ft-abnormality implies /-abnormality. Prove
that the formation function / is integrated.
7. Let ft = LF(f) and let $F(G) denote the set of subgroups of a group G defined as
in (5.11).
(a) If f(p) £ f*(p) for all p e P, prove that .T(Gj = Proj R(G).
(b) For p e P, let h(p) = (G : the system normalizers of G are p-groups). Prove
that 91 = LF(h). Let D = Dih(10), and let r be the permutation representation of
D in the cosets of a Sylow 2-subgroup. Let A = Alt(4), and set W = A Qj, D.
Finally let H = [B, £>]£>, where В is the base group of W. If T e Syl2(H), show
that T e 3F(H) and that T ф Proj ,,(//).
8. Define a formation function / as follows: /(2) = QR0(Sym(3)), /(3) = QR0(Sym(4)),
and f(p) = (1) for p > 3. Prove that / is an integrated local definition of ft =
LF(/j. Let IV = Z/ljna, Sym(4), let В denote the base group of W, and let
Z = Z(W\ Finally, let D e Syl2(Sym(4)), and let U be a U-projector of BD.
Prove that both U/Z and W/Z belong to ft and that U/Z is not ft-subnormal
in W/Z.
9. Let ft = LF(/*). If/* is integrated, show that ft = (1) or <5.
10. Let F = LF( F}, and let q e P. Prove that any two of the following statements are
equivalent:
(a) F(q) = f*(q);
(b) f*(q) 9ift;
(c) F(q) £ F(p) for all p e P.
11. Let ft = 9if> = LF(F) with & a formation, and let p e P. Show that if a group
has F*-central p-chief factors, then it also has g-central p-chief factors^
12. Let ft = LF(F), and assume that 91 £ ft. Show that any two of the following
statements are equivalent:
(a) ft = 91& for some formation &;
386
IV. The theory of formations
(b) f*(p)r,f*(q) £ s for all distinct pairs {p, q} £ P;
(c) F(p) n F(q) £ F(r) for all p, q, r e P with p q.
6. Theorems about /-hypercentral action
In this section we focus our attention on groups of operators acting hypercentrally
with respect to general formation functions. The main results are formulated in the
full generality of a treatment due to P. Schmid [1], [2], and from them we deduce
some established theorems about stability groups, local formations, and generalized
hypercentres due to Baer, P. Hall, and Huppert (i.a.).
Throughout this section an Л-group will mean a finite group G with a (not
necessarily faithful) group of operators A. For the basic definitions and facts about
operator groups we refer the reader to Section 2, Section 3, and Section 4 of Chapter
A. We begin with a technical lemma, which will be used several times in subsequent
proofs.
(6.1) Lemma. Let G be an А-group, and let К and N be А-invariant normal subgroups
of G such that К < N. Let В = CA(N) n CA(G/K), and let M denote the set of all maps
from G/N to CK(N).
(a) M is an abelian group with respect to pointwise multiplication, defined as follows:
(gNj(Afi) = (gN)A. (gN)p for all g e G and A, p e M.
(b) A may be regarded as a group of operators for M by defining the action of A as
follows:
(6.a) (gN)A° = (g° ‘N)A for all a eAandAeM.
(c) The map т: В -» M defined thus:
(g/V)(br) = [g, b] for all g e G and be В
is a homomorphism with Ker(r) = CB(G).
(d) If C < CX(CK(N)), and if we regard В as a C-group with the action defined by
conjugation (obviously В < Aj, then т is a C-homomorphism.
(e) If G centralizes CK(N), then Вт < Hom(G/7V, CK(N)).
Proof. The Statements (a) and (b) are obvious.
To verify Part (c) we first check that т has the stated target and is well defined. If
geG and be B, then gbK = gK by hypothesis, and so [g, b] e K. Moreover, using
the fact that beCA(N), we have [g, b]n = g~' b~‘ gbn = g~' b~'(gng~')gb =
9 *(gng 1 )b *gb = n[g, b], and therefore [g, b] e CK(N). Consequently br e M. If
gN = hN, then h = gn for some ne N, and if b e B, it follows that [h, b] = [gn, b] =
[9, b]"[n, b] = [g, b]. Thus r is well defined.
6. Theorems about /-hypercentral action
387
Next we show that t is a homomorphism. Let tie G and h h В ri.™
= [9, b b2] = [9, ь2][9, = [s, M[99 b2] ‘(bih4.
Furthermore, if b e Ker(r), then [9, b] = 1 for all g e G, and it follows that Ker(r) =
To prove Part (d) let be В and с e C. Then for all g e G we have (9А)((Ь/т) =
[g,bc] = [gc ,by = [g‘ ,bl = (gc ,А)(Ьг) = (9А)((Ьт)с). which yields (bch =
(br)‘, as desired.
Finally, for Part (e), suppose that [G, CK(A)] = 1. Let 91, g2 e G and b e B. Then
using the fact that g2 centralizes [91, b] e CK(N}, we have (017V)(ff2N)(br) =’
[0102, b] = [01,b]«2[92,b] = (91N)(bT).(92N)(br). Thus the map r is a group
homomorphism. I-,
(6.2) Definitions. Let f be a formation function and G an Л-group.
(a) We say that A acts f-centrally on an Л-composition factor H/K of G if
А/Сл(Н/К}е f(p} for all primes p dividing \H/K\, and otherwise that A acts /-
eccentrically.
(b) We say that A acts f-hypercentrally (resp. f-hypereccentrically) on G if it acts
/-centrally (resp. /-eccentrically) on every Л-composition factor of G. (Of course, by
the Jordan-Holder theorem it is only necessary to examine one Л-composition series
to verify such action.)
(6.3) Remarks. Let G be an Л-group.
(a) If /(p) = (1) for all primes p, it is usual to omit the prefix '/’ and to talk simply
of central and hypercentral action, etc.
(b) Let f and g be two integrated local definitions of a local formation Jy. Assume
that either G is soluble or A/Ca(G) contains Inn(G). If p is a prime divisor of the order
of an Л-composition factor H/K of G, then Op(AutA(H/K}} = 1 by A, 13.6, and since
by (3.8) (a) we have Sp/(p) = Spb(p), it follows that A acts /-centrally on H/K if and
only if it acts 9-centrally. Thus in this case the concept of /-hypercentral action does
not depend on the chosen integrated local definition, and so then we say that A acts
5-hypercentrally instead of f-hypercentrally on a group.
The assertions made in the following lemma are obvious in the light of A, 3.2.
(6.4) Lemma. Let f be a formation function, let G be an А-group, and let M and N be
А-invariant normal subgroups of G.
(a) If A acts f-hypercentrally (f-hypereccentrically} on G, then it acts similarly on
G/M. J , .
(b) If A acts f-hypercentrally (f-hypereccentrically} on G/M and G/N, then и acts
similarly on G/(M n N}.
(c) If A acts f-hypercentrally (f-hypereccentrically} on M and N, then it ac
similarly on MN.
(6.5) Lemma. Let G be an А-group, and let H/K be an А-invariant normal factor of
G such that [//, Л] < K. Then [G. Л] < CG(H/K}.
388
IV. The theory of formations
Proof. Clearly our hypotheses imply that [//, G, Л] < [H, Л] < K, and that
[Л, H, G] < [K, G] < K. Therefore by the Three Subgroups Lemma (A, 7.6) we have
[G, A, H] < K; in other words [G, A] < Cc(H/K). □
We need one further preparatory lemma before we come to the first substantive result
of this section.
(6.6) Lemma. Let peP. If A acts on G. centralizing all factors of an A-composition
series whose orders are divisible by p, then [G, О₽(Л)] < Opp(G].
Proof Let 1 = Go <1 <1 G„ = G be an Aut(G)-composition series of G. If p divides
|Gi+1/G,|, then p divides the order of each composition factor of the characteristic-
simple group Gi+1 /G\; therefore A centralizes each factor of an Л-composition series
of G,+1/G;. Thus, if Gj+i/Gj is abelian, then A/CA(Gi+i/Gi) is a p-group by A, 12.4(a),
and if Gi+I/Gt is non-abelian, then A = Q(Gi+1/G;) by A, 13.7. Hence, if we set
в = A {CitG.+t/G,): p| |G1+1/Gj|},
we have 0p(A) < B. Moreover, from (6.5) we can deduce that [G, B] is contained
in A {Cc(Gi+i/Cr,-): p| |G;+i/Gj|), which equals Op. p(G) by A, 13.8. Therefore
[G, О₽(Л)] < [G, B] < Opp(G). ’ □
We now come to the first significant theorem of this section; it shows that hypercentral
action ‘lifts’ to Frattini extensions.
(6.7) Theorem (P. Schmid [1]). Let f be a formation function, and let G be an A-group.
If A acts f-hypercentrally on G/<b(G), then A acts likewise on G.
Proof. We argue by contradiction. Let G be a group of minimal order with a group
A of operators acting f-hypercentrally on G/U>(G) but not f-hypercentrally on G;
then clearly <D(G) 1. First let N be a minimal Л-invariant normal subgroup of G,
and let T/N = ®(G/N). Since N®(G) < T. by (6.4) (a) the group A acts f-hyper-
centrally on G/T, and it follows from the minimal choice of G that A acts f-hyper-
centrally on G/N. We then conclude from (6.4)(b) that N is the unique minimal
Л-invariant normal subgroup of G. In particular, we have: N < ®(G); F(G) is a
p-group for some prime p; Op(G) = 1; and F(G) = Opp(G). Since <I>(G) 1, it follows
from A, 11.8(a) that p| |G/®(G)|, and hence that ftp) 0 since A acts f-hypercen-
trally on G/®(G). Let R = Af<p>. Since R centralizes all chief factors of G/O(G)
whose orders are divisible by p, we deduce from (6.6) (applied to G/®(G)) that
[G. O₽(R)]O(G)/O(G) < Op. p(G/O(G)). But by A, 13.4(f) we have Op p(G/O(G)) =
ОР-,„(С)/Ф(С) = F(G)/®(G), and so [G, OP(R)] < F(G). Furthermore, since A acts
.f-hypercentrally on the p-group F(G)/®(G), it follows that R, and hence certainly
OP(R), acts hypercentrally on F(G)/®(G). Let q e P, q * p, and let Q e Syl,(O₽(K)).
Then for suitable n e N we have [F(G), Q, • , Q] < ®(G), and therefore [F(G), Q] <
Ф(О) by A, 12.3. Now apply (6.1) to G/®(G), with the subgroups К and N of that
lemma both set equal to F(G)/®(G). Since Q centralizes G/F(G) and F(G)/®(G), we
6. Theorems about /-hypercentral action
389
can conclude from (6.1)(c) that the q-group Q/C0(G/®(G)) is isomorphic with a
subgroup of the p-group M consisting of all maps from G/f(G) to F(G)/<D(G)
Consequently Q centralizes G/<D(G). However, since O”(R] is p-perfect, it is generated
by its Sylow q-subgroups as q runs through P\{p}. Hence O'(R) centralizes G/®(G)
and И fo lows from A, 944 that O₽(R) centralizes G. Hence we may assume that
О (R) and therefore that A e Qpf(p). Since an Л-invariant composition factor X
of G below ®(G) is a p-group, by A, 13.6 we have О,(Л/Сл(Х)) = 1, and therefore
АиЦХ) e J(p). Thus A acts /-hypercentrally on <I>(G). q
Remark. From Theorem 6.7 we can retrieve the fact that a local formation is satur-
ated (cf. Theorem 3.3) by taking A = G, acting by conjugation.
(6.8) Notation and Definitions. Let ft = LF(f), and let G be an Л-group.
(a) By (6.4) (c) each group G possesses a unique maximal /-hypercentral normal
subgroup, which is denoted by Zy(G, Л). Similarly there is a largest /-hypereccentric
normal subgroup, which we denote by Ef(G, A).
(b) In the special case when A is the group G, acting by conjugation, we write Zf(G)
instead of Zs(G, G) and call Zf(G) the f-hypercentre of G. If f is integrated, by Remark
6.3(b) we can write ZB(G) instead of Zf(G) and talk of the ft-hypercentre of G without
ambiguity; similarly we can write ES(G) for Ef(G) in this case. We remark that the
hypercentre Z„(G) becomes Z4I(G) in this notation.
By (3.2) a group G belongs to a local formation ft if and only if G = ZB(G). The
next result shows, somewhat surprisingly, that the sufficiency of this condition holds
for a more general group of operators.
(6.9) Theorem (P. Schmid [1]). Let ft = LF(f) with f integrated, and let G he an
А-group such that C„(G) = 1. If Zf(G, Л) = G, then A e ft.
Proof. We suppose that the theorem is false and derive a contradiction. Let (G, Л)
be a counterexample with G of minimal order. Then obviously G 1. If G is itself an
Л-composition factor of G, then A ef(p} for all p||G|, and since by hypothesis/is
integrated, we conclude that A e ft. Thus we can assume that G has a maximal
Л-invariant normal subgroup N and a minimal Л-invariant normal subgroup К
such that К < N. Let В = CA{N)r. CA{G/K\, then by the choice of G we evidently
have A/В e Roft = ft. Since A £ ft, the group В is non-trivial, and so by (6.l)(c) the
group M of all maps from G/N to CK(N) is also non-trivial because by hypothesis
CB(G) = 1. Consequently CK(/V) 1, and since CK(/V)( = Kr\Z(N)) is Л-mvanant
and normal in G, by the choice of К we have К < Z(N). This has two important
consequences: first, it follows from the choice of N that Cc(K) = N or G; second,
is an elementary abelian p-group for some prime p, and. in particular, /(p) * 0-
Furthermore, M is a p-group by (6.1 )(a), and so В is a p-group by (6. l)(c).
Let R = Af,p'. Because R centralizes K, we conclude from (6.5) that R centralizes
G/Cg(K) and hence centralizes G/N in the case when Cc(K) = N. The other possibihty
is that Cc(K) = G, in other words that G centralizes К = CK(/V). Then from (6. )(
we conclude that 1 * Вт < Hom(G//V, K). Since К is a p-group, it follows that
390
IV. The theory of formations
Or(G/N) < G/N and hence that G/N is an elementary abelian p-group. Thus, in this
case too, R centralizes G/N because A acts /-centrally on G/N. It is clear from
Equation 6.a in (6.1) that R acts trivially on M, and since R centralizes CftN). it
follows from (6.1 )(c) and (d) that R centralizes B/CB(G) S B. Thus А/СЛ(В) e /(p), and
by (3.5)(c) we have A e ft. This is the desired contradiction. □
Let g = LF(F), and let X be a finite group. If we put G = ZS(X) and A =
X<Cx(Zs(X)). it is clear that Inn(G) < A < Aut(G) and that A acts E-hypercentrally
on G. Therefore we can deduce from the preceding theorem that Л e ft, in other words
that Xя < Cx(Zs(X)). Now writing G instead of X, we obtain the following result as
a special case of Theorem 6.9.
(6.10) Theorem (Huppert [6]). If ft is a local formation, then
[G»ZB(G)] = 1.
The next theorem shows how the existence of a suitable ft-group of operators gives
rise to a direct decomposition of the operator domain. It generalizes a decomposition
theorem of Baer’s for an ‘ft-embedded’ normal subgroup of a group; it is also useful
for finding decompositions of a finite abelian group with respect to a group of
operators.
(6.11) Theorem (P. Schmid [2]). Let ft = LF(f), and let G be an А-group such that
lnn(G) < A < Aut(G). If A e ft, then
G = Z/G, A) x Ef(G, A).
Proof. If the theorem is false, let (G, A) be a counter-example with |G| as small as
possible, and put N = Zf(G, A) x Ef(G, A) (this product obviously being direct by
the Jordan-Holder theorem). If X is any Л-invariant proper subgroup of G, then
the choice of G implies that X = Zf(X, A) x EftX, A). Since Inn(G) < A, we have
Zf(X, A) < G, and therefore Zf(X, Л) < Zz(G. A). Similarly EftX, A) < Ef(G, Л).
Hence N is the unique maximal Л-invariant subgroup of G, and certainly N # 1.
Next we assert that there exists a maximal Л-invariant subgroup R of N such that A
acts /-centrally on exactly one of the factors G/N and N/R. This is obvious if
Zz(G, A)^ 1 Ef(G, A). On the other hand, if Zf(G, Л) = 1 for example, then N =
Ef(.G, A) and A cannot act /-eccentrically on G/N because G # Ef(G, A). The asser-
tion is therefore clear. Now, if R 1, the choice of G then implies that G/R is the
direct product of two maximal Л-invariant subgroups, a conclusion which contra-
dicts the uniqueness of N as a maximal Л-invariant subgroup of G. Therefore N is
the only proper, non-trivial, Л-invariant subgroup of G.
Since G/Z(G) s Inn(G) < A e ft, the group A acts /-hypercentrally on G/Z(G).
Therefore Z(G) * 1, and so Z(G) = N or G. If Z(G) = N, then G' # 1, and so N < G'.
But in this case N < Z(G) r\G’ < Ф(С) by A, 9.3(d), and therefore by (6.7) we have
G = Zf(G, A), a contradiction. Hence G = Z(G), and G must be an abelian p-group
because of the uniqueness of N. If ftp} = 0, then G = Ef(G, A), which is not the
6. Theorems aboui /-hypercentral action 39,
case. Therefore /(p) * 0, and the subgroup Q = AW is weU defined. Moreover
since A e ft, it follows from (3.2) that Q is a p'-group. Then by A, 12.5 we have G =
[G, Q] x Cc(Q), and both direct factors are Л-invariant because В < A. Thus either
[G, ffl — 1, >n which case G = Zf(G, A), or CG(Q) = 1, in which case it follows from
Maschke s theorem (A, 11.4) that Q centralizes no Л-composition factors of G and
hence that G = Ef(G, A). In any case we have a contradiction, and therefore no
counterexample exists. p
Next we derive two corollaries from this theorem, the first being the Main Theorem
in Baer [6] and the second a deduction made from it in the same paper.
(6.12) Theorem (Baer [6]). Let 3 = LF{f), and let N he a normal subgroup of a group
G such that G® < CG(N). Then
N = Zf(N, G) x Ef(N, G).
Proof. If we substitute N for G in Theorem 6.11 and G/Cg(N) for A, then it is clear
that the hypotheses of that theorem are satisfied We therefore conclude that
N = Zf(N, G/Cc(N)) x Ef(N, G/Cc(N)) = Zf(N, G) x Ef(N, G).
□
(6.13) Theorem (Baer [6]). Let ft be a local formation and G a group. Then
Cc(G®) = ZS(G) x (£,(G)nZ(G’)).
Proof. Let ft = LF(F), and put N = Cc(G®). Since G® < Cc(N), it follows from (6.12)
that N = Z^(N, G) x £g(N, G). But we clearly have ZR(N, G) = ZR(G)nN, and
therefore ZR(N, G) = ZR(G) because ZR(G) < N by (6.10). Moreover, since G/G® is
F-hypercentral, it follows that £S(G) < G®, and therefore E^(N, G) = £,y(G)nN =
£R(G)n(G®nN) = £s(G)nZ(G®). □
The next result enables one in particular to derive the ft-hypercentre of a group from
knowledge of its ft-residual and an ^-projector (if one exists).
(6.14) Theorem (Doerk—see Huppert [6]). Let % be a local formation and G a group.
If U is an ^-maximal subgroup of G such that G = GG® (in particular, if U is an
'S-projector of G), then
Zn(G} = QfG®).
Proof. Let ft = ££(£). We first assert that ZS(G) < V and prove it by induction on
|G|. Clearly we can suppose that ZB(G) / 1 and choose a minimal norma! subgroup
N of G contained in Z„(G). If p is a prime dividing |N| then G/CG(N)16 F(p) ft,
consequently G = GG® = GCc(N) = UNCG(N), and it fo lows that UN/CGN(N)e
£(p). But UN/N S U/(U n N) e Qft = ft, and therefore GN e ft by (W ™ f
N < G Since G and G, and hence all intermediate groups, induce the same group
392
IV. The theory of formations
automorphisms on N, it also follows from (3.5) (c) that U/N is g-maximal in G/N. Ap-
plying the induction hypothesis to G/N, we then have Zg(G)/A = Zg(G/A) < U/N,
which proves the initial assertion. Hence we can conclude from (6.10) that Zg(G) <
OnCc(G*) = QG8).
To prove the reverse inclusion, first note that C[,(Gi’) < GGR = G. If X is a G-chief
factor below Cl/(GB), then G/Cc(A) = UCG(X)/CG(X) s U/CV(X). Hence X is a chief
factor of U and is furthermore g-central in G because U e g. Therefore CV(GS) is
contained in, and hence equal to, Zg(G). □
Finally we look at the question of when the g-hypercentre belongs to the local
formation g. The answer is fairly obvious.
(6.15) Theorem. Let s„g = g = LF(F), and let G be a group. Then ZS(G) e g.
Proof. First observe that F(p) = s„F(p) for all p e P by (3.16). Write Z for Zg(G) and
let X be a G-chief factor below Z. If p is a prime dividing |X|, by definition of the
g-hypercentre we have G/Cg(X) e F(p), and therefore Z/Cz(X) = ZCG(X)/CG(C} e
s„F(p) = F(p). It then follows that Z induces on all its Z-chief factors a group of
automorphisms belonging to QF(p) = F(p). Hence Z e g by (3.2). □
Remark. In Exercise 5 we suggest an example to show that the statement of (6.15) is
false without the hypothesis that g is s„-closed.
Exercises
1. Show that every Л-invariant subnormal subgroup of a finite Л-group G belongs
to an Л-composition series of G.
2. (Schmid [2]) Show that Zf(G, A) contains every Л-invariant subnormal sub-
group of G on which A acts /-hypercentrally, but that the corresponding state-
ment for EZ(G, Л) is false.
3. From (6.9) deduce the following theorems about an Л-group G with A < Aut(G).
(a) (P. Hall [7]) Assume that G has a chain of subgroups
G = G„>G„_,>->G0= 1
such that Л leaves invariant each coset of G;_f in G,fori = 1,.. ,,n. Then A e 91.
(b) (Baer [4]) Assume that G has an Л-composition series whose abelian com-
position factors are cyclic and on whose non-abelian composition factors A
induces a supersoluble group of automorphisms. Then A e U.
4. Let g = LF(F), and let G be a group with an g-projector E. Show that
(a) ZB(G) = Zg(G, E), and (b) Zg(G) Corec(E). Find an example for which the
inclusion in (b) is strict.
5. (a) If g is a local formation contained in 9l2, show that ZX(G) e g.
(b) Let p, q, r, and s be four distinct primes, and let S denote the formation of
all soluble groups with no central q-chief factors. Then define a formation function
F as follows: F(p) = S n G,Gr, F(q) = Gr, and F(t) = 1 for all t 6 P\{p, q}. Show
6. Theorems aboui /-hypercentral action 593
that g = LF(F) is a local formation contained in 9l3 with F canonical. Next let
S = £(r/s), Q = E(r/q), and let H be a subdirect product of S and Q with amalga-
mated factor group Zr (see A, 19.2 for definition). Then let N be an irreducible
FpH-module with Ker(H on N) e Syl,(H), and put G = [N]H. Show that
zB(G) ф g.
6. (Semetkov [1]) Assume that G® = £B(G) for some local formation Й and further
that the Sylow p-subgroups of G® are abelian for all primes p dividing |G/G®|.
Show that G® is complemented in G. (Compare with Theorem 5.18.) Can the
second assumption be dispensed with?
7. Let g be a formation and f some integrated local formation function. Define
HRes(g, f) = (G 6 S: G/G® is /-hypercentral in G).
Show that HRes(g, f) is a Schunck class but not in general a saturated formation.
(This is a dual of Construction C in Chapter IX, Section 2.)
Chapter V
Normalizers
1. Normalizers in general
All groups considered in this section are soluble.
To begin with we define the concept of a normalizer in its most general setting, and
explore some elementary consequences of the definition.
(1.1) Definition. Let G be a group, and let .1(G) denote the set of all normal
subgroups of G.
(a) A normal subgroup function v for G is a map
v.P-.r(G)u{0}.
The set it = |peP:v(p)^0} is called the support of v; thus v(p) is a normal
subgroup of G for all p Eit.
(b) Let v be a normal subgroup function for G of support it, let X be a Hall system
of G, and let G„ denote the Hall л-subgroup of G in E. Then the subgroup
D = D,(E) = G„ n ( П Ac(Gp. n v(p))
\pe«
is called the v-normalizer of G associated with E.
(1.2) Proposition. Let v he a normal subgroup function for G of support it.
(a) The v-normalizers of G form a conjugacy class of subgroups of G.
(b) The Hall system E reduces into DV(E); in particular, if p e it and if P( = Gp) is
the Sylow p-subgroup in E, then NP(Gp. n v(p)) e Sylp(Dv(E)).
Proof, (a) If E and E are Hall systems of G, then Ё = E9 for some g e G by I, 4.11.
Therefore D,(E) = DV(E9) = DV(E)9.
(b) Let p e it. Since v(p) <l G, the subgroup Gp. n v(p) is pronormal in G and has
E reducing into it; hence E reduces into Nc(Gp. n v(p)) by 1, 6.8. It then follows from
I, 4.22(a) that E reduces into
Gn n ( П Ng(G„. n v(p))') = DV(E).
\рея /
Since P < Gq < Nc(Gq. n v(q)) for all primes q A p, the final assertion is clear. □
1. Normalizers in general
395
N 1Ье°Г.£Г' \Ье ° nOrmal ^roup function of support л for a group G Let
N < G, and let v he the normal subgroup function for G = G/N defined hy
v(P)= V(P)N/N
10
forpe л,
for pin.
V D DV(S), then DN/N is the v-normalizer of G associated with X = Y.N/N.
Proof. Denote the v-normalizer «,(!) by D. Since for p g л we have
(la)
(v(p)N/A) n (Gp.N/N) = (v(p) n Gp.)N/N,
it is clear that DN/N < D. Let gN e D. Then from (l.a) we obtain (v(p)<sGp.yN =
(v(p) n Gp.)N and conclude from I, 4.13(b) that (v(p) r> Gp.'f = (v(p) n Gp.)" for some
neN, where n is independent of p e л. Thus gn'1 e f\pt „ NG(v{p} rx Gp.), and so gN
belongs to
GnA/Nnl Q NG(v(p)cxGp)\N/N = I G„rJ Q Ac(v(p)nGp)) ]N/N = DN/N.
\Pen / \ Veit //
Therefore D < DN/N, and equality holds.
□
(1.4) Lemma. Let N be a minimal normal p-subgroup of G, and let N < M < G. If M
has a p-complement Mp. such that NG(Mp)cx N =£ 1, then N < Z(M).
Proof: If n e NG(Mp.) n N, then [n, Mp] < N n Mp. = 1, and so n e CK(Mp ). If we
regard N as a simple Fp G-module, the hypotheses of В. 11.5 are satisfied, and we can
therefore conclude that M centralizes N. □
We now prove that v-normalizers are CAP-subgroups.
(1.5) Theorem. Let v be a normal subgroup function for G of support n,let D = DV(E),
and let H/K be a chief factor of G. Then
(a) D either covers or avoids H/K, and
(b) D covers H/K if and only if H/K is a п-chief factor whose centralizer contains
all the subgroups v(p) ex Gp. (p e л) which avoid H/K.
Proof, (a) By (1.3) we may clearly suppose that К = 1 and hence that H is a minimal
normal p-subgroup of G. Suppose, for a contradiction, that 1 < H cxD < H. Then
pen because D is a л-group by definition, and there exists a prime qen such that
H $ Nc(v(q) n G„f where G,. e X. If H $ ?(<?). then H n v(«) = 1. and so [H, vfa)] =
1. But in this case H certainly normalizes v(</) n G,-, and therefore H < v(q}. Ifq^p,
then H is contained in the subgroup v(q) n G„ and so again normalizes it. Therefore
q = p and we can apply (1.4) to obtain H < Z(v(p)). a final contradiction.
(b) Let H/K be a л-chief factor centralized by those subgroups v(p) rx Gp. whic
avoid it. Let q e n. If v(p) n Gp- n H < K, then [H. v(p) n Gp ] < K, and so
396
V. Normalizers
(1.Л
H < Nc((v(p) GP )K)-
On the other hand, if v(p) n G„. n H $ K, then H < (v)p)c\Gp c\ H)K because
v(p)n Gp. is a cover-avoidance subgroup, and again (l./l) holds. Since H < G„K, we
conclude that H/K is contained in the v-normalizer DJI) of G = G/K and hence from
(1.3) that H < DK. In other words, D covers H/K.
Conversely, suppose now that H < DK. Then H/K is certainly a л-group since
D < G„. Let дел, G, el, and suppose that v(<j) n Gq. n H < K. Since DK (and
therefore H) normalizes (v(q) n G,)K, we have [v(g) n G,-, H] < [(v(^)n Gg)K,H] <
(v(<j) n G, )K r> H < (v(g) r> G,- r> H)K < K, in other words, v(<j) n Gg < CG(H/K).
The correspondence between a normal subgroup function v and its conjugacy
class of v-normalizers need not be one-to-one, as the following observation indicates.
(1.6) Lemma. Let v and v' be normal subgroup functions of support л for a group G.
If v(p)v'(p)/(v(p) r> v'(p)) is a p-group for all p ел, then the v- and v'-normalizers of G
coincide.
Proof. This follows directly from the definition of a normalizer and the fact that,
under the stated hypotheses, HaHp.(v(p)) = Hallp.(v(p)n v'(p)) = Hallp.(v'(p)) for all
pe л. □
2. Normalizers associated with a formation function
All groups considered in this section are soluble.
Here we use a formation function f to define a normal subgroup function v = v( f)
and we relate the properties of associated v-normalizers to f
(2.1) Definition. Let f be a formation function and let л = {p e P ..f(p) 0} (we
call л the support off). Define a normal subgroup function v = v(f) by
v(p) = Gfw
for all p 6 л. Then we write Df(L) for D,,(X) and call Df(X) the f-normalizer of G
associated with the Hall system I.
(2.2) Remarks. It follows from the definition of D,(I) and from (1.2)(b) that
(2.a)
D/L) = G„ n I П Ac(Gp. n G™)) = П (b/c(Gp. n G/Ir”) n Gp),
where G„, Gp., Gp e I.
2. Normalizers associated with a formation function
397
Therefore, if p e it and Dp = Gp n D, we clearly have
(2.Д)
(2.3) Theorem. Let f be a formation function of support it.
(a) The f-normalizers of G form a conjugacy classs of G.
(b) If D = Df(L]andN < G,thenDN/N is the f-normalizer of G/N associated with
ZN/N.
(c) An f-normalizer of G covers each f-central chief factor of G and avoids each
f-eccentric chief factor of G.
Proof Assertion (a) follows from (1.2)(a), and since (G/Nf^ = Gr,p>N/N for all p e it
by II, 2.4, Assertion (b) is a consequence of (1.3).
Let H/K be a p-chief factor of G. In proving Assertion (c), we may assume, in view
of (b), that К = 1 and hence that H is a minimal normal p-subgroup of G. First
suppose that H is /-central and hence, in particular, that p 6 л. By definition GrM <
CG(H) and so H is certainly centralized by Gfw>r,Gp.. Now let </ел\{р}, and
suppose that H n G7”1 n G,. = 1, with Gq. e I. Since H is a p-group, H < 0, (G) < G,.;
consequently = 1 and hence Grw < CG(H). Therefore by (1.5)(b) the
/-normalizer D covers H.
Next suppose that H < D. Then H is a л-group (because D g G„), and since
HnG/lplnGp<HnGp. = 1, by (1.5)(b) we have Gr,'’) n Gp- < CG(H). Since
G^(pl n Gp. 6 Hallp.(G/1,’lH), we can conclude from (1.4) that H < Z(Gr,'”H). Thus
G/,',) < Cc(H), and H is /-central in G.
Finally, since D has the cover-avoidance property by (1.5)(a), Assertion (c) now
follows. □
We can now prove that /-normalizers are precursive subgroups for the /-hypercentre.
(Precursive subgroups are defined and discussed at length in Section 5 of this chapter.)
(2.4) Theorem. Let f be a formation function, let G be a oroup, and let D be an
f-normalizer cf G. Then
П D’ = Zf(G).
geG
[Here Z/G) denotes the /-hypercentre of G. defined in IV. 6.8(b).]
Proof By (2.3)(c) every /-hypercentral normal subgroup of G is covered by each
/-normalizer of G. Thus Z/G) < Q96C D9- But AseC De is a normal subgroup of G
covered by D. By (2.3) (c) once more, it is /-hypercentral and hence lies in Zf(G).
two normal subgroups therefore coincide.
(2.5) Proposition. Let / and h be formation functions, and let G be a group,
(a) If&pf{p) = <=ph(p) for all primes p, then the f- and h-normaltzers of
(b) i//(p) — h(p) for all primes p, then
398
V. Normalizers
D/E) < D„(L)
for all Hall systems X of G.
Proof, (a) The hypothesis implies that G~'/,rn < Gf,p> r\ G1"”' and hence that
(/<p>/(G/,'’) n and G'I<'’,/(G^,'’) r> G'1'1’1) are p-groups. Lemma 1.6 now yields the
desired conclusion.
(b) If л and p denote the support of f and h respectively, then it < p and G„ < Gp
for G„, Gp e I. Ifp 6 p\it, then G„ < Gp < NtfC1"”' r> Gp-). On the other hand, ifp e it,
we have
NC(G'"” n Gp ) < Ng((G^ n Gp.) n G^} = NC(G’"P1 n Gp.)
since Grw < Gw,’>. From (2.a) we therefore conclude DffL) < Dh(L). □
(2.6) Examples, (a) If f is the formation function defined by
,f(p) = (1) for all primes p,
then the f-normalizers of a group are just the system normalizers, described in
Chapter I, Section 5.
(b) Let F be the canonical local definition of a saturated formation g = LF(F),
and as in IV, 5.19 define
/*(₽) = (G : Proj R(G) <= F(p)).
By IV, 5.19(c) the f*-central chief factors of a group G are just those chief factors
which are covered by an ^-projector of G. Therefore, according to (2.3)(c), the
f *-normalizers of G cover the same chief factors as an g-projector and avoid the rest.
The next result relates the f-normalizers of a group to those of supplements to its
Fitting group and provides an iterative procedure for their construction. It may be
regarded as the analogy for normalizers of Theorem IV, 5.16(a).
(2.7) Theorem. Let V < G and UF(G) = G. Let X* he a Hall system of U extending
to a Hall system X of G. Let f be a formation function of support it, and let D* be the
f-normalizer of U associated with I*. Then
D = D* П COp(C)(G^'nGp.)
реи
is the f-normalizer of G associated with X.
Proof. If p e it. for brevity we set
Gp = U'1” n Gp. and Gp = Gf,p' n G„-.
From (2.a) we recall that
2. Normalizers associated with a formation function
399
°* " Д(Gp n Nv(V"° and D = П (Gp
pen
By IV. 1.17(a) we have U'>P>F(G) = G^'F(G) for all p e л. and tn consequence
<2 ’’> G₽ П O.(G) = G" П 0(G)
since both are p-complements of X n Gf,p}F(G}. It therefore follows that
(2'6) G0p(C)(GO) = COp(c,(G₽).
If an element x normalizes Up, by (2,y) it normalizes (С₽П O(G))nG/,pl
С'’(ПЧ*РО,(С)^СЯ'’>) = G”; hence Gp n Л/НЛ < Gpr NC(GP), ’and therefore
D* < D. But it follows from (2.3)(b) that D*F(G)/F(G) and DF(G)/F(G) are both
/-normalizers of G/F(G) associated with X*F(G)/F(G)= XF(G)/F(G) Therefore
D*F(G) = DF(G) and D = D*(F(G) r, D).
Finally, let P e Sylp(F(G) n D) for pen. Certainly, in view of (2.Й), we have
Gop(gi(G'’) = COp(C)(G'’) < Ng(Gp) n Gp n Op(G) = P. On the other hand, [P, G*1]
Op(G) nGp = 1, and therefore P = COp(C)( О"). □
(2.8) Lemma. Let f be a formation function, and let M he a maximal subgroup of a
group G such that G/Corec(M) / Sp/(p) if p is the prime dividing [G:M[. (In
the terminology of III, 4.13 the maximal subgroup M is 2pf(p)-abnormal in G). If
X is a Hall system of G reducing into M, then D = Df(L) < M. In particular,
Nc(Gp. n G'tp>) = GPDP < M iff(p} * 0.
Procf Let К = CoreG(M) and H/K = Soc(G/K). Let D* be the /-normalizer of M
associated with X n M; then D*K/K is the /-normalizer of M/K associated with
(X n M)K/K, and DK/K is the /-normalizer of G/K associated with LK/K. Apply-
ing (2.7) to the group G/K we see that if DK D*K, then D covers H/K, and
(M/KYW c: GpK/K < CG!K(H/K) = H/K since G/K is primitive. But this implies that
(M/K)f,p} is a p-group and therefore that G/K e &„f(p}. contrary to hypothesis. Hence
DK = D*K < M. The final assertion of the lemma follows at once from (2 /7). □
(2.9) Theorem. Let f be a formation function, let M be a maximal subgroup of a group
G, and assume that G/Corec(M) / ep/(p) for all primes p. Let Y.bea Hall system of G
which reduces into M. Then
Df(Y} < Df(L n M).
(Remark: We use the obvious convention that Df(L) denotes the /-normalizer of
whatever group X is a Hall system of; thus Df(L n M) is a /-normalizer of M in the
statement of the theorem.)
Proof Let л = {q: f(q) / 0|, and let p be the prime divisor of |G: M|. By the final
statement of (2.8) we have Ng(Gp- n G/,pl) < M if p e n.
400
V. Normalizers
Let qen. Since G/Corec(M) / S,/(?) by hypothesis, we have G = Gf^M, and
therefore M/(M n G^”’) S G/Gfw e f(q\ Consequently Mfw < M n G/(e), and with
q = p we obtain
Nc(Gp. n G™) = W„(G„. n G^')
< NM(G„. n GIW n M/(pl)
= ЩСр.оМП
On the other hand, if q # p, we obtain
M n Nc(Gq- n (X'«) = NM(G,. n G'«»)
< NM(G,. n GJW n МЛч>)
= NM(GqnM'«>).
It therefore follows from (2.a) that Df(E) < Df(L r> M).
3. {^-normalizers
All groups considered in this section are soluble.
The normalizers associated with a canonical formation function f (namely, one
satisfying/(p) = Gp/(p) £ LF(f) for all primes p) have many interesting additional
properties, which we describe in this section.
(3.1) Definition. Let g be a saturated formation, and let f be an integrated local
definition of g.
(a) An /-normalizer is called an ^-normalizer of G. [We remark that this definition
does not depend on the choice of an integrated /: for if / and g are integrated local
definitions of g, then G,,/(p) = Gpg(p) by IV, 3.8(a), and then by (2.5)(a) the /- and
p-normalizers coincide.]
(b) An /-central (/-eccentric) chief factor is called ^-central (^-eccentric). [Again,
these concepts do not depend on the choice of an integrated / because by A, 13.6
normal p-subgroups always centralize p-chief factors.]
The main properties of g-normalizers are listed in the following result.
(3.2) Theorem (Carter and Hawkes [1]). Let g = LF(F) be a saturated formation,
and let D be the ^-normalizer of G associated with the Hall system Г.
(a) If D is an ^-normalizer of G, then D = D" for some geG.
(b) The Hall system T. reduces into D.
(c) If G,,eLc~ Hallp.(G) and D„ e Sylp(D), then Ng(G„. n Gr'”') = G„D = Gp.D„.
(d) If c is an epimorphism of G, then e(D) is an ^-normalizer of e(G).
3. g-normalizers
401
(e) The ^-central chief factors of G are covered and the ^-eccentric chief factors
are avoided by D; in particular, G = DGS.
(f) If H/K is a chief factor of G covered by D, then (D
of D and G/Cg(H/K) s D/Cd((D r> H)/(D r> K)).
n H)/(D r\K) is a chief factor
(g) The ^-normalizer D is in ft, and G is in ft if and only ifD~G.
(h) The order of D is the product of the orders of the ^-central chief factors of G.
(i) If £> denotes the formation of groups without ^-central chief factors, then
• (? 6 = G®. If Char(g) = P, then = (1), and in this case G is generated by
its ^-normalizers.
Proof. Assertion (a) follows from (2.3)(a), Assertion (b) from (1.2)(b) and Assertion
(c) from (2./J). Assertion (d) is a consequence of (2.3)(b), and Assertion (e) is a
restatement of (2.3)(c).
Assertion (f): If H/K is an g-central chief factor of G, then G" < CG(H/K). Hence
D covers G/CG(H/K) by the final part of (e), and Assertion (f) is then clear by A, 13.9.
Assertion (g): If we intersect D with a chief series of G, it follows from (e) and (f)
that we obtain a chief series of D, all of whose chief factors are g-central. By IV, 3.2
it follows that D e g. If G e g, then G = DG" = D by (e).
Assertion (h) is obvious from (e), and Assertion (i) also follows from (e) since D
evidently avoids all chief factors of G above <D": g e Gy. □
If 5 is a saturated formation and if £ is an g-projector of G with the cover-avoidance
property, it is easy to see that the ^-projectors of G are characterized by this property
(in the sense that if U is a subgroup which covers and avoids the same chief factors
of G as £, then U is conjugate to £). The corresponding statement for g-normalizers
is false, as the following example shows.
Example. Let S = SL(2,3), let N be the natural module for S, and let G = [A]S.
Then G has a unique chief series
1 <3 N <3 NZ(Q) oNQ<iG,
where Q = O2(S), a quaternion group of order 8. However, NQ has three conjugacy
classes of complements in G (namely the Sylow 3-subgroups of S and the conjugates
of <yn>, where <</> e Syl ,(S) and n e CK(g) is a element of order 3); clearly these all
cover and avoid the same chief factors of G, but just one of the classes (namely Sy 13 (S'))
consists of the G ,-normalizers of G. (An example to show that even system normalizers
are not characterized by their cover-avoidance properties can be found in Huppert
[5], VI, 11.12.)
The next theorem shows that among system permutable subgroups the g-
normalizers are characterized by their cover-avoidance properties.
(3 3) Theorem (Gillam [1]). Let g be a saturated formation, let U be a subgroup of
a group G, and let E be a Hall system of G. Then the following statements are equivalent-.
(a) U is the ^-normalizer of G associated with E;
402
V. Normalizers
(b) U covers the g-central chief factors, avoids the g-eccentric chief factors of G,
and permutes with L.
Proof, (a) => (b): This follows immediately from (3.2)(e) and (c).
(b) => (a): Let Char(g) = it, let F be the canonical local definition of g, and let U
be a subgroup with the stated properties. Then U has the order of an g-normalizer
of G and is, in particular, a л-group. Since U permutes with the Hall p'-subgroup Gp.
of I, we have U < Gp. when pen' and therefore U < Q {Gp-: p e л'} = G„ e E. Now
let pen, and consider the subgroup
X = VG„..
The subgroup X covers the p'-chief factors and the g-central p-chief factors of G, and
since | UGp. |p = | L/|p, it avoids the g-eccentric p-chief factors of G. Let H/K be an
g-central p-chief factor of G. Since G/Cg(H/K) e F(p) £ g, the sections G/Cg(H/K)
and H/K are covered by X. Therefore by A, 13.9 the section (H n X)/(K n X) is a
chief factor of X, and X n CG(H/K) = CX((H r> X)/(K r> X)). If T denotes the inter-
section of the centralizers of the g-central p-chief factors of G, it follows that Tn X
is the intersection of all the centralizers of the p-chief factors of X, and so T n X =
Op- p(X) by A, 13.8(a). Consequently Gf|plnX (<TnX) is p-nilpotent and
Gp. nGF(P) = Gp.n GF(P) nX < X. Hence U < Nc(Gp.r>GF,pl), and we conclude that
U < Df(X) (see Equation 2.a). Since U and Dr(E) have the same order, we conclude
that U = Df (£). □
Our next major objective is to characterize g-normalizers as the minimal “sub-g-
abnormal” subgroups of a group. [The concept of an g-abnormal maximal subgroup
is explained in IV, 5.6(c).]
(3.4) Lemma. Let g = LF(F), and let M be a maximal subgroup of a group G into
which a given Hall system L of G reduces. Then
(a) if M is g-abnormal in G, then DF(E) < Df (L n M), and
(b) M contains an g-normalizer of G if and only if M is g-abnormal in G.
Proof, (a) Since F is the canonical local definition of g, the definition of g-abnormal
implies (if M has p-power index in G) that Af/Corec(Af) f F(p) = SpF(p) (cf. IV,
5.6(c)), and it follows that G/Corec(M) f SgF(q) for all primes q. Therefore DF(E) <
OF(I n M) by (2.9).
(b) Let К = Corec(M). If M is g-normal, then G/K e g, and therefore DK = G for
any g-normalizer D of G by (3.2)(e). In this case D £ M, and in view of (a), Assertion
(b) now follows. □
(3.5) Definition. Let § be a Schunck class. A maximal subgroup M of G is called
^-critical if
(i) M is ^-abnormal in G (see 111, 4.13), and
(ii) M is critical, in other words MF(G) = G.
3. Ff-normalizers
403
(3.6) Proposition, (a) (Forster [2]). For a Schunck class § any two of the following
statements are equivalent'. J
(i) Every group in S\§ has Sy-critical maximal subgroups'
00 5 = f®ORo(§n'F);
(iii) There exists a formation X such that § = ЕфХ.
(b) If g is a saturated formation, a group G has ^-critical maximal subgroups if and
only if G J
Proof, (a) First we prove that (i)=>(ii): Let GeS>. We will show that G/O(G)e
QR0(S Ф). In so doing, we can clearly suppose without loss of generality that
<&(G) = 1, in which case by A, 10.6 we have FIG) = Soc(G) = Nt x x N„ where
< G. Let 1 < i < i, and write Nf = N, ... Ni+1 ...N,. Since <D(G) = 1, all chief
factors of G below F(G) are complemented in G; in particular, F(G)/N, has a comple-
ment, M, say, and CoreG(M,) n F(G) = Nj. It follows that F(G) r> (QtJ CoreG(M,.)) <
Al=iw; = 1 and hence that P|'_, CoreJM,.) = 1. Therefore, since G/Corec(Mf) e
£> n ад we conclude that G e R0(f> n ад and hence that
§ £ e®R0(§ г, ЗД.
Conversely, let G 6 r0(§ n ад. Then G has normal subgroups Kt,..., Kr with
trivial intersection such that each G/K-, is in § n ад Since Ф(С)К,/К( < <b(G/Ki) = 1,
it follows that Ф(С) < K-, for i = 1, ..., r, and consequently Ф(С) = 1. Let N be a
minimal normal subgroup of G. Then N £ K, for some 1 < i < r, and so NKJK, is
the unique minimal normal subgroup of the primitive group G/K,. Thus NKt =
Cg(N), and (N](G/Cc(N)) S G/K, e f>. Since all critical maximal subgroups M of G
complement some minimal normal subgroup N of G when ®(G) = 1, and since
G/Corec(M) S [N] (G/Cc(N)), it follows that G has no ^-critical maximal subgroups
and so G e § by Hypothesis (i). Hence EeQR0(§ n ад £ = f>, and we have
shown that (ii) follows from (i).
(ii) =-(iii): To see this, simply take X = QR0(§ri'Jt). which is a formation by 11,
1.18(b) and 11, 2.2.
(iii) => (i): Let G f § = ЕфХ. We must find an ^-critical maximal subgroup of G,
and to this end can again suppose that Ф(С) = 1. Let F(G) = N, x • • x N„ and let
Mj be a complement of F(G)/Ni, as in the proof of (i) => (ii). If is ^-normal for
i = 1,..., t, then G/Corec(M,) e § n £ X, and so, as above, G e r0X £ Sy, contrary
to assumption. Therefore some M, is ^-abnormal and hence ^-critical.
(b) If G ф g, then G has ft-critical maximal subgroups by Part (a). On the other
hand, if G e g, then every maximal subgroup is g-normal by IV, 5.7 and therefore G
has no g-critical subgroups. □
(3.7) Lemma. Let g = LF(F), and let M be an ^-critical maximal subgroup of G into
which a given Hall system E reduces. Then Dr (X) = DFfE n M).
Proof. By (3.4) (a) we have DF(E) < DF(E n M). By III, 6.5 the intersection of M with
a chief series of G is a chief series of M with the same automorphism groups induced
on corresponding chief factors. Because M is g-abnormal, it covers the g-central chief
404
V. Normalizers
factors of G, and therefore |Df (DI = |Df (E n M)| by (3.2)(e). Thus D,(D = D,fL n M).
□
(3.8) Proposition (Carter and Hawkes [1]). A subgroup D of G is an ^-normalizer of
a group G if and only if (i) D e g and (ii) D can be joined to G by an ^-critical maximal
chain, namely a chain of the form
(3.a) D = G, < Gr_, <••< Gj <• Go = G,
where Gt is an ^-critical maximal subgroup of G,_, for i = 1, 2,.... r.
Proof. If D is an g-normalizer of G, then D = D,(D for some Hall system Г of G. If
G e g, then D = G, and there is nothing to prove. If G f g, then G has an g-critical
maximal subgroup Gj by (3.6)(b) and Gj may be chosen so that I \ G,. By (3.7) we
have D = Dr(Yr~G,), and therefore by induction on |G| we obtain an g-critical
maximal chain joining D to G1 and hence to G.
Now let D be a subgroup of G satisfying (i) and (ii). By (3.7) an g-normalizer of
G, is an g-normalizer of G,_j for i = 1, 2,..., r. Because Deg, the subgroup D is an
g-normalizer of D = G, and is therefore an g-normalizer of every subgroup G; in the
chain (3.a), including Go = G. □
(3.9) Corollary. Let D be an ^-normalizer of a group G.
(a) D is a well-placed subgroup of G; in particular, the intersection of D with a chief
series of G is a chief series of D, and the automorphism groups induced on corresponding
chief factors are isomorphic.
(b) DegnQR0(G).
Proof. Part (a) follows from (3.8) and III, 6.6. Part (b) is a consequence of IV, 1.15(a)
and (3.2)(g). □
The concept of an g-normalizer has been generalized in various ways since it was
first studied by Carter and Hawkes [1]. One such generalization was described in
Section 1, where the defining property as the normalizer of a system of p-complements
of normal subgroups is taken as fundamental. In [2] Avino’am Mann chose the
characterization in (3.8) as his starting point and was able to extend the normalizer
concept to certain Schunck classes fj. Thus he made the following definition:
A subgroup D of a group G is called an f>-normalizer if
(i) D can be joined to G by an ^-critical maximal chain, and
(ii) D has no ^-critical maximal subgroups.
Using this definition, Mann proved analogues of many of the properties of g-
normalizers, of which the following theorem is a sample.
Theorem (Mann [2]). Let § be a Schunck class with the property that each group
not in f) has ^-critical maximal subgroups.
3. ft-normalizers
405
(a) Each group has exactly one conjugacy class of ^-normalizers
(b) If D is an ^-normalizer of G, then e(D) is an ^-normalizer of e(G) for each
epimorphism e.
(c) An ^-normalizer D has the cover-avoidance property; a complemented chief
factor H/K of G with complement M is covered by D if and only if M is 5-normal.
[However, it can happen that a group G has a pair of G-isomorphic chief factors,
one covered and the other avoided by D, in contrast to the situation for normalizers
associated with saturated formations.]
In (3.6)(a) we gave Forster’s characterization of Schunck classes § which satisfy
the hypothesis of this theorem of Mann’s; they are those of the form § = ЕФХ for
some formation J. Whereas there are many such Schunck classes which are not
saturated formations, it is also not difficult to find Schunck classes which do not have
this form (The class Qr of p-perfect groups is a example.). If § = ЕФХ with JE = qr0JE,
then 911 is a saturated formation, and Forster [2] has characterized Mann’s
normalizers as the ^-projectors of 'JiJE-normalizers. (This description follows from a
combination of (3.11) and (4.2) below).
(3.10) Theorem (Carter and Hawkes [1]). Let J be a saturated formation and G a
group. Let У denote the set of all subgroups Sof G which can be joined to G by a chain
of the form
(З.Д) S = G,< Gr l <•-•<• G, <• Go = G,
where G{ is an ^-abnormal maximal subgroup of G,_1 for i = 1,..., r. Then the minimal
elements of if are precisely the ^-normalizers of G.
Proof. If S e //, there is a chain of the form (3./J), and by (3.4)(a) an ^-normalizer of
G, contains an g-normalizer of G(1 for i = 1,..., r. Thus S contains an g-normalizcr
of G. Since the g-normalizers of G belong to //’ by (3.8), the desired conclusion follows.
□
(3.11) Theorem (Carter and Hawkes [1]). Let g( and g2 be saturated formations with
g, E g2, and let X be a Hall system of a group G. If D2 is the g2-normalizer of G
associated with E and if Dt is the ^-normalizer of If associated with LnD2, then D,
is the (^-normalizer of G associated with E
Proof. Since gj E g2, every g2-critical maximal subgroup of G is also gccritical.
By (3.8) the g2-normalizer O2 can be joined to G by a g2-cntical maximal chain, an
D. can be joined to D2 by an g, -critical maximal chain, with £ reducing into bot
chains. Therefore Dt can be joined to G by an g.-critical maximal chain mto which
T. reduces, and since Dt e g,, we conclude from (3.8) that Dt is the g,-normalizer^
G associated with E
In Section 5 of Chapter VII we shall study a partial order « on saturated
formations defined by: g, « g2 if and only if for all G e G an giprojector
contained in some g2-projector of G. The preceding theorem shows that the partia
406
V. Normalizers
order similarly defined with normalizers instead of projectors coincides with the
ordinary partial order of inclusion between classes.
(3.12) Lemma. Let g, and g2 be saturated formations. If G e $2 a'id if D is an
^-normalizer of G, then D is an g, n ft 2-normalizer of G.
Proof. By (3.9)(b) we have Deg,ng2. Since an 5,-critical maximal chain is
obviously g, n g2-critical, Proposition 3.8 yields the desired conclusion. □
(3.13) Proposition. Let g, = LF(Ff for i = 1, 2, and let D/ = DFfL) for some Hall
system Lof a group G. Then D, n D2 is the g, n ^-normalizer of G associated with
L and for {i, j} = {1,2} also the ^/-normalizer of D} associated with LnDj.
Proof. Let D* be the g2-normalizer of Dr associated with LnOj. By (3.12) the
subgroup D* is the g, n g2-normalizer of D, associated with LnDj and is therefore
the g, n g2-normalizer of G associated with L by (3.11). Similarly, if D+ is the
gi n g2-normalizer of D2 associated with T. n D2, then D+ is the g, n g2-normalizer
of G associated with L. Therefore D* — D+ < D, n D2, and, in particular, D, n D2
covers all the g1 r> g2-central chief factors of G. Since the g, r> g2-eccentric chief
factors of G are gp and g2-eccentric, these are avoided by r> D2, and therefore
by order considerations D* = D+ = n D2. □
We show next that certain properties of g-normalizers may fail to hold for f-
normalizers when f is a non-integrated local definition of g.
(3.14) Examples, (a) To show that an /-normalizer need not be contained in LF( f),
define
/(p) =
(1)
• U
QR0(Sym(4))
for pf {2,5}
if p = 2
if p = 5.
Let G = Z5 Qj Sym(4), where the wreath product is taken with respect to the natural
permutation representation of Sym(4). All the chief factors of G, apart from those of
order 3, are obviously /-central, and so by (2.3)(c) the /-normalizers of G are just the
Hall {2,5}-subgroups of G. Hence, if В is the base group of G and if P e Syl2(G),
then BP is an /-normalizer of G. Since В = O5. 5(BP), we have BP f LF(f): for
P = BP/05. 5(BP) is dihedral of order 8, whereas /(5) n G2 is easily seen to consist of
just elementary abelian 2-groups.
(b) For each prime p let /(p) denote the formation consisting of all groups whose
Carter subgroups are p-groups. If G = Sym(4) and P e Syl2(G), then P is an
/-normalizer of G. If V denotes the normal four subgroup of G, then V is a minimal
normal subgroup of G but not of V. Thus, if D is an /-normalizer of G, it does not
necessary hold that (D n H)/(D r> K) is a chief factor of D when H/K is a chief factor
of G, and so /-normalizers are not in general well-placed subgroups.
3. Л-normalizers
407
If g is a local formation, we have already introduced three distinguished local
definitions for g:
(i) The smallest local definition /(cf. IV, 3.9(b));
(ii) The canonical local definition F, uniquely determined by the requirement
topf (p) = F(p) £ for ali primes p;
(iii) The local definition /* which describes exactly which chief factors are covered
by an ^-projector of G (cf. IV. 5.19).
We end this section with a description of yet another local definition f which turns
out to be a unique upper bound for all local definitions. We will refer to it informally
as the “largest” local definition.
(3.15) Definition. Let 5 and ® be formations with g saturated. Denote by gt6 the
class of all (soluble) groups whose g-normalizers belong to ®.
(3.16) Lemma. Let g and ® be formations with g = r.og.
(a) gi,-, is a formation:
(b) ® s gK;
(c) A formation X is contained in gE if and only if X n g £ g.
Proof. Assertion (a) follows easily from the epimorphism-invariance of g-normalizers.
If D is an g-normalizer of a group G in ®, then D e qr0(G) £ ® by (3.9)(b), and so
Assertion (b) is clear.
To prove Assertion (c) we observe that if X £ gp,, then X n g £ gE n g £ ffi.
Conversely, suppose that X n g £ ffi. Since gj = g(Ir,m by (3.2)(g), appealing toPart
(b) of this lemma, we have
I — g(ir^) — g<S-
(3.17) Definition. Let F be the canonical local definition of a saturated formation g.
Then, in the notation of (3.15), define
/(p) = gf(Pi
for all primes p, and note that У is a formation function by (3.16)(a).
The following theorem shows that f is the “largest” local definition of g.
(3.18) Theorem (Doerk [3]). Let % be a saturated formation, and let hbea formation
function. Then g = LF(h) if and only if
(З.У)
/(p)£h(p)£/(p)
for all primes p (where f and f are the formation functions defined in IV, 3.9(b) and in
(3.17) respectively).
Proof. Let F be the canonical local definition of g. IfP Ф W
gf(p) = 0- Suppose that p e Char(g). By (3.16)(c)the inclusion 6ph(p)n g - b(P)is
408
V. Normalizers
equivalent to the inclusion G,,h(p) S /(p). Therefore by IV, 3.8(a) the formation
function h locally defines g if and only if
(3.<5) F(p) £ <2,Л(р) s f(p)
for all p e Char(g). Since F(p) = GpF(p), we have gF(p) = GpgP(p) and so the inclusion
toph(p) £ f(p) is equivalent to the inclusion h(p) £ f(p). Finally, since Gp /(p) = F(p)
by IV, 3.8(a), the inclusion F(p) £ Gph(p) is equivalent to the inclusion f(p) £ h(p).
Thus (3.<5) is equivalent to (3.y). □
(3.19) Example. Let f be a formation function. Any two of the following statements
are equivalent.
(a) LF(f) = 91;
(b) 0 # /(p) £ 91 for all p e P;
(c) 0 # /(p) £ C for all p e P;
(d) For all p e IP, the formation /(p) is non-empty and comprises groups with no
central p'-chief factors;
(e) For all p e IP either /(p) = (1) or Char(/(p)) = p.
To justify the above assertion, first note that the equivalence of (a) and (b) follows
from (3.18). Next we prove that (b) =>(c). If D is a system normalizer (that is to say,
an 9l-normalizer) of G e 9l3 , then D is a p-group, and hence so is G/G' since G = G'D.
Thus G is p'-perfect. The implication: (c) => (d) follows from IV, 1.5, and the implica-
tion: (d) =» (e) is trivial. Finally, to see that (e) =» (a), note that (e) implies that a group
G in /(p) contains no central p'-chief factors and so by (3.2)(g) a system normalizer
of G is a p-group.
Exercises
1. Use (3.3) to give another proof of (3.11).
2. A chief factor H/K of a group G is called ^-critical if H/K is g-eccentric and if
every chief factor of G below К is either g-central or Frattini. Show that a
maximal subgroup of G is g-critical if and only if it complements an g-critical
chief factor of G.
3. Find formations g and (S with g saturated, such that (5 g | C5. Thus (3.16)(b)
fails when g-normalizer is replaced by g-projector in the definition of gE. Show
that it also fails when /‘-normalizer is used in this definition.
4. Let g and (f> be formations with g saturated. If the g-projectors of a group G
belong to (5, prove that the g-normalizers of G also belong to (S.
4. Connections between normalizers and projectors
All groups considered in this section are soluble.
The fundamental connection is that every g-normalizer is contained in an
g-projector.
4. Connections between normalizers and projectors
409
(4.1) Theorem (Carter and Hawkes [1]). Let g be a saturated formation, and let G
be « finite soluble group Then each ^-projector of G contains an ^-normalizer of G
and eat h ^-normalizer of G is contained in an ^-projector of G.
Proof. By IV, 5.11 an g-projector S of G can be joined to G by an g-abnormal
maximal chain (i.e. one of the form (З.Д)). Therefore S contains an g-normalizer of
G by (3.10). Since the g-normalizers of G form a conjugacy class of G, the rest of the
theorem is obvious.
(4.2) Theorem (Carter and Hawkes [1]). Let g be a saturated formation, and let
G e 9lg. Then the ^-normalizers and ^-projectors of G coincide.
Proof. Let E e ProjB(G). In a maximal chain
(4.a)
E = Gr <
<G, <G0 = G,
each G, is g-abnormal in G,-, by IV, 5.11. Since G e 'Jig, we have ££(G) = G; it
follows that Gj is a critical subgroup of G,_j for i = 1, 2,..., r and hence that (4.a) is
an g-critical chain. Since £ e g, we conclude from (3.8) that £ is an g-normalizer of
G, and the statement of the theorem is then clear. □
The next result shows that it is possible to define g-normalizers entirely in terms
of projectors. It plays an important part in the development of another generalization
of normalizers considered in the next section.
(4.3) Theorem. Let g be a saturated formation.
(a) The ^-projectors of the SHfi-normalizers cf a group G are precisely the g-
normalizers of G.
(b) Let G e 9lrg. Set G = G0 and let Gj be an 91(r“‘^-projector of С;_, for i = 1,
..., r. Then Gr is an ^-normalizer of G.
Proof, (a) Let £ be an 91g-normalizer of G, and let D be an g-projector of £. Since
£ e 9lg by (3.2) (g), the subgroup D is an g-normalizer of £ by (4.2) and it is therefore
an g-normalizer of G by (3.11).
(b) By Part (a) the subgroup G, is an 91(r_,,g-normalizer of G for i = 1, 2,..., r
(4.4) Remarks. Theorem 4.3(b) offers another point of departure for defining ft-
normalizers when § is a Schunck class. Since ^-projectors always exist in the universe
S, we have to find a substitute for the class 'Jift. (Recall from III, 5.13 that 9Iftis a
Schunck class if and only if § is a saturated formation.) One possible approach is to
define, for a Schunck class ft, a new class 91[ft] as the Schunck class generated by
ft vj b(ft) If ft happens to be a saturated formation, it turns out that 91 [ft] coma es
with the usual class product 91ft. We then define the Schunck class 9Г[ft] inductively
by
9T[ft] = 9l[91'-1[ft]],
410
V. Normalizers
and if G g 9F[§] we set Go = G and define Gf to be an 9F-1 [^/projector of G,_, for
i = 1,..., r. Then the ^-normalizers are defined to be the subgroups Gr of G obtained
in this way. It is then clear that these “^-normalizers” of G form a characteristic
conjugacy class of ^-subgroups of G, invariant under epimorphisms. Moreover,
if every group in <5\b> possesses ^-critical maximal subgroups, then these “bi-
normal izers” coincide with those defined by Mann described in Section 3.
(4.5) Example. Let G = Z5 rlinal Sym(4), the example analysed in detail in III, 4.34.
Let g = LF(F), where F(2) = S2, ^(3) = ®з> ^(5) = Ss^(2) and P(p) = 0 for
p > 5. (Here, 91(2) denotes the class of elementary abelian 2-groups.). Then G = BS,
where В is the base group of order 54 and S S Sym(4). Since g n S(2 3) с 9), an
g-normalizer of S is a system normalizer and has order 2. Therefore G/B e /(5) = gF(5)
in the notation of (3.15), and so В is /-hypercentral and is contained in an /-normalizer
of G by (2.3)(c). Now let P e Syl2(S). Then P Dih(8) and P, as a Carter subgroup
of S, is an g-projector of S. As we showed in III, 4.34, the FsP-module BP decomposes
thus:
BP S N, ф (V2 ф JV2*,
where P/Ker(P on (V;) 6 91(2) for i — 1,2, and where N* is irreducible and faithful for
P. It follows that NtN2P is an g-projector of G, and hence that no /-normalizer is
contained in an g-projector of G.
We shall now determine the formation functions h for which h-normalizers are
always contained in g-projectors, and for this purpose we begin with the following
lemma.
(4.6) Lemma. Let g be a saturated formation, and let f* be its local definition of the
form described in IV, 5.19. Further, let hbea formation function with h(p) £ /*(p) for
all primes p. Let E be an ^-projector of a group G, let E < U < G, and let Ebe a Hall
system of G which reduces into U. Then Dh(E) < Df-(E n U).
Proof. By (2.5)(b) it will suffice to show that DF»(E) < DF»(E n U), and by using an
induction argument we can assume that U is a maximal subgroup of G, of p-power
index say. Since £ < U, we deduce from IV, 5.11 that U/CoreG(U) £ f*(p) and then
appeal to (2.8) to conclude that < V. Let p 6 Char(g), and set R = Gr*w. Then
ER/R is an g-projector of G/R e f*(p) and so ER/R e F(p). Since ER/R is also an
g-projector of UR/R, we have UR/R ef*(p) and hence < R. Let Gp- be the
Hall p'-subgroup in Z. Since D = DF«(E) < U and D normalizes R n Gp., then D also
normalizes
R n Gp. n IV*"” = Gp. n
and so D < n U). □
(4.7) Theorem. Let g = ££(/*) be as in (4.6), and let hbe a formation function. Then
the following statements are equivalent :
4. Connections between normalizers and projectors
(a) For each GeQ,an h-normalizer of G is contained in an
(b) For all primes p we have h(p} c f *(p).
'S-projector of G;
Proof, (b) =» (a): If we take U - E in (4.6), we obtain ЩЯ < D,.(E r> 17) -
E e ProjR(C). M
(a) => (b): Assume that h(p) f*(p} for some prime p, and choose a group G of
minimal order in h(p)\f *(p). Then G has a unique minimal normal subgroup N, and
since ep/*(p) = f *(p), we know that N is a p'-group. By B, 10.7 there exists a faithful
irreducible G-module M over F„, and if H denotes the semidirect product [M]G. the
p-chief factor M is h-central and is therefore contained in each h-normalizer of G by
(2.3)(c). Since M is f *-eccentric, by IV, 5.19(c) an g-projector of G does not contain
M, and we conclude that an h-normalizer is contained in no g-projector of G. □
If F is the canonical local definition of g, then certainly F(p) c f *(p) for all primes
p, and so Theorem 4.7 gives another proof of Theorem 4.1.
In (3.14) (a) we gave an example to show that an f-normalizer need not be contained
in g = LF(f}. It is an open question whether an f-normalizer which is contained in
an g-projector necessarily belongs to g; also whether an f *-normalizer is necessarily
in g. In a positive direction we have the following:
(4.8) Proposition. Let hbe a formation function, and assume that h(p) = sh(p) for all
primes p. Then each h-normalizer of a group is contained in some LF(h}-projector and
also belongs to LF(h).
Proof. By IV, 5.19(b) we have h(p) £ f*(p) for all primes p, and so by (4.7) an
h-normalizer D is contained in an LF(h)-projector E of a group G. Since h is s-closed,
so also is LF(h) by IV, 3.14 and therefore D, like E, belongs to LF(h). □
Theorem 4.7 presents us with another characterization of projectors which are
CAP-subgroups.
(4.9) Theorem. An %-projector Eof a group G has the cover-avoidance property if and
only if E is an f*-normalizer of G.
Proof. If E is an f *-normalizer, then E has the cover-avoidance property by (2.2)(c).
Conversely, if E has the cover-avoidance property, by IV, 5.19(c) it covers the same
chief factors and therefore has the same order as an f *-normahzer of G. Since E
contains an f *-normalizer of G by (4.7), it is itself an f *-normalizer of G. □
(4.10) Theorem. Let g be a saturated formation of characteristic n, and let *_&п(Яе
the formation of all groups in which ^-normalizers are ^-projectors. Let G e JU ana
let D be an ^-normalizer of G. tnHrhi
(a) (Carter and Hawkes [1], D’Arcy [2]). The subgroup E = COpt0)(U ))
is the unique ^-projector of G containing D.
(b) (Hawkes [3]). If D is ^-subnormal in a subgroup L of G, then L _
412
V. Normalizers
Proof, (a) By (4.1) there is an g-projector £ of G containing D, and since G 6 911',
we have DF(G) = EF(G). Then by IV, 5.16 the subgroup E has the form stated in
Part (a) of the theorem.
(b) Since DF(G)/F(G) is an g-projector of G/F(G) by hypothesis. DF(G) is “sub-
g-abnormal" in £F(G) by IV, 5.11. If D is g-subnormal in L, then DF(G) is g-
subnormal in £F(G); consequently DF(G) = LF(G). Since £ = (£ ci F(G))£> with
£n F(G)a nilpotent normal subgroup of £, the definition of “g-subnormal” implies
the existence of a chain of subgroups
D = Do < • • < D„ = E
such that D,_, is/-normal and critical in D; (i = 1,..., n). By IV, 5.14(b) it follows
that £ c g and hence by IV, 5.15 that £ < £. □
(4.11) Theorem (D’Arcy [2]). Let g = LF(F] be a saturated formation of characteris-
tic n. Let G g 9lrg, set Go = G, and let Gt be an <iV'~,}’^-projector of G^t for i = 1, 2,
..., r. Let D = Gr (an ^-normalizer of G by (4.3)(b)), and set
(i) E,_t = Er = D, and
(ii) Ek = Ek+i(X„e,COptGk,(E^>))for 0 < к < r — 2.
Finally set E = Eo. Then
(a) Ek is an ^-projector of Gk for к = 0, 1,..., r, and
(b) EknGk+1 = Ek+1 for к = 0, ...,r — 1.
In particular, E is an ^-projector of G.
Proof. Since Gr = Er = D, Assertion (a) holds for к -= r. Suppose, inductively, that
we have shown that £k+1 is an g-projector of Gk+1 for some 0 < к < r. By definition
of Gk+1 we have Gk+lF(Gk) — Gk, and so Ek is an g-projector of Gk by IV, 5.16(a);
therefore Assertion (a) holds by reverse induction on r.
Furthermore, for a given к e {0, 1,..., r — 1} and p e л, let Kp denote the centralizer
ofE™ in Op(Gk\ and set К = XpenKp. Then
Ek Qk + 1 — Ek+i К ^k + 1 ~ Ek+i(K Ci Gk+1).
Applying IV, 5.16(a) to Ek+1(F(Gk)n Gk+1), we see that К n Gk+1 is contained in Ek+1,
and so Assertion (b) also holds. □
(4.12) Theorem (Hawkes [3J). Let E be a Hall system of a group G. If E is the
'^-projector of G into which E reduces, then the ft-normalizer D = DF(E) is an g-
subnormal subgroup of E.
Proof. Let G e9Fg, and for i = 1,..., r let G, be the (unique) 9i(r,|g-projector of
G,_, into which E ri G, , reduces. By (4.3)(b) the subgroup G, is the 91(r-,,g-normalizer
of G associated with £, and, in particular, G, = ЙД(Е).
Form the subgroups Ek (0 < к < r) described in the statement of Theorem 4.11,
and note that £ reduces into each Ek by construction. By IV, 5.16(b) each £k+1 is
4. Connections between normalizers and projectors 413
g-subnormal in Ek, and therefore E,( = Gt = DB(E)) is ^-subnormal tn £0, which by
(4.1 l)(a) is an g-projector of G, in fact the unique such into which L reduces □
!t?p^Gm™ 5 = LF(/)’ Ond E ^-pr°'eCtOr °f ° G-IfEe f(P),
Proof. We argue by induction on |G|. If G e g. then E = G, and the statement is true
Suppose G f g, and let G*/R be a chief factor of G. Since E covers G/GB, we have
G/G' 6 Qf(p) = /(p), and therefore G^/R, which is /-eccentric, is a p'-group Now E
avoids G"/R, and therefore |G:ER| = |G»/K|. Since £e ProjB(E£), by induction
\ER : £| is a p'-number; hence so is |G: £|. n
(4.14) Theorem (Hawkes [3]). Let g = LF(F) be a saturated formation of charac-
teristic n, and let D be an ^-normalizer of a group G associated with a Hall system X.
Then D is an ^-projector of G if and only if CP(Dr,t’>) < D for all pen, where P denotes
the Sylow p-subgroup in E.
Proof. Let D 6 ProjB(G) and let pen. Set N = Nc(DF<pl). Then D/DFW is an g-
projector of N/Dfw and belongs to F(p). Therefore by (4.13) the index |N : D| is a
p’-number, and it follows that NP(DF,r,f < D.
Conversely, suppose that CP(DFW) < D for all pen. Then applying Theorem
4.11 (a) repeatedly for к = r — 2, r — 3,..., 2, 1, 0, we obtain, in the notation of that
theorem,
D — Er — Er-i — • • • — Et+1 — E* — • — Eo,
where £0 is an g-projector of G by the final statement of that theorem. □
Exercises
1. (Carter and Hawkes [1]). If G 6 91g, and if a subgroup H of G covers all the
g-central factors of a given chief series of G, then H contains an g-normalizer of
G. If H furthermore avoids the g-eccentric factors of this series, then H is itself
an g-normalizer of G.
2. (Carter and Hawkes [1 ]). Let G 6 9lg, and let H be an g-subgroup of G satisfying
HF(G) = G. Then NC(H) is contained in an g-projector of G.
3. (Carter and Hawkes [1]). If G 6 9l2g, an g-normalizer of G is contained in a
unique g-projector of G.
4. (Carter and Hawkes [1]). Let D be an g-normalizer of a group G e Ji g, and
suppose that D < Ее ProjB(G). Then NG(D) < E.
5. (Alperin [1]). Let D, and D2 be system normalizers of G, both contained in a
Carter subgroup C of G. Then D2 = DJ for some x e C.
6. (Carter and Hawkes [1]). Let D, and D2 be g-normalizers of a group Ge 91 g,
and suppose that both are contained in an g-projector E of G. Then V2
some ,v e £. (For an example of a group G and a saturated formation g for which
this conclusion fails, see Hawkes [4]).
414
V. Normalizers
7. Let g) and Sr be saturated formations, and let G be a group for which РгорЛ1(6) =
ProjR,(G). Then the ft,- and ft2-normalizers of G coincide. In general, however,
the coincidence of the normalizers does not imply the coincidence of the corre-
sponding projectors.
8. (Carter and Hawkes [1]). The intersection of a supersoluble-normalizer of a
group with a suitable Carter subgroup is a system normalizer.
9. (Doerk and Hawkes [1]). Let ft be a saturated formation, and let 9)я denote the
class of all groups in which the g-normalizers coincide with the ft-projectors.
(a) The class 9) й is a formation.
(b) If F is the canonical local definition of ft, then the canonical local definition
У of the saturation 9)я of 9)B is given by
V(P) =
ft
for F(p) ft, and
for F(p) = ft.
(c) The class 9)я is saturated if and only if there exists a prime p for which
8 = SpF(P)-
10. (Doerk [2]). (a) The class 9lg is the largest saturated formation contained in the
class 9) я described in Question 9.
(b) If § is another saturated formation, and if 9)s != 9)$, then ft <= £>. However,
in general it cannot be deduced from g£ that 9)ff £ 9)6.
11. (Doerk [1]). Let ft = 91§, where § is a non-empty formation. If a group G has
p-length 1 for all primes p and if the ft-projectors of G are САР-sub-groups, then
Ge9)s.
12. If ft = 91§ as in the previous question, and if the g-normalizers and g-projectors
of G are Hall subgroups, then G e '1)5.
13. (Alperin, Thompson; see Huppert [5], VI, 13.7). Any finite soluble group can be
embedded as a subgroup of a group in ч2) ,л (i.e. 59),,= <5).
5. Precursive subgroups
All groups considered in this section are soluble.
(5.1) Definitions, (a) A mapping r which associates with every group G a subgroup
r(G) satisfying
(5.a)
r(G) = 0(r(G))
for every isomorphism в: G -» G is called a characteristic subgroup function. (It is clear
from this definition that r(G) is a characteristic subgroup of G. On the other hand, if
a characteristic subgroup r(G) of G is defined for just one representative G of each
isomorphism class (G), and if (5.a) is used to define r(G) for all G e (G), it is not difficult
to verify that r is a well-defined characteristic subgroup function.)
(b) A subgroup U of G is called t-precursive (or simply precursive when r is
understood) if the following condition holds
415
(5-P)
5. Precursive subgroups
r(e(G)) = Corer(G)(c(l/))
for all epimorphisms e: G -»c(G).
The functions т = O„ and t = Z„ are obvious examples of characteristic subgroup
functions which have associated precursive subgroups; for the Hall л-subgroups of
a group are O„-precursive, and by I, 5.9(b) the system normalizers of a group are
Z,„-precursive.
Our treatment here of the question: “For which characteristic subgroup functions
r does every group have r-precursive subgroups?” is based on an unpublished
manuscript of Fischer. There he handles the special case where the fixed, but un-
specified, saturated formation g that is built into our treatment coincides with 91 or
with (1), and he is able to characterize the functions r for which precursive subgroups
of a special form always exist. This theory includes functions of the form r = ZR
(O„ and Zx are special cases), and also r = Ф, the Frattini subgroup function, but does
not include the functions т defined by r(G) = G*, the residual of some formation jE.
It would be of interest to have a description of all functions r with precursive
subgroups.
For our treatment we first need the concept of an g-chain of a group.
(5.2) Definitions. Let g be a saturated formation. For each group G let m = m(G)
denote the smallest integer such that G6 9lmg; thus m = 0 if G e g, otherwise
G 6 9lmg\9lm-1g. A chain of subgroups
f:G = Gm>Gm_1>
> G, > Go > 1
is called an g-c/ioin if Gi is an 91'g-projector of Gj+1 for i — 0, 1,..., m — 1. We say
that a Hall system of G reduces into I if it reduces into each subgroup of the chain.
Remark. Since G ^9lm-1g, the Э!”’1 g-projector Gm_j is a proper subgroup of G.
Since Gm_j covers the quotient G/G41 ®, it follows that Gm_j e 91” g\91 g
and hence by induction that the proper inclusions indicated in the g-chain t are
justified.
(5.3) Lemina. Let t:G = Gm> - >GO> 1 bean %-chainofG
(a) If e: G -> e(G) is an epimorphism, then e(t) is an fi-chain of c(G) (after deletion
of the redundant terms in e(t)).
(b) If t* is an %-chain of G, then t* = I9 for some geG.
(c) Gj is an 91‘^-normalizer of G for i = 0, 1, ...,m.
Proof, (a) The epimorphism-invariance of projectors ensures that e(GJ1 is> an 91g-
proiector of e(G +1). Of course, m(s(G)) may be smaller than m(G) so tha the first
m(G) - m(c(G)) lerms of c(t), all of which equal e(G), have to be deleted to obtain
- > GS > 1. By the conjugacy of projectors G*-i=G^'-i for
some 91 g G. Thus G*_. >° '> 1 and G9L. > - > G^ 1 are g-cha.ns of
416
V. Normalizers
G* ,, and by induction on | G| we can find an element g2 e G*_, such that Gf = G?'"1
for i = 0,..., m — 1. Therefore f9192 = t*.
(c) This was proved in (4.3)(b). □
(5.4) Definitions. Let 5 be a saturated formation and r a characteristic subgroup
function.
(a) If t: G = Gm > Gm_t > > Go > 1 is an g-chain of a group G, we define
r(t)= r(Gm)r(Gm_1)... r(G1)r(G0).
(Since t(G,) normalizes r(Gl+1), r(G,+2),..., and r(Gm). the product r(t)is a subgroup
of G.)
(b) A subgroup U of a group G is called an (g, r)-precursive subgroup if and only if
(i) U = r(t) for some g-chain I of G, and
(ii) U is r-precursive (in the sense of (5.1)(b)).
(c) A chief factor H/K of G is called т-central if H/K < t(G/K) and т-eccentric
otherwise.
(5.5) Proposition. Assume that a group G has (g, r)-precursive subgroups. Then they
form a characteristic conjugacy class of G, cover the т-central chief factors, and avoid
the т-eccentric chief factors of G.
Proof. Let I be an g-chain of G, and let в e Aut(G). Then 0(f) is an g-chain of G,
and so 0(f) = t9 for some g 6 G by (5.3). Thus, applying (5.a) twice, we obtain
0(r(f)) = T(0(t)) = r(f9) = r(t)9.
Since by (5.3)(b) the group G acts transitively by conjugation on the set of g-chains
t, it also acts transitively on the subgroups r(t), and so these form a characteristic
conjugacy class of G.
If H/K is a chief factor of G, the stated cover-avoidance property for V = r(f)
follows from (5./J) with e taken to be the natural homomorphism from G to G/K.
□
(5.6) Lemma. Let G £ g and let t: G„ > Gra l > • be an %-chain of G. If Gml <
M < G, then M is ^-critical in G (in the sense of (3.5)).
Proof. Since m > 1 and Gm , covers the g-residual quotient of G, it follows that
G/Coreo(M) f: g. Therefore M is g-abnormal in G. Since Gm_lF(G) = G because
G 6 91mg, we have MF(G) = G, and so M is a critical maximal subgroup of G. □
We now justify the first of the examples mentioned at the beginning of this section.
(5.7) Example. Let g be a saturated formation, and for each group G define
r(G) = ZK(G),
5 Precursive subgroups
417
the g-hypercentre of G. Clearly
that if t is an g-chain of G, then
(g, r)-precursive subgroup.
t is a characteristic subgroup function. We assert
r(t) is an g-normalizer of G and is consequently an
We will prove this assertion by induction on |G|. If G e g, then t(I) = G which is
an g-normalizer of G. Therefore suppose that m = m(G) > 1, and let
t:G>G„ _,> >GO>1
be an g-chain of G. By induction r(t n G^-,) is an g-normalizer of Gm r Since by
(3.8) and (4.3)(b) there is a chain of g-critical maximal subgroups joining Ст_, to G,
it follows from (3.11) that r(t n Gm,) is an g-normalizer of G. Thus t(G) < tllnG _,)
by (2.4), and so
r(I) = r(G)r(!nGm_1) = t(lnGm J.
We have therefore shown that r(t) is an g-normalizer of G, as asserted. Finally we
observe that by (2.3) (b) and (2.4) an g-normalizer U of G satisfies (5.0) when r = Zs,
and so g-normalizers are (g,Zs)-precursive subgroups.
The next result yields a sufficient condition for a precursive subgroup to be
preserved under epimorphisms.
(5.8) Proposition. Let r be a characteristic subgroup function, and let
t:G=Gm>G„_1>- >GO>1
be an fi-chain of G. Assume that is precursive in G, for i = 0, 1, ..., m.
Then e(r(f)) = r(e(t)) for all epimorphisms r. of G. and e(r(I)) is precursive in e(G).
Proof. Let в: G -> 0(G) be an isomorphism. Since projectors form a characteristic
conjugacy class, clearly 0(t) is an g-chain of 0(G). Moreover, r(0(t)) = 0(r(f)) because
r satisfies (5.a). In particular therefore, by the isomorphism theorem we can restrict
attention to natural epimorphisms e: G -► G/N when proving this proposition.
We will argue by induction on |G|. Let 1 / N < G, and write U = Gm_,, the
91m-1 g-projector of G. Since U < G and since the hypotheses of the proposition are
satisfied for the g-chain t n U of U, by induction we have
r((I n U)N/N) = r(t n U)N/N.
Since by hypothesis r(t) is precursive in G, from (5.0) we have r(G/N) =
Corec/N(r(t)fV/fV). Thus
tfJN/N) = v(G/N)v((W/N}r>(UN/N))
= Corec/f,(r(G)r(t n G)N//V)r((t n U)N/N)
= (r(G)N/N)r((t n U)N/N)
418
V, Normalizers
= (r(G)7V/7V)r(t n U)N/N
= r(t)N/N.
For the natural epimorphism a: G -> G/N. we have shown that r(e(t)) = e(r(t)).
It now follows easily that e(r(I)) is precursive in e(G). For let K/N <! G/N. By
hypothesis r(G/K) = CoreG/K(r(t)K/K), and therefore by applying the standard iso-
morphism from G/K to (G/N)/(K/N) we see that
t((G/N)/(K/N)) = CoretoimnKINi((t(l)K/N)/(K/N))
= Соге£(С),с(К/е(г(1))е(К)/е(К)).
Thus e(r(f))) is a precursive subgroup of e(G). □
(5.9) Corollary. If every group has (8, t)-precursive subgroups, they are preserved
under epimorphisms.
We are interested in conditions satisfied by r and 8 which guarantee the existence
of (8. t)-precursive subgroups. The next results yields three necessary conditions.
(5.10) Theorem (Fischer). Assume that every group in G has (8. tfprecursive sub-
groups. Then for all G 6 G the following conditions are satisfied.
WD1: r(G/r(G))= 1.
WD2: If N < G, then t(G)N/N < t(G/N) with equality when G e 8-
WD3: If U is an 91m-1 ^-projector of a group G in 9lm8\9lm-18 and if r(G) = 1>
then CoreF(G)(r(l/) n F(G)) = 1.
Proof. Let T = r(t), where 1 is an g-chain containing U. By Property (5./J) we have
r(G) = CoreG(T), and therefore r(G/r(G)) = CoreG/t(G)(T/r(G)) = r(G)/r(G). Thus
WD1 holds.
If N <! G, by (5./J) once more we have r(G)N/N = Corec(T)N/N <
Coreclfl(TN/N) = r(G/N); furthermore, if G e g, then T = r(I) = r(G), and therefore
r(G/N) = CoreG/N(T(G)N/N) = t(G)N/N. Thus WD2 is satisfied.
Finally we consider Condition WD3, first noting that G = UF(G). Then by (5./J)
we have 1 = CoreG(T) > Corelf (o|(r((7)) = П„Г(0|г(С)х > Coref(C)(r(G) n F(G)),
and so WD3 holds. □
By adding one further condition, we obtain a set which is sufficient to ensure the
universal existence of (8, r)-precursive subgroups.
(5.11) Definition. A characteristic subgroup function r is said to be well disposed for
8 if it satisfies Conditions WD1, WD2, WD3, and the following additional condition:
WD4: If G 6 91'"g\91'"18 with r(G) = 1, and if Gm_, < M < G for some 8-chain
t:G = Gm>G„_1> >GO>1,
5. Precursive subgroups
419
then
(i) t(M) < r(t), and
(ii) r(G/N) < z(M/N) for each minimal
normal subgroup N of G contained in M.
(5 12) Theorem (Fischer). Let g be a saturated formation, and let r be a characteristic
subgroup junction which is well disposed for g. Then every group has (g, rfprecursive
subgroups.
Proof. Let
t:G = Gm>Gm _,>
> Gj > G0 > 1
be an g-chain of G, set V = r(t) and Ц = r(t n Gm ,). We aim to show that V is
precursive in G. Arguing by contradiction, we suppose that V is not precursive in G,
but that for all groups X with |X| < |G|, a subgroup of the form r(t*) is precursive
in X for each g-chain I* of X. If N < G e g, then V = r(G) and VN/N = c(G/N) by
WD2. Therefore we can assume that G j g, and, in particular, that the subgroup
U = Gm_, is a proper subgroup of G.
Consider the possibility that CoreG(F) = r(G). Since by assumption V is not
precursive in G, there exists a non-trivial normal subgroup К of G such that
t(G/K) / CoreG/K(CK/K).
If N is a minimal normal subgroup of G contained in K, it follows from the well-
known isomorphism between (G/N)/(K/N) and G/K that
(5.y) VN/N is not an (g, r)-precursive subgroup of G/N
in this eventuality.
We first suppose that r(G) = 1, and claim that in this case CoreJF) = 1. For, if
r(G) = 1, then V = Ц = r(t n U), and since the theorem is >rue for the g-chain I n V
of U, we have Corel;(F) = But UF(G) = G, and so
CoreG(P)= A t(Uy.
xeFiG)
Therefore, were Corec(L) non-trivial, we should obtain
Coref,G)(r(C) n F(G)) = CoreG(V) n F(G) * 1,
contrary to the hypothesis that WD3 holds. Therefore CoreG(C) = 1 = r(Qas
claimed, and so G has a minimal normal subgroup N s^ylnf be a
maximal subgroup of G containing U = G„t. Then r(M) < У ’
V = r(t n M), and from the choice of G as a minimal counterexample we condude
that V is a precursive subgroup of M. If N $ M, then / (
precursive subgroup of G/N& M), against (5.y). Thus N < M, and by WD4 we
420
V. Normalizers
r(G//V) < t(M/N).
Since V is an (g, r)-precursive subgroup of M, it follows from Theorem 5.8 that
r(M/N) < VN/N. This theorem applied to U also gives t((I n U)N/N) = Vx N/N =
VN/N. Thus
r(W/N) = r(G/N)r((f n U)N/N) < VN/N < t{W/N),
and we conclude that VN/N = r(tN/N), which is precursive in G/N because G is by
choice a counterexample of minimal order. This contradicts (5.y) and shows that
t(G) # 1.
Now t(G/t(G)) = 1 by WD1, and therefore
r(h(G)/T(G)) = T(G/T(G))r((tn U)t(G)/t(G))
= l^TfOMG) = V/t(G)
by (5.8) applied to the g-chain t n U of U. Thus, since G is a minimal counterexample,
F/t(G) is an (g.r)-precursive subgroup of G/t(G). In particular, CoreG/t(G|(l//r(G)) =
r(G/r(G)) = 1, and consequently CoreG(k) = r(G). Therefore G has a minimal normal
subgroup N satisfying (5.y). Suppose first that N < r(G), and set W/N = i(tN/N), a
precursive subgroup of G/N. By WD2 we have r(G)/N < t(G/N) < W/N, whence
t(G) < W, and therefore W/r(G) = Wr(G)/r(G) = t(It(G)/t(G)) by the now familiar
argument involving (5.8). Thus V = W, and so VN/N) = W/N) is an (g, r)-precursive
subgroup of G/N, against (5.y). Hence we may suppose that N £ r(G) and that
Nt(G)/z(G) is a minimal normal subgroup of G/r(G). Since r(G/r(G)) = 1 by WD1,
we may apply the argument used earlier in the proof to deal with the case r(G) = 1 and
conclude that VN/t{G)N = t(1N/t(G)/V), whence VN/t(G)N is an (g, r)-precursive
subgroup of G/r(G)N. Let X/N be an (g, r)-precursive subgroup of G/N. Then by
WD2 and (5. fl) we have t(G)N/N < t(G/N) < X/N, and so X/r{G)N is an (g,r)-
precursive subgroup of G/t(G)N by (5.8). But then by (5.5) the subgroups VN/r[G)N
and X/t(G)N are conjugate in G/t(G)N, and so VN/N, as a conjugate of X/N, is an
(g, r)-precursive subgroup of G/N. This final contradiction of (5.y) shows that no such
counterexample G exists. □
As another illustration of this theory, we end this section with an account of
prefrattini subgroups. These were discovered by Gaschiitz. who proved their exis-
tence and main properties in [7]. In order to fit them into the framework of Fischer’s
theorem, we need a preliminary lemma.
(5.13) Lemma. Let N <3 G = NH, where N n H = 1. Assume that N < Soc(G). Then
®(G) = СФ(И)(А) = П„„Ф(НГ.
Proof. If T denotes the intersection of the maximal subgroups of G containing N,
then Ф(С) < T and T/N = ®(G/N). But from the standard isomorphism H -»HN/N
we deduce that O(G/A) = ®(H)N/N, and therefore Ф (G) < <b(H)N. If IK is a minimal
5. Precursive subgroups
421
normal subgroup of G contained in N, then N = W x W* with W* < G because N
is completely reducible by hypothesis. Thus W is complemented in G by W*H and
it follows that O(G) n N = 1. Therefore O(G) < Cc(N) n Ф(Н)И = r.(N) x W.
Set К - СФ1И)(Л/), and note that К < G. We suppose that G has a maximal
subgroup M which does not contain К and derive a contradiction. In this case
KM = G. If N < M, then M/N < G/N, and so M n H <• H. But then К < Ф(Н) <
M, contrary to supposition. Therefore N £ M. Let J = Mr N. Since N < Soc(G),
there is a minimal normal subgroup V of G such that N = V x J. Furthermore, M
and JH are maximal subgroups of G complementing Ц and are not conjugate in G
because JH contains the normal subgroup K, whereas M does not. It follows from
A, 16.9 that M rJH < JH and hence that M nH < H. But then К < Ф(Н) < M,
against supposition. Hence every maximal subgroup of G contains К and К < Ф(С).
Finally, we have
O(G) = O(G) rNK = (O(G) n N)K = K,
as claimed. The second equation follows immediately from A, 16.3(b). □
(5.14) Example. We begin by showing that the Frattini subgroup function Ф is well
disposed for (1), the trivial saturated formation. By A, 9.2(e) Conditions WD1 and
WD2 are certainly satisfied.
Let G be a soluble group with Ф(С) = 1. Then F(G) = Soc(G) and G = F(G)H with
F(G) r H = 1. Let G e 9T”\9T”’1, and let U be an ^"“'-projector of G containing H.
(Such a choice is always possible by III, 3.23.) Then U = NH, where N = F(G) r U.
Clearly N < Soc(C), and therefore Ф(17) < H by (5.13). Hence Ф(U) n F(G) = 1, and
consequently Condition WD3 holds.
To verify Condition WD4 again assume that Ф(С) = 1, and let U < M <• G, where
as before U is an 9Im“1 -projector of G containing H, a complement of F(G) in G. If
N = F(G) r U and N* = F(G) r M, then Ф(Л1) = CmH}(N*) < СФ(Н)(А) = Ф(С) by
(5.13). Thus Ф(М) < Ф(1) for any (l)-chain I through U.
It remains to show that if N is a minimal normal subgroup of G contained in M,
then
(5.<5)
Ф/G/N) < Ф/M/N).
Since MF(G) = G and F(G) = Soc(G), there exists a minimal normal subgroup Lof
G which is complemented by M. Let
У = {L<G :L/N <-M/N}.
Then, via the isomorphism from M/N to G/NV, we obtain LV/NV<-G/NV and
hence LV/N < G/N for all LeJF. Therefore
O(G/W < M/N r ( П LV/n) = П L/N = Ф(М/№).
' ' \Le У / Le-y
422
V. Normalizers
Thus (5.<5) holds, and the characteristic subgroup function Ф satisfies WD4 when
S = (1).
From Theorem 5.12 we can now conclude that every soluble group has ((1),Ф)-
precursive subgroups. These are called prefrattini subgroups, and in each group they
form a characteristic conjugacy class. If W is a prefrattini subgroup of a group G and
if N < G, then WN/N is a prefrattini subgroup of G/N and <b(G/N) = CoreG(WN)/N.
Furthermore, 11V| is the product of the orders of the Frattini chief factors in any chief
series of G.
The following characterization of prefrattini subgroups highlights the role of Hall
systems in their construction.
(5.15) Theorem (Gaschiitz [7]). Let L be a Hall system of a group G, and for each
prime divisor p of \G\ set
Wp = W£(G,E) = p {M: Gp < M < G},
where Gp denotes the p-complement in E. Then
П и; = Ф(1),
pIIGI
where t is the (l)-chain of G into which L reduces.
Proof. Set W = p Wp. Let G e and let
pIIgi
t:G>Gm_1>->G0=l
be the (1 (-chain into which E reduces. We will prove that W = Ф(1) by induction on
|G|. If Ф(С) 1, evidently Ф(С) < Wp for all prime divisors p of |G|, and so by
induction
W/O>(G) = П 1Т„/Ф(С) = Ф(1Ф(О/Ф(С))
e
= Ф(1)Ф(С)/Ф(О = Ф(1)/Ф(С)
by (5.8), which yields W = Ф(1) in this case.
Now suppose, on the other hand, that Ф(С) = 1. By I, 4.20 there exists a maximal
subgroup M of G containing Gm j such that E M. Since M contains Gm j, it
supplements F(G) (= Soc(G)) in G and therefore complements some minimal normal
subgroup N of G. Cortsequently, if L <-G and L is not conjugate to M, it follows
from A, 16.6 and A, 16.9 that Lr> M < M. (We note that M is its only conjugate into
which E reduces.) Moreover, if L* < M, then NL* < G. It now follows easily that if
p is the prime dividing |/V| = |G : M|, then
Wp(M,YoM)= IVP(G,E),
and
423
5. Precursive subgroups
Wg(M, Z n M) = M n Wg(G, Z)
T^US У' л E n M), and so by induction w = Фр n Л1) -
Ф(М)Ф(1 n Gm_,) = Ф(1), since Ф(М) < Ф(1) by WD4 “
This theorem shows how to associate a unique prefrattini subgroup with a given
Hall system Z of G and how to describe it without recourse to (l)-chains. The
following fact about prefrattini subgroups is an easy consequence of the proof of (5.15).
(5.16) Corollary. If W is a prefrattini subgroup of a critical maximal subgroup of G
then IVO(G) is a prefrattini subgroup of G.
We also have the following characterization.
(5.17) Corollary. Let Lbea Hall system of G, and for each complemented chief factor
of a fixed chief series of G, choose a complement into which Z reduces. Then the
intersection U of such a set of complements is a prefrattini subgroup of G.
Proof. By (5.15) the subgroup U contains some prefrattini subgroup W of G. Since
U avoids the complemented factors in some chief series of G, its order is at most the
product of the orders of the Frattini chief factors in this series. Therefore |U| < | tV|,
and consequently U = W. □
Postscript
A database search in 1990 brought to light some 38 articles having “prefrattini
subgroup” either in their title or cited as a key phrase, which is convincing evidence
that Gaschiitz’s original idea has been widely investigated and variously generalized.
One particularly interesting generalization, due to Kurzweil [1], goes as follows:
Let H be a subgroup of a finite soluble group G, and in a given chief series
1 = No < Nt < < Ni < < 4 = G
of G, let IH denote those suffices ie{l,...,n} such that the set of maximal
subgroups of G that complement NJNi-i and contain H is non-empty. If H = G, the
group G itself is the unique H-perfrattini subgroup of G. If H < G, then IH is non-
empty, and the H-prefrattini subgroups are then defined as the subgroups D of the form
D = P Mh
where one maximal subgroup Л7, is chosen from each set ЛЦ in this intersection. This
definition proves to be independent of the choice of the chief senes. The H-prelrattini
subgroups of G are conjugate in G, and if H = 1, then they coincide with the prefrattim
subgroups discovered by Gaschiitz [7]. If H e Hall -(G), the "-Р^Хо^сЦП
are the p-local prefrattini subgroups W„ investigated by Hawkes [Ц К X “ nsl’
and Brandis [1] (these are even defined in the larger class of p-soluble groups).
424
V. Normalizers
Furthermore, if g is a saturated formation and H is an g-normalizer of G, then the
//-prefrattini subgroups coincide with the analogues described in Hawkes [1] and
denoted by I' in Exercise 2 below.
In the cited paper, Kurzweil characterizes his H-prefrattini subgroups in terms
of the Euler characteristic of the simplicial complex associated with the interval lattice
[H,G]
of all subgroups lying between H and G (the simplices are the chains of distinct
subgroups different from H and G). In a subsequent work [1], he and Hauck give the
following elegant description: the H-prefrattini subgroups of a finite soluble group G
are precisely the minimal elements (in the partial order of inclusion) of the set of
subgroups U of G satisfying
(i) H <U, and
(ii) the interval lattice [If, G] is complemented.
Exercises
1. (Fischer). Let т be a characteristic subgroup function, and g a saturated forma-
tion. For each group G define r*(G) = CoreG(r(f)) for some g-chain I of G. Show
that r* is a characteristic subgroup function, and that r*(f) = r(t) for each
g-chain I in each group G.
2. (Hawkes [1]). Let f be a formation function, and let E be a Hall system of G. If
p| |G I, let Vr denote the intersection of the set of /-abnormal maximal subgroups
of G which contain the p-complement G’eX (with Vp = G if this set is empty),
and set
F=I/(E)= П Vf.
rl|G|
Show that
(a) V avoids the complemented /-eccentric chief factors of G and covers the rest.
(b) Let Л denote a set of /-abnormal maximal subgroups of G into which E
reduces. Assume that Л contains at least one complement of each complemented
/-eccentric chief factor in a given chief series of G. Then V = Q {M : M e Л}.
(c) If W is the prefrattini subgroup of G associated with E and D = D/(E), the
corresponding/-normalizer, then V = DW = WD.
(d) If r(G) = ®(G)Zg(G), and if g = LF(F), then V = VF(E) is an (g, r)-precursive
subgroup of G.
(e) If U = Dr W, then U covers the /-central Frattini chief factors of G and
avoids the rest.
(f) If <r(G) = Ф(С) n Zf(G), then If is a ((1), <r)-precursive subgroup of G.
3. Do the characteristic subgroup functions r and a defined in Exercise 2 satisfy
Condition WD4?
4. (Gillam [1]). Show that a subgroup U of G is the prefrattini subgroup associated
with E if and only if (1) U covers all Frattini chief factors and avoids all
complemented chief factors of G and (ii) U permutes with E. Find a group G
5. Precursive subgroups
425
which has a subgroup U satisfying (i) such that U is not a prefrattini subgroup
of G.
5. (Doerk [2]). Let f be a formation function, and let 3EZ denote the class of groups
whose Frattini chief factors are all /-central. If 8 is a class of groups, let SBя
denote the class of groups whose prefrattini subgroups are in 8-
(a) If 8 = LF(/). then 8 is the largest saturated formation contained in
(b) If / is a full local definition of 8> then 'IBft = Xe, where g(p) = SB/(p) for all
p e P.
(c) If § is the intersection of all formations 8 for which 5ВЯ = S, then 'Jl§ is the
saturation of § as a formation.
(d) If 8 6 8 S, then there is a group G whose prefrattini subgroups W are
not in 8 is defined in VII. 3.1).
6. (Forster [10]). Let SB denote the class generated by all prefrattini subgroups of
soluble groups. Then qSB = S.
Chapter VI
Further theory of Schunck classes
General hypothesis for the chapter
All groups under consideration will be assumed to be soluble unless the contrary
is explicitly stated.
1. Strong containment and the lattice of Schunck classes
We saw in Chapter III that Schunck classes are precisely those classes which guarantee
the universal existence of a single conjugacy class of associated projectors. However,
the fact that one Schunck class is contained in another by no means implies a
corresponding inclusion between their projectors. For example, in Sym(4) an 91-
projector has order 8 and cannot be contained in a U-projector, which has order 6,
although, of course, 91 £ U. It was Cline [1] who first thought of using inclusion
between projectors to define a new relation between their specifying classes, and
although his paper deals only with saturated formations, his idea fits very well into
the larger framework of Schunck classes.
(1.1) Definition. Let § and Я be Schunck classes. We say that is strongly contained
in Я, and write
&«Я,
if, for each G e 6, an ^-projector of G is contained in some Я-projector of G. It is
clear that “«” is a relation of partial order on the set of all Schunck classes. This
relation is called strong containment (or sometimes strong inclusion elsewhere in the
literature). As an illustration we mention the obvious fact that
6„ « 6t if and only if n £ t.
(1.2) Theorem (Hawkes [9]). With respect to the partial order «, an arbitrary non-
empty set {§>/ Л e A} of Schunck classes has a least upper bound, namely the composite
Л 6 A) (see Definition 5.6 of Chapter Ill). The set of all Schunck classes derives
from this partial order the structure of a complete lattice Ж in which the join and meet
operations, v and c. respectively, are given by:
§ v Я = Я), and
& л Я = <C 6 : £ « fj and £ « Я>.
1. Strong containment and the lattice of Schunck classes
427
7 6 Л'>г and note that is a Schunck class by III. 5.8(b) But by
III, 5.8(a) an ^-projector of an arbitrary group G is generated by a certain set of
§ ...-projectors, one for each Ze Л, and consequently fjA « § for all 2 e A Now
suppose fj* to be a Schunck class satisfying
(la)
5л « $>* for all 2 e A.
Let G be a group, let Z be a Hall system of G, and let E* denote the &*-projector of
G into which Z reduces. If 2 e A, by Supposition (La) the subgroup E* contains an
§A-projector, FA say, of G, and FA e Proj6,(E*) by III, 3.22(a). Let EA be the unique
conjugate of FA into which the Hall system Z n E* of E* reduces. Then EA must be
the unique §>A-projector of G into which Z reduces. Since the subgroup <EA: 2. e A>
of E* is an ^-projector of G by III, 5.8(a), we conclude that § « §»*, and our assertion
that § is the least upper bound of the set {§A: 2 e A} is justified.
It is well known and straightforward to verify, that a partially ordered set У which
has a least element and in which every non-empty subset has a supremum is a
complete lattice, with
x v у = sup{x, y}, and
x л у = sup{s e У: s < x and s < y}
for all x, у e У Since the set .Lofall Schunck classes has a least element (1), it follows
that the partial order « induces on the structure of a complete lattice whose join
and meet operations are as stated in the theorem. □
In this section we shall take a closer look at the partial order of strong containment,
while at the same time developing some notation and machinery for the next
section, in which we shall concentrate on aspects of the lattice Ж The properties of
boundaries of Schunck classes play an especially important part in all this, and it is
convenient to introduce at this stage some boundary-related concepts and notation
which will prove useful throughout the chapter.
(1.3) Definitions and Notation. Let § be a Schunck class, and let я S P.
(a) We recall that the boundary b(£>) of § comprises all primitive groups G whose
^-projectors are stabilizers, that is. complements of the socle; from III, 2.9 we also
recall that ® is the boundary of some Schunck class if and only if ® is a class of
primitive groups such that B, (Q - 1 )(B2) for all B„ B2 e ®. It will be useful to have
the concept of the n-boundary bn(S}) of 45 defined thus.
bn(ty = Ь(Ь) n F = (G 6 b(£>): Soc(G) e S„),
and also to have the notation to denote the Schunck class
= ВДМ
whose boundary is />,(§>)
428
VI. Further theory of Schunck classes
(b) We recall from III, 4.15 that the avoidance class a(§) of & is defined
a(Jj) = (G e : H n Soc(G) = 1 for all H e Projs,(G)).
The л-avoidance class is defined analogously thus:
u,(S) = «(S) n = (G e a(H): Soc(G) e GJ.
(c) Set c,(&) = (G/Soc(G): G e b„($j)).
(d) We shall call n-separated if <„($>} £ Q„-.
(1.4) Remarks, (a) It is obvious that b(Jj) £ a(Jj). If a group G belongs to a(§)\b(§),
then G/Soc(G) ф §>, and therefore G has a proper epimorphic image in i>(Jj), a simple
remark that has several implications. It follows, for example, that a group is a(Jj)-
perfect if and only if it is h(§»)-perfect, and hence that h(a(Jj)) = f>. Thus the maps a
and h are mutually inverse bijections between the set Iff of Schunck classes and the
set {«(§>): & e -ff} of avoidance classes. Moreover, if the elements of an avoidance
class a(§) are partially ordered by -<, where X -< Y if and only if X e q(T), then the
groups in /)(§>) may be characterized as the minimal elements of the poset (u(f>), -<),
and so it is easy to retrieve a Schunck boundary from its avoidance class. The reverse
process, however, is a much harder nut to crack; for although the theory is clear, in
practice it is difficult to determine with confidence all the elements of a(Jj) when
presented with a specific b(§), unless b(f>) has a very special shape. It is another
implication of the earlier remark that b(Jj) is a maximal Schunck boundary in o(f>)
(with respect to inclusion). In general there are many maximal Schunck boundaries
in an avoidance class, and these will determine which Schunck classes the given
Schunck class is strongly contained in. Avoidance classes, unlike boundaries, do not
seem to admit of a simple, self-contained characterization, and yet they play a central
part in the study of strong containment, as will soon become evident. It is therefore
important to establish connections between a(Jj) and and herein lies the fascina-
tion of this particular topic.
(b) As we pointed out in Section 2 of Chapter III, a subclass of a Schunck
boundary, and therefore in particular the л-boundary is again a Schunck
boundary.
(c) The class c„(§) consists of the ^-projectors of the groups in the л-boundary of
§ and is therefore a subclass of Sy
(d) If G e a(Jj), then each complement of Soc(G) in G contains an ^-projector of
G; this follows from III, 3.25(d) and the conjugacy of projectors and complements.
(e) Let H be an ^-projector of a group G. By III, 4.18 there exists a chain
H = M,< М,_г < < Mi < M0 = G
such that for i — 1,..., r the subgroup Mt is a />(§>)-normal maximal subgroup of
Af,-!, which means that M.^/Core*,. JMj) e /)(§>). Since |M^, : AfJ is therefore the
order of the socle of a group in b(§), we can conclude that if b(f>) = b„(S) for some
л — P, then |G: H| is a л-number, and, in particular, that a(f>) = «„(§>)•
1. Strong containment and the lattice of Schunck classes
429
HThe'n011 °f‘ЬеPreCedinedefmitiOnS’COnSidertheclass°'of*Wect
b(^') — (Zp: p e t), and
a(D') = $ n
For P we have c.(D') s (1), and therefore Q* is л-separated. Furthermore, we
have 5" = h(Zr:petnn) = £1™.
The following simple result provides the key to the study of strong containment
between Schunck classes.
(1.5) Lemma (Doerk [4]). Let §» and Я be Schunck classes. Then any two of the
following statements are equivalent:
(а) Я « Sj;
(b) a(£) s а(Я);
(с) ад)с0(Я).
Proof. It follows at once from the relevant definitions that Statement (a) implies (b),
and since i>(5) s a(5), it is also clear that Statement (b) implies (c). It remains to prove
that Statement (c) implies (a).
To this end assume, by way of contradiction, that Statement (c) holds and that
Statement (a) does not. Let G be a group of minimal order subject to the condition
that an ^-projector H of G contains no Я-projector of G, and let N •< G. The choice
of G means that HN/N contains a certain K*/N e Proj «(G//V), and by III, 3.25(c) we
can write K* = KN for some К e Proja(G). Since К e Proja(HW), the ^-projector
H of HN contains a conjugate of К if HN < G. Therefore HN = G; moreover
H r- N = 1, for otherwise H coincides with G and then certainly contains а Я-
projector of G. Since this is true for an arbitrary minimal normal subgroup N of G,
it follows that CoreG(H) = 1 and hence that G e h(f>) s а(Я). Thus KoN = 1. But
then by 111, 3.23(a) all complements to N in KN are Я-proiectors of KN and hence
Я-projectors of G. Since H n KN is just such a complement, we therefore conclude
that H contains a Я-projector of G. This is the desired contradiction. □
Of the following elementary observations perhaps Statement (c) is the least expected,
for it is easy to find examples to show that an X n 9)-projector need not be contained
in either an X-projector or a 9)-projector (see Exercise 1).
(1.6) Remarks. Let §, X, and 9) be Schunck classes.
(a) = 5»\(&) = 0oS„ « §.
(b) <6.-, S> « 5".
(c) If S> « X and 5 « 9), then Sj « 3E n 9).
(d) If X « § and 9) « §, one cannot conclude that X n 9) « S>-
Proof, (a) If 6„&\§ is non-empty, it contains a group of minimal order which
belongs to h,(5) by III, 2.2(a). Since s 6,5, the first equivalence is clear. For the
430
VI. Further theory of Schunck classes
second simply observe that />„(§) = 0 if and only if />(§) S tp*' and that by (1.5) this
last statement is equivalent to the assertion: <S„ « § because a(<S„) = ф’’.
(b) Since b„.($j”) = 0 by definition. Part (a) yields to„. « Since />(£>") = />„(§) s
a($j), we have $ « $>” by (1.5). Then (1.2) implies that <<£„., §> « f>".
(c) This follows from (1.5) and the obvious fact that h(X n9)) S h(.X) и h(9)).
(d) Let X = Sip.,) and?) = 91|3 7). Let N be a group of order 7, let A = Aut(N)
Z6, and form the semidirect product В = [N]/1. Thus В = in the notation of
B, 12.5. It is easy to see that Projj(B) = Syl2(B), that Projs(B) = Syl3(B), and hence
that В e o(JE) n a(9)). Therefore if £> denotes the Schunck class whose boundary is
h(&) = (B), then by (1.5) we have
3E « f) and 9) « fl.
However, JE n 9) = <S7, and so ProjIr,s(B) = {N}. Therefore />(§) o(JE n ?)), and
we conclude, again from (1.5), that JE n 9) is not strongly contained in Sy □
Clearly S is the unique upper bound for the lattice Ж The next set of results is
devoted to Doerk’s analysis, later extended by Forster, of properties of the partial
order « just below S. It leads to Doerk’s characterization of the maximal elements
of Ж and to an interesting, but seemingly difficult, open question about the nature
of «-maximal elements. First we prove some lemmas of a technical nature.
(1.7) Lemma. Let G be a group with a normal p-subgroup N possessing a complement
К in G such that CK(N) = 1. Let Sy he a Schunck class, let H e Projf/K), and assume
that H e Proj6(G). Then H/Op(H) e Rocp($j), and, in particular, H belongs to the forma-
tion SpQR0(cp(f>)).
Proof. Let
1 = No < N, < • < N, = N
be an H-composition series of N. Let i e {1,..., r}. Since H e Proj 6( G) and H nN = 1,
we infer from III, 3.22(a) that HN,-1/N,_I is an Sj-projector of HNi/Ni_1 comple-
menting the minimal normal subgroup NJN,-,. Let Cf = C^Nj/N^j). Then
G bp(f>), and therefore H/C, = e cp(Jj). Appealing to A, 12.4
and B, 3.12(b), we have H/OffL) = H/(Q'=1 C,) e Rocp(§), and the rest of the state-
ment is clear. □
(1.8) Lemma. Let § be a Schunck class, and suppose that G e r„op(§>), i.e. that G
has normal subgroups N3, .... Nr with G/N, e op(f>) and N, = 1. Let MJN, =
Soc(G/N;) for i = 1,..., r. Then
(i) (%i Mi = F(G) = Op(G) = Soc(G),
(ii) <I>(G) = 1,
(iii) H r\ F(G) = 1 for each H e Proj6(G), and
(iv) H e SpQR0(cp(§)).
Proof. Set F = F(G) and R = Qf=1 M,. Since M,/N, — F(G/N,), we have FN, < M,
and hence F < R. But R/(R n N;) s RN./Nj < M,/N, e Vl(p), the class of elementary
1. Strong containment and the lattice of Schunck classes
abelian groups; therefore R e R0W(P) = Ш(р). Hence R, as a normal p-subgroup of
G, is contained in F and consequently R = F = 0„(G). Since <H(G)W,/W( /®(G/W.)
by A, 9.2(e), we have ®(G) s Q;=1 N, = 1. Therefore F(G) = Soc(G) by A, 10.6(c) and
we have proved Parts (i) and (ii). 1 *
г H VLr°yfG)’ let T= HPF(G}’ and supP°se for a contradiction that
Т/ 1. Then T,t N, for some ie{l...........r}, and so TN,/N, is a non-trivial H-
invanant subgroup of G/A,. contained in F(G)N,/N,. It follows that 1 1 TN/N =
(H n F(G))N,/N, < (HN/N,) n (F(G)N,/N,) < (HN,/N,) n (M,/N,), which contradicts
the fact that HN,/N, e Proj 6(G/N,) and G/N, e a(f>). Therefore T = 1, and Part (iii) is
justified.
Finally, since it is clear, in view of Part (iii), that the hypotheses of (1.7) are satisfied
with HF(G), F(G), and H in the roles of G, N, and К respectively, we can now deduce
from that lemma that Part (iv) holds. r-i
The proof of the next result, although somewhat technical, requires ways of relating
the subgroup to the quotient structure in a given group, and these methods could
have applications elsewhere in this area.
(1.9) Proposition (Doerk, Hawkes—see Hilfsatz 2.5 of Doerk [4]). Let § be a Schunck
class. Let G e a(§), let N = Soc(G), and let H be an ^-projector of G contained in a
complement К to N in G. Further assume that the following condition is satisfied:
(1-Д)
If H <, L < К and if V is an L-composition factor of N,
then [F](L/Cl(F)) e q(G).
Then И = К and therefore G e b(Sj).
Proof (Forster [6]). We suppose that the lemma is false and derive a contradiction.
Let G be a counterexample of minimal order, let p denote the prime dividing | N |, and
note that O„(K) = 1. Consider the set //(G) of all groups which have the form of a
semidirect product
[P](L/Ct(F)),
where L runs through all proper subgroups of К containing H, and V runs through
all L-composition factors of the socle N of G. The choice of G clearly implies that
//(G) is non-empty. Let B,_B„ be a full set of representatives of the isomorphism
classes of the groups in //(G) and observe that each of them belongs to ap(§) because
Gea (fl). A similar construction can then be applied to each of the groups B„ and
it is important to notice for later reference that by the nature of the construction an
from the properties of ^-projectors we evidently obtain
ЯА)£(Вп • > Д)\(Д)
f°Our first goal will be to show that n = 1. Since L < К we have B^:Gfor each
i, and therefore Hypothesis (1-jB) implies that each B, belongs to q(G/N)
432
VI. Further theory of Schunck classes
For i = 1, ..., n the complement К therefore has a normal subgroup S, such that
К/S, S Bj e tp, and К/S, has a unique minimal normal subgroup, R,/S, say. Set
C = Q Rt and D = Q S,.
i=l i=l
Lemma 1.8 now applies to the group К/D, and we can deduce from Part (i) that C/D
is the Fitting subgroup of К/D, from Part (ii) that C/D has a complement, U/D say,
in К/D, and from III, 3.24 and Part (iii) that 17/D contains an ^-projector H*/D of
К/D. Furthermore, without loss of generality we may suppose that H* = HD, where
H is the ^-projector in the statement of the proposition. Thus 17 contains H and also
contains F(K) because F(K) is a p'-group and |K: 17| = |C: D| is a power of p.
Now let 17* be a proper subgroup of К containing 17. Since F(K) is a normal
p'-subgroup of U*, we have ОД17*) < CK(F(K)) < F(K) by A, 10.6(a), and therefore
Op(l7*) = 1. Let kj,..., Ц be the [/’-composition factors of N; then by definition
of У(С), for each i = 1, ..., к we have [Ц/КС^/С^Р,)) B7 s K/Sj for some
j = j(i) e {1,..., n}. The subgroup 17* of К acts faithfully on N = Cc(/V), and
thus, appealing to B, 10.1, we obtain U* = U*/OP(IJ*) = U*/([yi=lCv.(Vl})e
r^K/Rj: j = 1,..., n). Denoting by g the formation generated by K/C, we then have
V* e R0Q(K/C) e g.
By Part (i) of (1.8) we know that К has p-chief factors TJD,.... Tm/D such that
C/D=T1/Dx ••• x TJD.
Suppose first that m > 2. With the usual notational convention for omitting a term,
let Uj = (7]T2 ... f;... Tm)U, a complement to TJD in K. Since 17 < 17, < K, the
above analysis applies with 17, as I/*, and therefore I7, e g for i = 1, ..., m. Since
K/T, = Ui/D and Ti = D, it follows that К/D e R0Qg = g. Tf I = l(K/C), the
nilpotent length of K/C, then g = qr0(K/C) e qr091‘ = 91', and so l(K/D) < I. But
C/D = F(K/D), and therefore l(K/D) = ЦК/С) + 1 = 1 + 1, a contradiction. Hence
m = 1, and К/D is a primitive group with socle C/D. By definition of D it now follows
that S, = D for just one value of i, which we may take to be 1 without loss of generality.
Thus B, = К/D, and we have C S S, for i = 2,..., n, whence the groups B2,..., B„
belong to q(K/C). Since C/’(B1) e (B2,..., B„) £ q(B, ), we conclude that B2 satisfies
the hypotheses of the proposition and therefore that Bt e b(Jj) by the choice of G as
a minimal counterexample. But then K/C e 5, and for i > 2 we have B, e § n op(£j) =
0. Thus n = 1, as desired.
Write В for the group Bt and R/S for the p-chief factor of К such that K/S = B.
Then by now we know that SH complements R/S in K, and that for every L such
that H < L < К and for every L-composition factor V of N we have [ P](L/C,( P)) S
B. Let
F(K) = Ft x • • • x Fr
be a decomposition of the Fitting subgroup of К into a direct product of its Sylow
1. Strong containment and the lattice of Schunck classes 433
ОДК) = 1we have P* {P1,....p,} and therefore f(K) s
Let C; = ОД <IK. Then Q'=1 q = CK(F(K)) < F(K) by A, 10.6(a). It then follows
from IV, 1.3(a) and an easy induction on r that for some к e {1,..., r) the groun K/C
has a chief factor which is К-isomorphic with R/S. Consequently K/Q has ад
epimorphic image isomorphic with K/R. Let c denote the nilpotency class of a Sylow
Pk-subgroup of В (s K/S). Then the Sylow ^-subgroups of K/R also have class c
whence those of K/Ck have class at least c. Let Pk 6 Syl^JSH); since SH complements
the P-chief factor R/S of K, we have Pk e SylJK). Since Fk = OJK) < Pk, we have
S Pk n Ck, and it follows that Pk has class at least c + 1 because Pk/(Pk n Ck)
TkQ/Ck e SylPk(K/Ck). Thus if Q denotes K,(Pk), we have Q # I. Since p J |Q| and N
is faithful for the subgroups of K, there is at least one SH-composition factor of N
call it К such that Q £ Ker(SH on V) by A. 12.3. Thus the Sylow p-subgroups of
SH/Csh(V) have class at least c + 1. On the other hand, we know that
[И] (SH/Csh(V)) s B, which is a group whose Sylow Pk-subgroups have class c. This
final contradiction completes the proof. q
(1.10) Definitions, (a) Let § and £ be Schunck classes such that § « £ # f>. We say
that § is maximal in £ (with respect to «) if whenever § « Я « £ for some Schunck
class Я, then either § = Я or Я = £.
(b) Let § be a Schunck class. We call § maximal if is maximal in S. A strictly
ascending chain of length n for £> is a sequence of pairwise-distinct Schunck classes
5 = £>o, Si, • ••,£>„ = ® such that
So « Si « • « S„-i « S„ = >=
We call such a chain maximal if SS,i is maximal in & for i = 1,..., n. Finally, for a
natural number n we call $ n-maximal if
(i) the lengths of strictly ascending chains for § are bounded above, and
(ii) every maximal chain has length n.
Remark. A different concept of n-maximality is considered by Doerk [4]. According
to this, a second maximal Schunck class is one which is maximal in a maximal
Schunck class. But there exist second maximal Schunck classes in this sense which
have strictly ascending chains of unbounded length.
Now if $ is a given Schunck class, and we wish to identify all Schunck classes Я
such that § « Я, it is sufficient first to find the avoidance class o(g) and then to
identify those subclasses ® of a(£>) which are Schunck boundaries; for by III, 2.9 and
Lemma 1 5 such ® are in one-to-one correspondence (via the maps h and b) with t e
classes Я that strongly contain $. If the boundary h(S) contains at least n distinct
isomorphism classes, we can find a chain of subclasses of b($) thus.
0,4®, <= • • •<= »„ = «>($).
Since subclasses of Schunck boundaries are also Schunck boundaries, this chain gives
rise to the following strictly ascending chain for g:
434
VI. Further theory of Schunck classes
6 = h('8„)« )«•••« b(® t)« to
The preceding elementary observations yield at once two conditions for n-maximality,
one necessary and one sufficient, contained in the following Proposition.
(1.11) Proposition. Let S) be a Schunck class.
(a) If Sy is n-maximal, then |b(6)l < n.
(b) If o(6) = b(Sy) and | b(6)l = n, then 6 is n-maximal.
(1.12) Theorem (Doerk [4]). A Schunck class 6 is maximal if and only if |b(6)| = 1,
and in this case o(6) = b(Sy).
Proof. The necessity is contained in (1.1 l)(a). To prove the sufficiency, assume that
b(6) = (B). Suppose, if possible, that the class o(6)\b(6) is non-empty, and let G be
a group of smallest order contained in it. By III, 2.2(a) a group of minimal order in
the class Q(G)\6, which is obviously non-empty, belongs to b(6), and therefore
Bcq(G). Let N = Soc(G), let К be a complement to N in G, and let H be an
^-projector of G contained in K. If H < L < K, and if V is an L-composition factor
of N, then the semidirect product [F](L/Ct(T)) belongs to o(6) and hence to b(6)
by the choice of G. Thus G satisfies Hypothesis l.p of (1.9), and from that Proposition
we then conclude that G e b(6). This contradiction shows that o(6) = b(6), and
therefore $ is (l-)maximal by (1.1 l)(b). □
A lattice with a greatest element is called dually atomic if every member of the lattice
is contained in a maximal element. From (1.12) and (1.5) the following statement is
clear.
(1.13) Corollary. The lattice (Ж, «) is dually atomic.
We now present two characterizations of second maximal Schunck classes due to
Forster.
(1.14) Theorem (Forster [6]). Let Sy be a Schunck class. Then any two of the following
statements are equivalent:
(a) Sy is 2-maximal;
(b) o(6) = b(Sy) and |b(6)| = 2;
(c) b(6) = (Bn B2) with В, f Вг and 6 maximal in both h(Bt]and h(B2).
Proof, (a) => (c): If 6 is 2-maximal, then we have |b(6)l < 2 by (1.11)(a) and therefore
Ib(6)l = 2 by (1.12). Let b(Sy) = (Bt, B2), and set 6< = b(B;) for i = 1,2. Since b(6<) =
(B;) S b(6) £ o(6), it follows from (1.5)(c) that
6 « 6.-« S, 1 < i < 2.
By III, 2.9 we have 6 / 6,- S, and therefore 6 must be maximal in 6i by definition
of 2-maximality.
1. Strong containment and the lattice of Schunck classes
435
(c)=>(b): Assume that Statement (c) holds. Then we must show that fl(ft) = ЫЯ1
Suppose not, and let G be a group of minimal order in a(ft)\b(ft) If В and В are
both in q(G) then as in the proof of (1.12), the group G saiisfoStol of
Proposition 1.9, which then yields the contradictory conclusion that G e b(ft) There
fore suppose that В , say is not in q(G). Since B./SocfB,) e $> and G e «(^evidently
G Q(Bl), and so by Definition III, 2.6(b) the class SB = (Bt G) (c js a Schunck
boundary. Since (BJ s Ж £ a(f->), by (1.5)(c) we have
& « b(S3)« (i(BJ,
where by III, 2.9 the consecutive pairs of Schunck classes in this chain are distinct.
Since this contradicts the assumption that § is maximal in A(BJ, our supposition
must be false, and therefore a(f>) = b(§).
(b) => (a): This is clear from (1.1 l)(b). r-i
A challenging unsolved problem in the theory of Schunck classes is whether Forster’s
first criterion for 2-maximality (Statement (b) in Theorem 1.14) remains valid when
2 is replaced by an arbitrary natural number n.
Open Question. Is the condition:
|a($)| = |b($)| = n
necessary and sufficient for a Schunck class § to be «-maximal?
Even the case n = 3 is difficult and seems to require some deep results from represen-
tation theory. A counterexample might be found by looking for a Schunck class §
with the following make-up:
(i) |b(§)| = 2,sayb(S) = (B1,BJ;
(ii) |o($)\b($)| = 2, say o(g) = (Л,, A2, Bj, B2);
(iii) The classes 'B, = (Л,, Bj, »2 = (A2, BJ, and ®3 = (Л,. A2) should each be
Schunck boundaries. (Note that, if ®3 and ®2 are to be Schunck boundaries, then
one must have Q(d,)nb(§) = (BJ for {i,j} = {1,2}, and in that case ®3 is auto-
matically also a Schunck boundary.)
(iv) Each of the Schunck classes f), = fi(®,) (i = 1, 2, 3) must be 2-maximal.
The chance of finding such a configuration seems remote. With our present knowl-
edge even the following question remains unanswered (cf. Condition (n) above).
Open Question. Does a Schunck class ri exist with o($)\b($) non-empty and finite?
It is clear from (1.12) and (1.14) that Schunck classes & for which a(§) coincides
with b(ft) have a special significance for the partial order «; in view of (1.5) t ey are
certainly much easier to handle. The next set of results is concerned with criteria for
this to happen.
(1.15) Lemma (Forster [6]). Let § be a Schunck class and p a prime.
(а) а(&’}\Ь№) s a(W(S)-
436
VI. Further theory of Schunck classes
(b) If G is a group of minimal order in the (by assumption non-empty) class
ap(£>)\bp(£>), either G is a group (of minimal order) in a(f>p)\b(§p) or G e SSL
Proof, (a) Since by definition b(§₽) = bp($>) £ b($>), by (1.5) we have § « $)<’, and
then by (1.5) we have a(f>₽) £ a(§). If В e b(S5), then an ^-projector of В is a maximal
subgroup of В contained in some J^-projector of B. Therefore either Be f>’ or
В e b(f>p), and it follows that a group in a(§p)\b(§p) is not in b(§).
(b) Let N = Soc(G), let Le Comp(G). and let H e Proj6(L). Since G e ap(F>), we
have H e Proj6(G) and N e Because f> « $jp, there is an §₽-projector, К say, of
L containing H.
First suppose that К = L. By III, 3.24 we know that К is contained in some
^'’-projector, K* say, of G, and since К = L <• G, this K* is either К or G. If К* = K,
then G e bp(§₽) = bp(§) £ b(§), contrary to the choice of G. Therefore, when К = L,
we must have G = K* e $>p.
Now suppose that К is a proper subgroup of L, and let V be a К-composition
factor of N. Certainly G f b(f>p). Since H is contained in К and avoids N, the
semidirect product [F](K/CK(V)) belongs to ap(§) and hence to bp(F>) (= b(§₽)) by
the minimal choice of G. Therefore G e a($yp) by III, 4.18. The minimality of G among
the groups in a($>p)\b($>p) follows from Assertion (a). □
We now present a set of sufficient conditions for a(f>) to coincide with b($>).
(1.16) Proposition (Forster [6]). Let § be a Schunck class with the following two
properties:
(i) Spqr0(cp(§)) n c,($) = 0 for all pairs of distinct primes {p,q};
(ii) a($j₽) = b($j₽) for all primes p.
Then a(5) = b(§).
Proof. We suppose this proposition to be false and derive a contradiction. Choose
a group G of minimal order in a(§)\ b($j), let H e Proj 6(G), and let H < L e Comp(G).
Let p be the prime such that G e ap(J>). Then by (1.8)(iv) we have
(1-7)
H e topQR0(cp(§)).
In view of property (ii) for F>, we conclude from (1.15)(b) that G e Sjp and hence, in
particular, that q(L) n bp(£>) e q(G) n bp(£>) = 0. Since G f b($j), we have and
therefore q(L) n b,(S5) 0 for some prime q / p. If L/K e bg(f>) and S/K = Soc(L/K),
then L/S e c,(§) £ §. Since H e Projs(L), we have HS = L and therefore H/(H n S)
s HS/S e <„(§). But by (l.y) the group H belongs to the formation Spqr0(cp($)) and
therefore so does its quotient H/(H n S), thus contradicting Property (i) for F>. □
(1.17) Corollary (Hawkes [9]). The following two conditions for a Schunck class §
are together sufficient to ensure that a(§) = b(F>):
(i) <SpQR0(cp(f>)) n c,(f>) = 0 for all pairs {p, q} £ P;
(ii) | bp($>) | < 1 for all p e P.
1. Strong containment and the lattice of Schunck classes
437
Proof. If bp(ft) (= is empty, then a(ft") is also emntv If lb iwi _ t .u
Hypothesis^) of (1 16)’isautin eihher “Se.Hypothesis (“) °f (l b) is satisfied.’sinw
Hypothesis (!) of (1.16) is left unchanged, that proposition now implies that a(§) =
□
Although the applications of this corollary are mostly to be found in the next
section, we can put it to immediate use in the following result, which is concerned
with the question of how far a Schunck class is determined by what lies strictly above
and below it in the partial order of strong containment
(1.18) Proposition (Forster [6]). Let 'J and 3 be Schunck classes.
(a) Assume that 'J is neither maximal nor equal to S. If the set of maximal Schunck
classes strongly containing V) coincides with the corresponding set for 3, then V) = 3
(b) If 9) and 3 are maximal and if
{X: X * 9) and X « 9)} = {X: X * 3 and X « 3},
then 9) = 3.
Proof, (a) Let b(9)) = (B,: i e I), and for each i e I set X, = /i(B,). Then X, is a maximal
Schunck class by (1.12), and furthermore a(X,) = b(X,) = (BJ. Applying (1.5), we then
obtain 9) « X,; therefore 3 « X; by hypothesis, and hence a(3) contains b(Xt) = (B,).
Consequently bfl)) S o(3), and once more by (1.5) we have 3 « ?). Since the hypoth-
esis clearly implies that 3 is not maximal, a similar argument shows that 9) « 3, and
the desired conclusion follows.
(b) By (1.12) we may take b(9)) = (Y) and fc>(3) = (Z). Suppose first that Y is not
cyclic, and choose X to be another non-cyclic primitive group with its order coprime
with |Y||Z|. Then (X, Y) is a Schunck boundary, and if X = h(X, Y), we have
a(X) = b(X) by Corollary 1.17. Since X « 9) and X # 9), our hypothesis implies that
X « 3 and therefore by (1.5) that (Z) = b(3) S o(X) = (X, Y). But Z X by choice
of X. Therefore Z =Y, and 3 = 9)-
There remains the case where Y = Zp and Z = Z, with p, q e P. In this case, if p
were different from q, then the class S, would be strongly contained in 9) — Dp but
not in 3 = O’, contrary to hypothesis. Hence p = q, and again 9) = 3- О
It is not known whether the dual of (1.18) (a) holds, that is, whether a non-minimal
Schunck class is uniquely determined by the set of minimal Schunck classes which
are strongly contained in it. In fact, it is not even known whether a non-identity
Schunck class always contains a minimal one; thus
Open Question. Is the lattice (JY1, «) atomic?
Generally there seems to be a dearth of information about Schunck classes which
are minimal with respect to «. In [9] Hawkes shows that non-identi , Schunck
classes ft with I ft n <₽p| finite for all primes p are minimal and that so too are the
classes 9V for r = 1,2,..., observations which seem to indicate that a charactenzatio
I. Strong containment and the lattice of Schunck cluses
439
4,-[ОД>
* . assert that В, c a(S). Let H, « O1(D„) c Sylj(D.). Then evidently H. c Proj^D.).
« -ее H. < D„ the subspace [ I'. D,] is a submodule of V„ and is non-trivial because
•,:s faithful for D.. Hence [ V„ D.] = V, because К is irreducible, and it follows from
». 12.6 that every //.-composition factor U of V, is non-trivial. Since H, e 91(3), we
-:er from B. 9.8 that H, induces on U a cyclic group of order 3 and that |l/| » 22.
-•terelbre
[U](H./C„_((7)) s Alt(3) e ИеК
J so by III. 3.25(b) wc have B, e u(J>). as asserted.
Let 91, = (B,....B.). Since evidently 91, <~i U = 0 and (q - I )S. S U. it is clear
-at 93, is a Schunck boundary. Set = Л(93.) and apply (1.5): Since h(J>.) = 93, C
. £».wchave$ « and since/>($,) - 93, s 93,., - />(£>,.,) s a(J>,.,). we also have
? -i « 5i for i = 1.2,... Thus Statement (2) is justified.
3) For each n e N there exists a strictly ascending chain of the form
§ « Я, « Я2 « « Я, « S.
in the notation of the proof of Statement (2) above, define a Schunck class Я, by setting
Я, = h(B„ B,......)
?r i = I, 2,..., and apply the arguments used there.
The preceding example fails to be n-maximal for any natural number n because it
• lolates the bounded chain-length requirement (cf. Definition I. IO(b)(i)). Although
:ts boundary is finite, its avoidance class contains an infinite Schunck boundary. This
raises the following question.
Open Question. If an avoidance class is infinite, does it necessarily contain an infinite
Schunck boundary? A ‘yes' in answer to this question would also resolve the following
question affirmatively.
Open Question. Let & be a Schunck class whose strictly ascending chains have
bounded length. Obviously b(&) is finite, but does it follow that a(J>) is also finite?
It might pay to study the foregoing questions in a more restricted framework. We
therefore mention two further possible criteria for 2-maximality.
Open Question. Let & be a Schunck class with |b(&)| = 2. Does cither of the following
two conditions imply that £> is 2-maximal? (i) |a($)| is finite, (ii) The strictly ascending
chains for f> have bounded length.
438
VI. Further theory of Schunck closet
of minimal Schunck classes in the same spirit as Doerk's Theorem 1.12 for maximal
classes may be hard to find. Clearly there is scope for further study here, and we
propose the following as a test problem.
Open Question. Which primitive saturated formations are minimal in (JF. «fl
We round this section oft with the promised example of a Schunck class which is
maximal in a maximal Schunck class, but which is not 2-maximal. It illustrates well
the part played by the avoidance class in the study of strong containment and
provokes some interesting questions, which we discuss at the end.
(1.19) Example (Forster [6]К Let $ be the Schunck class with boundary b(£>) =
(Z2. Alt(4)). We now state and prove certain properties of this class $>.
(I) § possesses a maximal strictly ascending chain of length 2 (and is therefore
second maximal in the sense of Doerk [4]). The chain in question is
5 « Q2« S.
ThedassC2of2-perfect groups has boundary b(C2) «(Z2)and is therefore maximal
by (1.12). Since b(C2) <= b(J>) £ a(J>). it follows from (1.5) that £> is properly strongly
contained in C2. It remains to show that £> is maximal in C2. Suppose then that
$ « Я « C2 with Я / C2. By (1.5) we have (Z2) £ а(Я) and Л(Я) £ o(J>). Since
groups in а(Я)\Л(Я) possess proper epimorphic images in Ь(Я), it follows that
Z2 e Ь(Як moreover, because Я # Q2. wchave(Z2) # Ь(Я) and so can find a primitive
group. В say. in b(H)\(Z2) Having seen that Be u(£>), we now aim to show that
Be h(f>). Let Л' = Soc(B), let C be a complement to N in B. and let H e Proj^lC)
(£ ProjjJB)). Because bff>) = h}(f>), it follows that S2- « J>. Therefore (i) a(£>) -
n2(J>), (ii) N is a 2-group. (iii) 02(C) = l,(iv)|C: HI is a power of 2, and (v) by (1.8) we
have H e 329l(3), where 91(3) denotes the class of elementary abelian 3-groups.
Because F(C) is a 2'-group. it is contained in H, indeed in a Hall 2'-subgroup of H.
Since Hall 2'-subgroups of H belong to 91(3). and since CH(F(C)) £ F(C), it follows
that F(C) is a Hall 2-subgroup of If. Hence |C: F(C)|(= ]C: H||H: F(C)|)isa power
of 2. Since Ь(Я) contains Z2 as well as B. the group В (and hence О has no non-trivial
2-quotient groups, and we conclude that C = F(C) = H; therefore В e b(f>) It follows
that В - A11(4) and hence that Л(Я) = Ь(Ц Thus = Я. and we have finally proved
that b is maximal in C2.
(2) For each neN there exists a strictly ascending chain of the form
« £>, « «•••«§,« S.
To see this, let K, denote a faithful, irreducible F2-module for the nth direct power
D, = (Sym(3)f (such a module exists by B. 10.7), and let B„ denote the semidirect
product
440
VI. Further theory of Schunck classes
Exercises
1. Let p, q, and r be 3 distinct primes, and set I = 91)p g) and 9) = 91)pShow that
in the group E(qr/p) an (I n ?))-projector is not contained in either an I- or a
9)-projector.
2. Prove that C’ is strongly contained in a Schunck class f> if and only if = Cf,.
3. (Hawkes [9], (4.2)) Let f>, Я e JT, and let n = P\Char(f>). Show that the follow-
ing 3 conditions are together both necessary and sufficient for a Я-projector of
G to contain an ^-projector of G as a normal subgroup, for all G e £:
(i) h(f>) = (Ь(Я) n С) и (Zp: p e л);
(ii) If G e Я, then O’(G) e Я;
(iii) If S is a stabilizer of a group В in Ь(Я), then O’(S) is a Я-projector of O"(B).
4. A Schunck boundary ® is called maximal if ® is a maximal element of the set of
Schunck boundaries partially ordered by inclusion. Prove that
(a) Each of the following two conditions is necessary and sufficient for 8 to be
maximal: (i) Л(®) £ pq®; (ii) 9ih(®) £ pq®,
(b) if f> (# S) is a saturated formation, then b(§) is maximal, and
(c) if f> = pq(G), then h(Sj) is maximal.
5. If a Schunck class f> is maximal (w.r.t. «) in a maximal Schunck class (call such
a class 2-step maximal), then |h(f>)| = 2.
6. (Doerk [4], Theorem 2.14) Let G, H e ® with G £ H, and set f> = h(G, H). Define
P(G, H) to be the class of all groups P in o(f>) such that (i) G e (Q — 1)(P), and
(ii) H i Q(P). Prove that S is 2-step maximal if and only if one of P(G, H) and
P(H, G) is empty.
7. (Doerk [4], Theorem 2.15) Let p, q, and r be primes. Show that h(E(r/p), E(s/q))
is not 2-step maximal if and only if p # q and either r = s or r = q, s = p.
8. If f> « Cp, p e P, prove that f> is not 2-maximal.
9. Let p and q be distinct primes > 5, let S = Sym(3), and let P and Q be faithful, irre-
ducible S-modules over the fields Fp and F, respectively. If h(f>) = ([P]S, [Q]2?),
prove that |<i(f>)\h(Sj)| is infinite.
10. Let p and q again be distinct primes > 5, and this time set S = SL(2, 3). Let
T e Sy 1 j (S). Show that S has faithful, irreducible modules U and V of dimension 2
over Fpand F„ respectively such that CV(T) = CV(T) = 0. If b(f>) = ([t/]S, [V]S),
show that n(f>) = h(f>). [This example shows that Property (i) of (1.16) is not a
necessary condition for a(f>) and b(Sj) to coincide.]
11. Show that Schunck classes of the form 91' (r = 1,2,...) and pq(G) are minimal
with respect to the partial ordering «.
2. Complementation in the lattice
In this section we focus on the lattice structure of the family of Schunck classes. This
is the structure obtained from the partial order of strong containment whose meet
and join operations are described in (1.2). The first milestone is the theorem that the
lattice is complemented; then we describe and justify necessary and sufficient condi-
tions for a formation in the lattice to be complemented by another formation. Both
2. Complementation in the lattice
441
undertakings are labour-intensive and call for the applieation of representation
theory to techniques for constructing groups with prescribed properties
First we describe the avoidance class and boundary of the join of two Schunck
classes.
(2.1) Lemma. Let § and Я be Schunck classes. Then
(a) a(f> v ft) = a(&) n а(Я), and
(b) b(f> v Я) = (G: q(G) n o(f>) n а(Я) = (G)).
Remark. Analogous results are true for the join of an arbitrary collection of Schunck
classes; therefore, in particular, the family of avoidance classes is closed under taking
intersections.
Proof, (a) Since & and Я are strongly contained in & v Я, by (1.5) we have a(& v Я)
£ a(f>) n а(Я). On the other hand, if G e a(f>) г, а(Я), let Z be a Hall system of G, and
let S be a stabilizer of the primitive group G into which Z reduces. If H and К are
respectively the f>- and Я-projectors of S into which Z reduces, then H e Proj6(G),
К e Proj«(G), and by III, 5.8 the join <H, K> is an (f, v R)-projector of G. Since
<H, K> Soc(G) < Sn Soc(G) = 1, it follows that G e <i(f> v Я). Consequently
a(f>) n а(Я) S a(f> v Я), and Part (a) is proved.
(b) This statement is an immediate consequence of the fact that the boundary of
a Schunck class f> consists precisely of those groups in a(f>) whose proper epimorphic
images all lie outside a(f>) (cf. Remark 1.4(a)). □
The avoidance class and boundary of a meet § л ft do not admit such simple
descriptions as the ones just given for a join. However, algorithms for finding
b(f> л Я) from b(f>) and Ь(Я) are known, and one is described in Hawkes [9], Section
2.'
Let I be a Schunck class. If a(X) = 0, then b(3E) = 0 and I = S. Therefore by (2.1)
we have v Я = S if and only if a(f>) r> а(Я) = 0. Furthermore, if a group of
prime order belongs to a(JE), then Zp I, and therefore Zp e b(3t). Consequently the
two conditions: (Zp: p e P) £ a(JE) and (Zp: p e P) £ b(J) arc equivalent and are each
necessary and sufficient for JE to be (1). Putting these facts together, we obtain the
following criterion for complementation.
(2.2) Lemma. Let f> and Я be Schunck classes. Then Я complements $j in the lattice
Ж if and only if a(&) n а(Я) = 0 and (Zp. p e P) £ a(& л «)•
(2.3) Lemma (Forster [6]). Let & and Я be Schunck classes. Let Beb(Rf let
К e Proj «(B), and let H e Proj6(K). Further, let M be an (& л ^-projector of K. If
U and V are respectively M- and H-composition factors of Soc(B), then
(a) [<j](M/C„(C'))e b(f> л Я), and
(b) [Р](Н/Сн(Р))еа(&лЯ).
follows that M e Projfe/ «(B). Then, as
a(f, л Я), and therefore by III, 4.18
Proof. Since л Я « Я and К e Proj«(В), it
M n Soc(B) < К n Soc(B) = 1, we have В e
Statement (a) holds.
442
VI. Further theory of Schunck classes
Since f) л Я « f>, on replacing H by a conjugate we may assume that M < H. We
then have M e Proj6zS)(Soc(B)H). Consequently the subgroup of
H/CH(V) is an (Jr, л Si)-projector of the primitive group [F](H/C„(F)), and since it
avoids V, the truth of Statement (b) is clear. □
This completes our preparations, and we can now prove
(2.4) Theorem (Hawkes [9]). The lattice .Vf of Schunck classes is complemented.
Proof (Forster [6]). Let $j e G}. We shall find a Schunck class Я such that
v Я = G and л Я = (1). First we deal with two special cases.
(1) The case where b(f>) = hp(f>) for some prime p: If G 6 G and H e Projs(G), then
by (1.4)(e) the index |G: H| is a power of p. Therefore G = <H, P> for any P e Sylp(G),
and it follows that <£>, Gp> = G. Since (1) and Gp are the only Schunck classes
contained in Gp, we have л Gp = (1) or Gp. If f> л Gp = Gp, then Gp « f>, and we
get f> = <f>, Gp> = G, contrary to the choice of f>. Therefore § л Gp = (1), and
Я = Gp is the desired complement.
(2) The case where f> = C" for some it £ P: In this case the subgroup On(G) is the
unique f>-projector of G for each GeG. Since G = O"(G)Gn for any G„ e Hall„(G), we
have <f>, G„> = G. On the other hand we have f> л G„ S C" n G„ = (1), and so we
can take Я = G„ for the complement in this case.
Thus we may now exclude both of the possibilities arising in Cases (1) and (2) and
may therefore suppose from now on that there exist distinct primes p and q such that
hp(f>) contains a non-abelian group, В say, and b,(f>) contains a group B*.
Let у = Char(f>) = {r: Z, e §} = {r,, r2,...}, and note that the primes p and q may
or may not belong to /. Let re/, say r = r„, the nth prime in the enumeration of the
distinct primes in /. Corresponding to r„ we shall construct a primitive group T„ with
r„||Soc(T„)|, such that for X e Proj&(T„) we have
(2.a) [Soc(T„), X] < Soc(T„).
It turns out the class ® = (7j, T2,...) is a Schunck boundary, and that the Schunck
class Я = li(25) has all the requirements of a complement to in the lattice Ж
Let H e Proj6(B) and H* e Proj6(B*); set IV = Soc(B) and N* = Soc(B*); and
form the direct product
D„ = В x ” x В x H*
of n copies of В with H*. Consider the following subgroups of D:
Y = H x x H x 1 g X = В x x В x 1.
Since the socle of a direct product is the product of the socles of the direct components,
we can write Soc(X) = x - x where Ni is the socle of the ith component В
of X, and since CB(N) = N < B, the minimal normal subgroups N,,..., N„ of X have
pairwise distinct centralizers in X and are therefore pairwise non-isomorphic as
"I. Complementation in the lattice
443
X-modules. Furthermore, KnSoc(X) = 1, and so we can apply в, 11 6todeducethe
existence of a simple F,X-module M, faithful for X, such thatM has the tririal
F.r.modU;T.ta«^
foTo ’( X Th ’ г19(Ьк hau eVery COmposition faclor of Л/ ® IV* is faithful
f i n 7 ОГе У the Jordan-H61der Theorem A, 3.2 there exists
a faithful simple D„-module U over F, such that the restriction UE of U to the
subgroup
E = Y x H* e Proj 6(Dn)
has a factor module, U/U say, which is isomorphic with 1 x N* /V*, when viewed
as an E-module with Y as its kernel.
Now let denote the semidirect product R„ = [C]D„. Since E e Projfe(D„) and
UE/UC^U/U) S N*H* =B*e h(f>), the subgroup UE contains an f>-projector F
of R, and we may assume that E < F. Then by III, 3.24 we can write F = UE with
U = U nF < UnUE = U.
We now repeat the above process, using the prime p and the subgroup H in place
of q and H* respectively as follows. If £ denotes the subgroup
£ = 1 x H x x H x H*
of £, evidently UE/UE = H. and via this isomorphism, the FpH-module N can be
viewed, by inflation, as a simple F,,!/E-module with Kerft/E on N) — UE. Since U
(= Soc(R„)) is the only minimal normal subgroup of R„ and U &UE, the hypotheses
of Theorem В, 11.6 are again fulfilled, and we can deduce that there exists a simple
FpR„-module V which is faithful for Rn and whose restriction VEE contains a quotient
module V/V isomorphic with N. As before, there is an f>-projector of the semidirect
product S„ = [F]R„ of the form VUE with V < V < V.
First suppose that p # r (= rn). Then, repeating the exercise, we can find a faithful,
irreducible ^„-module IT over Fr whose restriction ITPE,E has a trivial quotient module
W/W of order r. Set
T„ = M = WVUD„.
Then T„ has an f>-projector of the form X = WVUE with W <. W, and therefore
[Soc(t"), X] = [IT, X] = [IV, VUE] <, W< W, whence (2.a) is satisfied in this case.
On the other hand, if r = p, as before we can find a faithful, irreducible S„-module
Q over F, such that has a quotient module isomorphic with N* (considered as
an F FFE-module by inflation), and so the semidirect product [Q]S„ has an Л-
projector of the form QVUE with Q < Q. Repeating the argument once more we
can conjure up a faithful, irreducible ^'„-module W over F, (= F ) such that
[IK QVUE] < IK In this case set T„ = [IT](QS„), and °bserve that <
satisfied because Soc(T„) = W and T„ has an f>-projector of the form X = WQVUE
f°Now«r/=7Ti T2. ...)andft = A(S).Sincelv(«)sraforr = nexandb,(«) =
0for76P\Z,tJe Schunck classftsatisfies Condition (ii)of(l.17). We wifi now show
444
VI. Further theory of Schunck classes
that it also fulfils Condition (i) of that Corollary. From this it will also follow that
Tj $ Q(T}) f°r all distinct pairs {i,j} £ M and hence that ® is a Schunck boundary.
For i e N let C, denote a stabilizer of 7J; thus С,- s Sf = FLO, if f p, and
Ci = QVUDi when r, = p. If сг(Я) / 0, then r = r, e x and сг(Я) = (C,). Therefore
Condition (i) of (1.17) will be satisfied if we can show that for all pairs of distinct
natural numbers {m, и} the following holds:
(2./<) C„ t Sr„QR„(C„).
We consider the p-chief factors of C„. These are of two kinds: either they lie in the
quotient of C„ which is isomorphic with £>„, or else they are isomorphic with V (and
then, in fact, equal to V or VQ/Q). Let / denote the nilpotent length /(/)„) of £>„; in
fact, / = max{/(B), l(H*)} and is independent of n. Then, because UD„ and VUD„ are
primitive, we have /(l/D„) = I + 1 and l(VUDn) = 1 + 2, and a p-chief factor of C„ is
either 9l'-central or is 9l'-eccentric and has induced on it by C„ the automorphism
group C„/CCn(V) s UD„.
Suppose that r„ # p, and let g denote the class of all groups whose p-chief factors
are either 9l'-central or else have UD„ as their group of automorphisms. By IV, 1.3
this class is a formation and therefore S,mQR0(C„) £ $5- Now the group C„ has an
91'-eccentric p-chief factor with UDm as its induced automorphism group. Since
U„Dm gt U„D„ if m # n, it follows that Cm £ $5, and therefore (2.J3) is satisfied in this
case.
For the case where r„ = p, a similar argument applies using (/-chief factors: The
class Gpqr0(C„) is contained in the formation ® comprising those groups whose
<;-chief factors are either 9l,-1-central or else have one of {Dn, UVD„] as induced
automorphism group. Since Dm $ (D„, UVD„) when m # n, it follows that C„ ф ®, and
so (2.fl} holds in this case as well.
We have now verified the hypotheses of (1.17) and can therefore conclude that
а(Я) = Ь(Я). We have also incidentally shown that Ь(Я) is a Schunck boundary. For
if Bm e q(B„), then Cm e Q(C„), which contradicts (2./J) unless m = n. If и e N and
X e Projg(Tn), then we know from the construction of T„ that Soc(T^) has an X-
central composition factor, Z say. Because
(2-7) [Z]WQ(Z))S<e&
by III, 4.16 we have T„ $ a(F>), and so a(F>) n а(Я) = a(F>) Ь(Я) = 0. Let r e P. If
r i x, then Z, £ F>, and therefore Zr e b(f>) £ a(F> л Я). If r e /, say r = r„, we apply
Lemma 2.3(b) with T„, Z, and X in place of В, F, and H to conclude from (2.y) that
Z, e a(F> л Я). Hence (Zr: r e P) £ a(f> л Я), and the criteria of (2.2) for Я to comple-
ment § are satisfied. □
We now look at the set of all complements in .iff of a particular Schunck class F>
(# (1) or 6). By the previous theorem this set is non-empty, but we now show that
in general it lacks the two obvious candidates for a distinguished element, namely a
unique maximal element and a unique minimal element.
2. Complementation in the lattice
445
(2.5) Notation. If § e we shall use Jf(ft) to denote the set of all complements of
& in Ж. Further, if denotes the set of formations in Ж (we shall see later in (2 71
that & is not a sublattice of Ж), then we shall write Ж,(&) = Ж(й) n.'^ the sL of
complements of ft which are saturated formations.
The methods of construction which we used in the proof of Theorem 2.4 enable us
to prove the following result.
(2.6) Corollary (Forster [6]). Let ft e Ж\{(1), S}.
(a) There exist Schunck classes Я, and R2 in Jf(ft) such that Я, v ft2 = ©• jn
particular, the partially ordered set (jT(ft), «) has no uniquely determined maximal
element.
(b) The infimum of the subset Ж(&) of Ж can sometimes belong to jF(ft); on the
other hand, it may also coincide with the identity Schunck class (1).
Proof, (a) We adhere closely to the pattern of the proof of (2.4).
Case 1: b(ft) = bp($f) # (Zp) for some prime p. We must find complements Rj and
Я2 of & such that а(Я,)пп(Я2) = 0. First we choose a group В in b(ft)\(Zp) and
ensure that B/Soc(B) e Gp. if this is possible.
Let ®0 denote the class of all primitive sections S of B/Soc(B) with the property
that Soc(S) e Sp. and S/Soc(S) e <3p. and let ®, = ®0 о (Z,: p ф q e P), which
is evidently a Schunck boundary. Let S3, =Ii(®,). Then a(ft)na(RI) =
Qp(f>) n Ор.(Я ,) = 0, and consequently ft v Rj = S. If q^p, then Z, e ®, =
b(R,) £ a(ft л R,). On the other hand, if H e Projf,(B), we have H s B/Soc(B), and
from the construction of ®0 it follows that a R, -projector of H is a p-group. Therefore
Zp e a(ft л RJ by (2.3)(b). Thus (Z,: r e P) s a(ft л RJ, and R, e JF(ft) by (2.2).
We now describe the groups which will make up the boundary of a second
complement R2. If q Ф p, let B, denote the semidirect product [F,]B, where F, is a
faithful, irreducible В-module over Fq such that
(2.6) [F„ Я] < К
(recall that H is a stabilizer of the primitive group В and that such a module always
exists by B, 11.7). Let S = (Bq: p Ф q e P). Let r be a prime not dividing |B|, and let
E = £(r/p) the primitive group of p-power degree with stabilizer of order r. If H e <Sp.,
set ®2 = ® and if H $ Sp, set ®2 = Su(£) Then it is clear that ®2 is always
a Schunck ’boundary. Let R2 = h(®2). In view of (2.6) it follows from (23)(b)
that Zq e a(ft л R2) for all primes q * p. If H e we a»dJ* thi“ 1
is an (ft л RJ-projector of H and can apply (2.3)(a) to Beb(ft) to deduce that
Zp e b(ft л RJ; and when H ф we obtam the same conclusion by applying
(2.3) (a) to £ e b(RJ. Thus (Z,: r e P) s a(f> л R2). and to prove that Я2 e Ж(&),
by (2 2) it remains to verify that Q(ft)nQ(RJ = 0. For a contradiction. let
Zf /о i _ „ Jr i and let К e Proj„ (G). Observe that in this case
6P(R2 )'J 0Qand that therefore b(ft) contains no ©„©„-groups by the choice of B. By
111, 4.18 there exists a chain of subgroups
446
VI. Further theory of Schunck classes
К = Mr <•••• <Mt <-M0 = G
such that the group belongs to Ь(Я2) for i = 1, .... r.
Let J denote the inverse image in Л/,., of Soc(£,_,). If for some i there were
a prime q such that L,_, = Bq, then the S)2-group would be isomorphic
with Bq/Soc(B,) S B. Since К is a B2-projector of Lr_1( it covers and
would therefore have a quotient isomorphic with B. But by (1.8)(iv) we have
К e ерокоср(Я2) £ SpSr and could then conclude that В e GpGr, which is not the
case by the choice of r and our initial hypothesis that В £ Zp. Therefore L,._, s £
for i = 1,..., r, and it follows that G e a(h(E)) by III, 4.18 once more. But then by
(1.12) we have G e b(h(E)) = (E), in other words G = E, and hence £ e o(S>) Therefore
q(£) n b(f>) 0, and since b(f>) = b4(f>), we must have £ e b(f>). But then b(f>)
contains an Gf,Gp.-group, and we have a contradiction. Consequently a(f>) n а(Я2) =
0, and we have proved that Я2 e
We now justify the claim that ft, v ft2 = £by showing that а(Я1)с> а(Я2) = 0.
This we do by supposing that there is a prime q such that ajftjna,^) # 0
and deriving a contradiction. Let A e а,(Я,) а?(Я2), and note that q p because
о„(Я1) = Ьр(Я1) = 0. Denote by c the class of a Sylow p-subgroup of H, recalling
that Я is a stabilizer of the primitive group В defined at the outset. Because A e а,(Я2)
and Ь,(Я2) = (lj,B), a B2-projector of A contains a quotient isomorphic with B.
Denoting the class of a Sylow p-subgroup of a group X by yp(X), we therefore have
)’р(Л) > yp(B) > Ур(Я) + 1 = с + 1. On the other hand, we have Sp « Я,, and so a
Si,-projector K, of A contains a Sylow p-subgroup of A. From (1.8)(iv) we know that
K, e G4qr„c4(S<1 ), and because the non-trivial groups of с,(К,) are isomorphic with
sections of H and therefore have class at most c, we therefore conclude that K,
belongs to the formation (X: yp(X) < c), which implies that yp(>4) < c. This
contradiction shows that <!(&,)о а(Я2) = 0, as desired.
Case 2: = C” for some n £ P (this includes the case b(f>) = (Zp), which was
disallowed in Case (1)). Since § £ {(1), £}, note that л Por 0. Let pen, and let
Я, and Я2 be the Schunck classes defined by the boundaries Ь(Я,) = (Zr: r e n') and
Ь(Я2) = (E(p/r): r e n').Thena(f>) = ф n S„а(Я,) = ф n S„.,and а(Я2) = а„(Я2),
and so certainly a($j)а(Я,) = 0 for i = 1, 2. Moreover, the class (Z/. re n) is
obviously contained in a(f> л Я,), and by a simple application of (2.3) one easily sees
that it is also a subclass of a(f> л Я2). By (2.2) we therefore know that Я, and Я2 are
complements of f>. Since a group in а(Я2) has a quotient in Ь(Я2), it has order divisible
by p and hence cannot lie in £„. Therefore а(Я,) n а(Я2) = 0 and Я, v ft2= G,
as desired.
Finally we follow the procedure described in the proof of (2.4) for finding a
complement to when neither Case (1) nor Case (2) applies. We denote the com-
plement Я found in that proof by Я, and define a second Schunck class Я2 by using
the same construction but substituting the group D2„ = В x • x В x H for £>„ in the
notation of that proof (O2n, like D„, is associated with a prime r„). By exactly the same
arguments we see that Я2 e JC(f>) and that а(Я2) = Ь(Я2). If n e N, the group in ЬГп(Я)
has UD„ as a primitive quotient, and this does not appear as a quotient of the groups
WVUD2„ or WQVUD2„ in Ь,п(Я2). Therefore «(Я,)г«(Я2) = Ь(Я,)пЬ(Я2) = 0,
2. Complementaiion in the laltice
447
and we have v == (5 in this final cae#»
of Corollary 2.6. Th,S COmpletes the Proof of Part (a)
(b) First consider the Schunck class f> = S,.. We assert that S is the uniquely
determined minimal element of the partially ordered set (X(S .) <<) Certaink we
ft6jr(S-> Then = and^onVqiS НЯ)"
a(f>) n а(Я) - 0. Thus 6(Я) c = a(<5p), and therefore S„ « Я by (1.5). Hence our
assertion is justified.
To show that the infimum of JT(&) may be (1), take 3 distinct primes Pl, p2, and
p3,let E, - E(p;/p3)fori = 1,2, and set B, = [fj]E„ where Vt is a faithful, irreducible
module for E, over Fp(. Clearly, ® = (B,, B2) is a Schunck boundary, whose asso-
ciated Schunck class fi(®) we denote by Then it is easily checked that this & fulfils
the hypotheses of (1.17), and therefore a(f>) = 6(f>). Let {/, j} = {1,2}, and let Я,-
denote the Schunck class whose boundary is '
b(R,) = (Zp ) ufEfpj/q): q e P\{pj, p2}).
By arguments similar to those that have gone before one readily verifies that Я„
Я2 e jT($j) and that Я2 л Я2 = (1). □
It is clear from the examples described in the preceding Corollary that the particular
complement Я of f> which we found in the proof of Theorem 2.4 has no special status
in the set JT(f>) of all such complements. For example, if the same construction is
used to find a complement of this Я, then it does not usually lead us back to f>. Indeed,
on the evidence of Corollary 2.5, it seems unlikely that any kind of ‘canonical’
complement can be found. The picture improves, however, in this respect at least, if
attention is confined just to the saturated formations of JF, and the rest of this section
is largely devoted to supporting this view. But, before getting under way, we would
point out one respect in which the situation then becomes less satisfactory, namely
the fact that the set & of saturated formations does not form a sublattice of jf, as
the following example shows.
(2.7) Example (Forster [6]). Let p, q, and r be 3 distinct primes, and set
3 = OW9W-
We assert that the join 3 of these two saturated formations is not a saturated
formation. Suppose, for a contradiction, that 3 6 then 3 — ^(/) f°r son]e
formation function f by IV, 4.6. Since E(q/p> and E(r/p) obviously belong to3,
it follows that Zo and Zr belong to /(p) and hence that ZqreDof(p) - ЦрУ
Consequently the group E = Efcr/p) belongs to 3- However, since the 5>W and
91,. ,.-projectors of E are clearly its Sylow q- and Sylow r-subgroups respectively w
conclude that the 3-projectors of E are its Haljq, r}-subgroups, whence E*3- Th.s
contradiction justifies our assertion that 3 Ф J'-
Our next goal is to characterize pairs of saturated formations which complement
one another in the lattice Ж
448
VI. Further theory of Schunck classes
(2.8) Proposition (Forster [6]). Let 5, and 82 be saturated formations. Then
6i v 62 = ® if and only if there exists a set n of primes such that G„6i =61 and
= "Si-
Proof If 3„8i = 31» then S„ « 31 by (1.6)(a), and since <£„, S,.> = S, the stated
condition is therefore clearly sufficient. To prove that it is also necessary, it will be
enough to show that for each p e P either Sp8j = 81 or Gp82 = 82- Suppose that
81 v 82 = and recall from (2.1) that we then have a(8i)<~' a(Si) = 0- We will
argue by contradiction, supposing that there exists a prime r such that
(2.c) <=,8A8.- # 0
for i = 1, 2. Let F; be the canonical local definition of Bi (cf- IV, 3.9), and recall
that G,F,(r) = Ff(r) £ 3i for all r e P. The first step will be to prove the following
statement:
(2.0 Fori = 1,2 there exists a primitive group G{ in whose stabilizer St belongs
to Bi and satisfies SJZ(S^ f Ffr).
Let i e {1, 2}. If Bi = Ffr), then Sr8i = Bo which is supposed not to be the case.
Therefore we can find a group Lr of minimal order in 8ДТ;(г). К Ffr) = 0, then
L,- = 1, and Condition 2.f is satisfied by setting Gf = Zr. Otherwise L,- has a unique
minimal normal subgroup, Nj say, and if q is the prime dividing |7Vj|, then FCLJ
is a q-group. Observe that q =£r because FJr) = £,Р;(г). Let Pt, ..., Ps denote a
complete set of representatives of the projective indecomposable F,L,-modules, and
set C4 = Pj ф • ф Ps. Since the regular F, L-module, which is a direct sum of copies
of these modules Pj, is faithful for Lit evidently Ц is faithful for 0. Let S; denote the
semidirect product [l/JL,-. Then clearly Z(SJ < U„ and therefore SJZ(St) f Ffr]
because SJlf f Ffr) = QF;(r). Because Oq.,,(L.) = F(L,) and Lf e 8,-, it follows that
LJFiL^ e F((q) and hence that S, e 8, because F(L,-) centralizes each composition
factor of Uj by B, 3.12(b). Now by B, 4.7 and B, 4.10(b) the module Pt has a unique
minimal submodule Soc(Pj), which uniquely determines Pt as a projective inde-
composable F,L,-module, and since Ц is faithful for L,, it follows that Soc(S,) =
£si=1 Soc(Pj). Thus the minimal normal subgroups of S; are non-isomorphic in pairs,
and by B, 10.7 there exists a faithful, irreducible S,-module Ц over Fr. It is then clear
that the semidirect product Gf = [10S, is the desired group satisfying the require-
ments of (2.0.
We now complete the proof by deriving a contradiction from (2.0. Let D = S2 x S2,
and regard V = 0 ® V2 as an (S, x S2 (-module in the usual way. If H2 is an 81 -
projector of S2, then the subgroup H = S2 x H2 is an 8i-Pr°jector °f F>. Let W be
an H-composition factor of V. Since VSi is a direct sum of copies of the faithful
S,-module 0, so also is WSt, and it follows that S, n C„(iV) = 1. Hence CH(W) <
CH(Sj) = Z(Sj) x H2, and consequently H/CH(W) has the group H/(Z(S,) x H2)
(=S1/Z(Sl) f Fj(r)) among its epimorphic images. Therefore H/C„(W) e Si\Fi(r),
and it follows that the semidirect product [B/](H/C„(B/)) belongs to h(8i)- Thus,
if U is an irreducible D-submodule of V, it follows easily from III, 3.25(b) that
2. Complementation in the lattice
449
[G](D/Cd(G)) belongs to a(8|); similarly it belongs
a final contradiction that a(8i)r>a(g2) 0.
to a(82), and we have reached
□
(2.9) Theorem (Forster [6]). Let 51 and 82 be saturated fortnations distinct from (1)
and S. Let Z‘ = {p 6 P : = SJ i = 1, 2. Then the following statements are
equivalent:
(a) 8i and Sj complement each other in the lattice
(b) The sets r, and r2 form a non-trivial partition of P. (Thus 0 г, / P =
Proof. Assume that Statement (a) holds. Then 81^82 = ®, and consequently
t, ut2 = P by (2.8). On the other hand, if p e t, n r2, then Sp « 8. for i = 1, 2,
contradicting the assumption that J, л g2 = (1). Hence t, nt2 = (?, and we have
shown that P = t, vjr2. Were t, equal to P, then it would follow that S = Spg, = g,,
contrary to hypothesis. Therefore Statement (b) holds.
The reverse implication requires more work. Assume that Statement (b) holds.
Then certainly v g2 = S by (2.8). The burden lies in proving that 8, л g2 = (1).
To this end we set
Л = Char(8i)\T|,
and we let Fj denote the canonical formation function which locally defines 8;
(i = 1, 2). We also set
® = 81 n 82
and recall from IV, 3.5(b) that G = LF(G), where G(p) = Fl(p)r F2(p) for all p e P.
We shall need the following hypothesis:
(2-4)
Sp® Ф G for some p e t2.
Our next goal will be to prove the following statement:
(2,0) Assume that (2.1/) holds, and let q e ф2. Then there exists
a group H e 8i\F,(q) with the following properties:
(a) Soc(H) is a minimal normal p-subgroup of H:
(b) Z(H) = 1;
(c) If F2 e ProjRj(H), then F2 r.Soc(H) = 1.
Hypothesis 2.>, implies that G * G(p\ and therefore we can find a group, R say of
minimal order in G\G(P). If R * 1. then R is monolithic, and we denote the umque
prime dividing |Soc(R)| by r. Let q e , and note that q e P\T}J
S and we can find a group, T say, of minimal order in 8i\ 2(q). У
J? we ha7e FAq) * 0 and hence T± 1. Again, T is monolithic, and we daiot
byt'the unique prime dividing |Soc(T)|. At this point we make some observation
for later reference.
450
VI. Further theory of Schunck classes
(A) Since G and are canonical formation functions, and are in particular full,
we have r Z p and t # q.
(B) Since pet,, we have Si = F,(p). If R belonged to F2(p), we should have
R e F\(p) n F2(p) = G(p), contrary to the choice of R. Therefore
R e %2\F2(P),
and in particular, if R = 1, then F2(p) = 0 and g2 £
(C) If R jt 1, we have Or.(R) — 1. In this case Or. r(R) = F(R) and R./F(R) e
F\(r) FiW-
In proving Statement 2.в we distinguish two cases.
Case 1: We have t p. If R = 1, we set R* = Z,, and note that R* e g, n g2 because
Fi (q) Z 0 and because q e r2. On the other hand, if R # 1, we let N denote the direct
sum of the projective indecomposable IF, R-modtiles, including only one representa-
tive of each isomorphism class, and let R* denote the semidirect product
R* = [7V]R.
Then R* e g, n g2 by Observation (C). Furthermore, since N is faithful for R, we
have Z(R*) < N (in fact, Z(R*) is the socle of the principal indecomposable summand
of N and has order r). Because R*/Z(R*) therefore has R as an epimorphic image
and because R F2(p) by (B) (whether or not R = 1), we can conclude that
R*/Z(R*) F2(p).
If R = 1, then R* is primitive, and if R / 1, then the minimal normal subgroups
of R* are the (pairwise-non-isomorphic) socles of the projective indecomposable
summands of N. Since in either case Op(R*) = 1, by B, 10.7 there exists an irreducible
FpR*-module, F, say, which is faithful for R*.
Working analogously with T (Z 1), we let M denote the sum of the projective
indecomposable IF, T-modules and set T* = [M] T. By repeating the above arguments
we can deduce that T* e g2, that T*/Z(T*) ф Ft(q), and also that there exists an
irreducible FpT*-module, V2 say, which is faithful for T*.
Let V be an irreducible submodule of the Fp(R* x T*)-module fj ® V2. Let L be
an g2-projector of T*, and note that R* x L is an g2-projector of R* x T*. Let V
be an (R* x L)-composition factor of V. Since UR. is a direct sum of copies of
the faithful module Ц, we have Ker((R* x L) on V) < Z(R*) x L. Therefore
(R* x L)/Ker((R* x L) on U) e g2\F2(p), and we conclude that the semidirect pro-
duct [U]((R* x L)/Ker((R* x L) on U) belongs to b(g2). Let C = CR.xT.(P), and
let H denote the semidirect product
H = [F]((R* x T*)/C).
Now (R* x L)C/C is an g2-projector of (R* x T*)/C, and the preceding argument
shows that (R* x L)C/C is also an g2-projector of H; m fact, we have shown that
H e ap(g2) since H is primitive. Thus Properties (a) and (c) of (2.0) hold for this group
H. Since VT. is the sum of copies of the faithful module V2, we have T* n C — 1.
2. Complementation in the lattice
451
Similarly R* n C = 1, and consequently C < Z(R*\ x ZiT*\ Since T j i „j
fore T*/Z(T*) # 1, it follows that (R* x r»i/r -A J t ’S t T # 1 and there‘
D e . \ " lnatx Г )/C A 1 and hence that Z(H) = 1 Thus
Property (b) also holds. Finally, since R* x T* e D„g, = g., we have H e £ n't
gi. But H has T*/Z(T*) ($ F,(q}} among its epimorphic images, and so Hi Ftq]
Thus Statement 2.0 is fully justified in this case * '™'
Case 2: We have t - p. Let R*, T*, and F, be as in Step 1, let Q e Hall (T*), and if
I/ is the trivial FpQ-module, let V2 = U^. By B, 6.16 the module F2 is ?he principal
indecomposable FpT*-module, and, in particular, Soc(F2) is the trivial module. If
P e Sylp(T*), then (V2)p is isomorphic with the regular module IF P. Since t = p, we
have Op(T*) = F(T*) < P; therefore V2 is faithful for F(T*) and hence for T*.
We set V = F, ® F2, viewed as an R* x T*-module, and assen that
(2-0 Soc(F) = Soc(F,) ® Soc(F2) s F,.
In order that the isomorphism in (2.1) should make sense, we must view Ft as an
R* x T*-module by letting 1 x T* act trivially. But since Soc(fj) = V2 and Soc(F2)
is the trivial T*-module, the isomorphism is self-evident; in particular, we see that
F, ® Soc(F2) is an irreducible submodule of F, ® P2. However (Ц ® V2)T. is a direct
sum of Dim (F,) copies of Fj, and so the socle of (F, ® V2)r. is a direct sum of Dim( F,)
copies of Soc( F2) and therefore coincides with (F, ® Soc( F2))r.. If W is an irreducible
submodule of Vt ® V2 different from F, ® SocfF,), then 5ос((Ц ® F2)r.) n И2. # 0,
and from this we conclude that (lj ® Soc(F2)) n W # 0, which is impossible because
the modules are irreducible and supposed distinct. Hence F2 ® Soc(F2) is the unique
minimal submodule of P, ® F2, and Assertion 2.i is therefore justified.
Now let C = Cr.. 7 .(F, ® F2), and set
H = [Ц ® F2]((R* x T*)/C).
Evidently Soc(H) coincides with Soc(F, ® P2), which we have shown to be iso-
morphic with F1; viewed as an R* x T*-module. The arguments used in Case 1 are
equally applicable here and show that
(i) the g2-projectors of H avoid Soc(H), and
Moreover, since Soc(H)(= Pj) is faithful for R* A 1, we have [SocfH), R ] — Soc(H),
and consequently Z(H) = 1. Therefore once again the group H we have constructed
fulfils the requirements of (2.H), and the proof of that Statement is complete.
Next we aim to prove the following assertion.
(2.x)
If (2.q) holds, then g, л g2 E
To do this, we suppose that (2.x) fails and derive a contraAction. The^ore let D be
a group of minimal order in (g, л g2)\5 Then D is a prmntiv group, O,(D) -
Soc(P) for some and if К is a stab.lizer of D,
Since P = T! vj r2, we have Ti)n(l/'2 и тг) V'l'-'l/'2 1
452
VI. Further theory of Schunck classes
81 n g2 S Consequently q e and there exists a group H satisfying the
requirements of (2.(1}.
Let Wt = O,(£>), viewed as an irreducible F,X-module, faithful for K, and let W2 be
a faithful, irreducible Я-module over IF, such that (Hj)^ has a trivia] factor module,
where F2 6 ProjgJH); according to B, 11.7 the Properties (a) and (c) for H in
(2.0) ensure that such a module exists. Let W be an irreducible submodule of the
F,(X x H)-module И] ® W2, and observe that W is faithful for К x H because
Z(H) — 1. Because К e g, л g2 £ g, and H e gi\F,(q), it is clear that К x H e
g,\Fi(q) and hence that [И/](К x Я) belongs to b(g,), which is a subclass of
e(8i A g2) by (1.5).
Let F be an gj л g2-projector of Я contained in the g2-projector F2 of Я. Then
К x F is an g, л g2-projector of К x Я. Since WH is a sum of copies of W2, the
construction of W2 implies that [И7, F] < [И7, F2] < И7, and therefore, if is viewed
as a К x Я-module in the obvious way (with Я acting trivially), then W/[W, F] is a
sum of copies of W2. Consequently has a factor module U such that the
semidirect product
L = [C](K x F/CKxf(U))
is isomorphic with W, К = D e g, л g2. But since [IF](K x Я) 6 a(gj л g2), by
III, 4.16 we have Le £>(g, л g2), which is a contradiction. Therefore (2.k) holds.
We are now close to completing the proof that 3, л g2 = (1). Since S / ©
(= 8i n g2)and Tj и т2 = P, the equation St|© = © holds for at most one i e {1, 2}.
Without loss of generality suppose that it does not hold for i = 1, and note that then
Hypothesis 2.q is satisfied. Consequently 3, л $2 E by (2.k).
Now let s 61//2; then s r2, and so g2\F2(s) is non-empty and therefore contains
a group, J say, of minimal order. Let q e r2 (/ 0), let M denote the sum of the
projective indecomposable F,J-modules, and set L = [M] J. Since g2 = S,g2, we
clearly have Le g2\F2(s). Choose an S in Hall1>2(J) (£ HalHJL)), and let U be the
trivial FsS-module. It is straightforward to verify that the hypotheses of B, 11.6 are
satisfied and hence to deduce that the module UL has a faithful, irreducible quotient
module V such that [ИХ] < И The semidirect product [F]L therefore belongs to
h(g2), and by (1.5) we obtain VLe a(g, л g2).
Let X be an g, л g2-projector of S, and observe that X is then an gj л g2-
projector of VL because g, л g2 E S1>j. From the fact that [И X] < [И S] < Ц we
conclude that Vx has a trivial composition factor IV. Because FLea(g, л g2), we
know that the class £>(g, л g2) contains [И/](Х/Сх(И/)) = li'EZ,. Thus we have
shown that g] л g2 c £T. Since this is true for all s e 02, it follows that
8i л gj^S^nQ^-d),
as desired. □
3. D-classes
453
3. D-classes
Our theme in this section is Schunck classes which have the property that their
projectors, like Hall subgroups, satisfy a D-theorem (cf. I, З.З^ДисЬ Schunck
classes were first investigated by Wood [2]. ' ’ scnunck
(3.1) (a) A class X is said to have the D-property in a universe D, if every
group G in SB has a unique conjugacy class of maximal X-subgroups; then it is called
a D-class (in SB).
(b) A Schunck class which is a D-class in 6 will be called a Schunck D-class, and
the set of all Schunck D-classes will be denoted by the script letter Q).
(3.2) Examples, (a) By Sylow’s theorem S„ is a D-class in G for all pe P.
(b) If л £ P, by Hall’s theorem S, is a Schunck D-class.
(c) Let M £ hi. The class ®M comprising groups which can be generated by a set
of elements whose orders belong to Л/ is a D-class in any universe. For an arbitrary
group G the subgroup
<S e G: o(g) e M)
is obviously the unique largest ft „-subgroup of G. If M = {4} and SB = 6, the cyclic
groupZ4oforder4isin ®A/. But Z2 and Z4/Z2 Z2 are not contained in ®M. Hence
®M is neither Q-closed, s-closed nor s,-closed. Since Z16 ф ®A/, we have (®M)2 A ®M,
and since Z8 ф ®M, also еф®м # ®M.
(d) In any universe the class of л-perfect groups, namely groups X with X = On(X),
is a D-class. For any group G the subgroup O"(G) is clearly the unique maximal
л-perfect subgroup of G. We have denoted this class by Q” in the universe 6, and
then it is an N0-closed Schunck D-class, which is neither s„-closed nor rc-closed.
(3.3) Remarks, (a) Let X be a D-class, and suppose that a group G has X-subgroups
H and К with G = HK. By definition of a D-class, К is contained in a maximal
X-subgroup M of Gand His contained in Me for some g e G. ThusG = HK = MM'
and there exist such that g = nig 1m'g. Therefore g eM and G = M e X.
Thus in particular, a D-class is always N0-closed.
(b) If X is a D-class and X = e»X, then X2 = X. To see this, let G e X2; then G has
a normal subgroup К e X such that G/K 6 X. Let H be a minimal supplement to К
in G. Then H e ефХ = X (see A, 9.2(c)), and as in Remark (a) it follows that G e X.
(c) Let X and 9) be D-classes in a subgroup-closed universe ®. Then X n 9) is a
D-class in ®: Let D,, D2 be two X n ?)-maximal subgroups of a group Ge®. Since
X is a D-class, we can assume that D„ D2 are both contained in an X-maximal
subgroup X e ® of G. IfX < G, by induction on the order of G, the subgroups D, and
D2 are conjugate in X. Therefore we may assume that G = X e X and similarly that
G WM^chunck class § "contained in a Schunck D-class T is clearly strongly
contained in T'.
454
VI. Further theory of Schunck classes
(3.4) Lemma. Let Sy be a D-class in <S, and let Char(g) = n.
(a) If g is a Schunck class, then « Sy « Q”'.
(b) If Sy is a Fitting class or a saturated formation, then § = S„.
Proof If Sy is a Schunck class, a Fitting class, or a saturated formation, then Sp c §
for all pen. Consequently a maximal g-subgroup M of a group G contains a Sylow
p-subgroup of G for each pen, and so if G e S„, then G = M e g. Thus <5n c g.
Since § is contained in the Schunck D-class Q"’, Statement (a) now follows from
(3.3)(d). Statement (b) follows from the fact that, under the given assumptions g s
(see IX, 1.9 and IV, 4.3). □
We will be mainly concerned with Schunck D-classes from now on. Therefore
for the rest of this chapter we confine ourselves to the universe S.
Of course, even with this restriction, not every D-class is a Schunck class (e.g. the class
®12| defined in Example 3.2(c)). However, when a D-class X is indeed a Schunck class,
then the maximal X-subgroups must obviously coincide with the X-projectors; thus
a Schunck class 3E is a D-class if and only if every X-subgroup of a group G is contained
in an X-projector of G.
In attaining our first objective, which is to prove that the set 2 of all Schunck
D-classes forms a sublattice of JF, the following criterion will be helpful; it shows that
in order to establish that a Schunck class is a D-class, it is sufficient to check the
boundary groups.
(3.5) Lemma (Doerk, [4]). A Schunck class g is a D-class if and only if each group
В e b(g) satisfies the following condition:
(3.a) Every ^-subgroup of В is contained in an ^-projector of B.
Proof. The necessity is clear. To prove the sufficiency we assume that (3.a) holds, we
suppose that § is not a D-class and finally derive a contradiction. Under this
assumption there is a group G of smallest order which contains an g-subgroup X
not contained in any g-projector of G. Let N be a minimal normal subgroup of G.
The choice of G means that the g-subgroup XN/N is contained in some g-projector
H/N of G/N, where by III, 3.25(c) we have H = HN for some H e Proj&(G). Since
X < HN and H e Proj^HN), again the choice of G forces HN = G. If Corec(H) were
non-trivial, we could choose N inside Corec(H) and conclude that G = HN = H,
which is clearly not the case. Hence Corec(H) = 1, and G is primitive; moreover, since
G = HSoc(G), it follows that G e b(g), and so by (3.a) the g-subgroup X is contained
in a conjugate of H, contradicting the choice of G. □
(3.6) Theorem (Wood [2]). Let Sy and R be Schunck D-classes. Then
(a) g v Я is a D-class, and
(b) g л R is a D-class and coincides with g n R.
In particular, & is a sublattice of Xf.
3. D-classes
455
«ыв! в» м-Лк *" 5 ” “Ьв«~Р "Г С. .«1 »Ле
Д , S’ 'Я, “°""' ,о Р™ ' В «mined In от
complement to N in G. Now by III, 3.24 there is a complement C to N in G such that
J = (J n C)(J n N).
Let H be an ^-projector of J n C. By III, 5.8 there is an g-projector L of G contained
in the § v Я-projector C of G, and because § is a D-class, every g-subgroup of G is
contained in some conjugate of L and hence avoids N. It follows at once that H is
g-maximal in H( J n N) and is therefore an g-projector of H(J r-, N) by III, 3.14. Thus
by III, 3.25(b) the subgroup H is an g-projector of (J r-, C)(J n N) and hence of J.
Similar reasoning shows that J r.C also contains a «-projector of J, call it K. Thus,
as J e g v « = <g, «), we have J = (H, К) < C, as desired.
(b) By (3.3) (c) the Schunck class g «is a D-class. To complete the proof we show
that g л « = g n «. The inclusion g л « £ g n « is obvious. By (3.3)(d) we have
g n « « g and g n « « «. Thus g n « « Sup{£: £ « g, £ « «} = g л «. □
In (2.4) we showed, with some effort, that the lattice JF is complemented. We shall
now show, with great ease, that the sublattice & is also complemented; however, like
Xf, it is not modular and therefore not distributive.
(3.7) Proposition (Wood [2]). The lattice & is complemented but not modular.
Proof. Let g be a Schunck D-class, and let n = Char(g). Then by (3.4) (a) we have
S = Qn v £ g v S„-, and therefore g v 6„. = 6. On the other hand, g л 6Ж.
= g n £ El”' n S„- = (1). Since (5,. is a Schunck D-class, it is a complement to
g in
To see that @ is not modular, take g = 62> Я = S3, and £ = Q3, and note that
these are all Schunck D-classes. Then clearly Sym(3) 6 (g v «) n £ = (g v Я) л £.
However, g л £ = g n £ = g and « л fl = (1), and therefore (g л £) v (« л fl) =
g, which does not contain Sym(3). О
The next result provides another criterion for a Schunck class to be a D-class; like
(3.a) in Lemma 3.5, it refers to the structure of boundary groups.
(3.8) Lemma (Forster [6]). Let $bea Schunck class. Then f>isa D-class if and only
if for all В 6 b(g) the following condition holds:
(3./J)
If X is an ^-subgroup of a complement of Soc(B) and if Vis
an X-composition factor cf Soc(B), then [F] (X/Cx(F)) 6 b(g).
456
VI. Further theory of Schunck classes
L = Lc\ NX = (Ln N)X = X. Hence X is ^-maximal in NX and is therefore an
^-projector of NX by III, 3.14. Proposition III, 4.18 implies that [F](X/Cx(F)) e
b(§) for all X-composition factors V of N, and therefore Condition 3./J holds.
To prove the sufficiency of Condition 3./J, assume that it holds for all В e b(§). We
aim to show that Condition 3.a of (3.5) holds. Let В = NH e b(§) with N = Soc(B)
and H e Proj6(B), and let X be an ^-maximal subgroup of B. By III, 3.14 this X is
an ^-projector of XN, and in view of (3./J) it follows from III. 4.18 that X n N = 1.
Hence by III, 3.24 we have X = X n He, that is X < H" for some g e B. Thus (3.a)
holds, and § is a D-class by (3.5). □
The criterion of this Lemma affords a procedure for constructing a variety of Schunck
D-classes.
(3.9) Example(Doerk [4]). Let n £ P. Let g = Qg = sg s S„., and let SB = SB(n, g)
denote the class of all primitive groups G with G/Soc(G) e g and Soc(G) e S„. Then
SB is evidently a Schunck boundary. Furthermore, if В 6 SB and 17 is a subgroup of a
stabilizer of B, then V e sg = g. If V is a 17-composition factor of Soc(B), then
U/Cv(V) 6 Qg = g, and therefore [FKG/C^fF)) e SB. Consequently the class § =
h(SB) is a Schunck class whose boundary SB satisfies (3./J), and therefore f) e @.
As a special case of this method, let л = {3}, and let g denote the class of elementary
abelian 2-groups. Then the class SB (= SB(n, g)) is (Z3, Sym(3)) in this case. If
§ = li(SB), then G e § if and only if G has no maximal subgroup of index 3, and
H e Proj6(G) if and only if the following two properties are satisfied:
(1) He6;
(2) There is a chain of subgroups
Н = Н0<Я1< -<H„ = G
with | Hj: | = 3 for i = 1,..., n.
(This is a special case of the situation described in III, 4.18.)
To end this section we consider two generalizations of Schunck D-classes.
(3.10) Definition. A class X of groups is called idempotent if
I1 = X.
In terms of closure operations this is equivalent to saying that J is E-closed (cf. II, 1.5).
(3.11) Remarks. (a) If 9) is a class of finite simple groups, the class of all finite
groups whose composition factors are in 9) is evidently an idempotent class.
(b) By (3.3)(b) any Enclosed D-class is idempotent, and so in particular Schunck
D-classes are idempotent.
(c) In Chapter IV, Section 1 we studied the formation product of two formations.
Pense [1] has contrasted idempotence for formation products with the concept
defined in (3.10) by means of the following characterizations (valid in any s„-closed
universe):
3. D-classes
457
ж! И I h °5 fThen S2 = S if and only if N ф 5 Whenever 1 # N < В e b(R)
(n) Let g be a formation. Then g о R = ft if and onlv if A t
g e b(^) ° ° ° 11 anQ ОП1У “ Soc(B) ф ft whenever
(d) In the same work Pense proves the following analogous criteria for the idem-
potence of Fit ting class products (defined in IX, 1.10) vis -a-vis ordinary class products
(for a Q-closed universe): и
£ Г S"5'Then ®2 = 5 if and оп|Уif B/H18 whenever N < В e
(,1)BLe‘J^e “ F,tt'ng Class' Then ft О ft = ft if and only if B/Cosoc(B) ф ft when-
ever В 6 b(ft). (Here Cosoc(B) denotes the unique maximal normal subgroup of the
single-headed group B.)
The proofs of Pense’s criteria in (c) and (d) are straightforward.
The next result resembles (3.5) and (3.8) in providing a test for the idempotence of
a Schunck class in terms of its boundary.
(3.12) Lemma (Forster [6]). A Schunck class § is idempotent if and only if the
following condition is fulfilled by all groups В e b(Sy):
(З.у)
If X < J e Proj ;,(B) with X e §, and if V is an irreducible
X-submodule of Soc(B), then [ИЩСИЮ) 6 b(6).
Proof. First assume that §2 — f>. Let В e b(£>), and let X be a normal ^-subgroup
of a stabilizer J of B. Write N = Soc(B), and let У be an ^-maximal subgroup of XN
with X < Y. By III, 3.14 we have Y e Proj 6(XN). Since YN = XN < JN = B, we can
apply the Frattini argument to conclude that В = NG(Y) YN = NG(Y)N, and because
У nN is normalized by NG(Y) and centralized by N, it follows that Yr-.N<B
Thus У n N = N or 1. If У n N = N, then У is a normal ^-subgroup of B, and
B/У e Q(B/W) S §. But this implies that В e §2 = §, which is not the case. Hence
У n N = 1, X = У, and from III, 4.18 we can deduce that [F](X/Cx(V)) e b(&) for
all X-composition factors V of N', in particular, Condition 3.y holds.
Conversely, assume now that (З.у) holds. We suppose that §2 # 6 and derive a
contradiction. Since § ez 5j2 = q§2, by III, 2.2 a group В of minimal order in § \F>
belongs to b(§). By definition of §2 the group В has a normal subgroup Ref) such
that G/R e f), and since В is primitive and R # 1, it follows that Soc(B) < R. Let
H e Proj 6(B), and let X = R n H. Since R = R n Soc(B)H = Soc(B)X, we have X S
R/Soc(B) 6 Qf) = f) Let V be an irreducible X-submodule of Soc(B). Since Soc(B) is
irreducible as an H-module, it is completely reducible as an X-module by Clifford s
theorem, and so has an X-submodule W such that V © W = Soc(B). But then
[F](X/Cx(F)) S RJWCX(V) e Qf> = f),
contradicting (З.у). It follows that our supposition is wrong and hence that f)2 = &
In the following example we describe a pair of idempotent 6 and
г such that s л г is not idempotent. Thus the set of idempotent Schunck classes
458
VI. Further theory of Schunck classes
does not form a sublattice of Ж, and since the class T is actually in the sublattice @,
it follows that § ф &. Consequently & is a proper subset of the set of idempotent
Schunck classes.
(3.13) Example (Forster [6]). Let S = Sym(4), and let N denote an irreducible F5S-
module, faithful for S, of dimension 3. (We could take the module N described in HI,
4.34 for example.) Thus NS is primitive, and if A = Alt(4) < S, an easy application
of Clifford’s theorem shows that NA is also primitive. Thus the class
b(§) = (Z5, Dih(10), NA, NS)
is the boundary of some Schunck class, § say. Since b(fN) = b5(f>), the ^-projectors
of a group have 5-power index, and it is clear by inspection that each group in b(§)
satisfies Condition 3.y. Therefore § is an idempotent Schunck class. If we now take
T to be the Schunck class whose boundary is given by
b(T>) = (Z3, Sym(3)),
then T is a Schunck D-class as pointed out in (3.9) and is therefore idempotent.
Since the §- and T'-projectors of a group have respectively 5- and 3-power index,
it follows that 62 « § л T>. But the f>- and T'-projectors of NS are respectively
the Hall {2, 3}- and {2, 5}-subgroups of NS, and so Proj6„ j,(NS) = Syl2(TVS). Let
T e Syl2(TVS). Since TV has a faithful, irreducible T-submodule V of dimension 2 (V
is L2 in the notation of III, 4.34), and since T is an § л D-projector of VT, it follows
that VT e b(§ л T). Let
IB = b(T) и (Z5, Dih(10), VT).
Since SB is clearly a Schunck boundary, we have SB = b(£) for some Schunck class £.
The calculations of III, 4.34 show that
[N/U](T/Cr(W/t/)) s Dih(10),
and consequently NS e a(£). Since O2(A), a four group, is the £-projector of A, and
since it induces on the 02(/)-composition factors of N cyclic automorphism groups
of order 2, it follows that NA e a(£), and hence that b(§) £ a(£). Furthermore, we
have b(T) £ b(£) £ a(£), and therefore С « § л t (in fact, it can be shown that
li = § л ®). Since VT e b(S?> л T>) and Z4 <3 T(~ Dih(8)), it would follow from (3.y)
that VZ4 e b(§ л t) if§ л t were idempotent. But because b(§ л t) £ a(£) and
Z4 6 £, we could then conclude that VZ4 e b(£), which is not the case. Therefore
§ л T) is not idempotent.
The boundary of the Schunck class § in the preceding example satisfies the
following condition:
(3.b) If Be b(§), К < J e Projs,(B), and X e Proj^fK), then
[F](X/Cx(I )) 6 b(§) for all X-composition factors V of Soc(B).
3. D-classes
459
It is clear that Condition 3.<5 implies Condition 3.y and is implied by Condition 3 Й
Consequently, the set of Schunck classes § whose boundary groups satisfy (3 6) lies
between У and the set of idempotent Schunck classes; it is characterized by the
following property. 7
(3.14) Definition. A Schunck class § is said to have the 1Г-property if, for all G e G
an g-projector of each normal subgroup of G is contained in some g-projector of G.
(3.15) Theorem (Forster [6]). A Schunck class g has the D~-property if and only if
b(g) satisfies Condition 3.6.
Proof. Let § be a Schunck class with the £>”-property. Let В e b(g), let К < J e
Proj6(B), and let X e Proj6(K). Write N = Soc(B), and let Y be an g-maximal
subgroup of XN with X < Y. By III, 3.14 we have Y e Proj6(XA), and since YN/N =
XN/N e Projb(KN/N), it follows from III, 3.25(b) that Y e Projs(KA). Since
KN < B, the D”-property implies that Y < J* for some J* e Projs(B). Therefore
Yc\N < J*oN = 1, and so Y= Y nXN = X(YnN) = X. Thus X 6 Proj6(X7V),
and since X n N = 1, it follows from III, 4.18 that (3.<5) holds.
Conversely, now let § be a Schunck class whose boundary groups satisfy Condition
3.<5. We suppose that § does not have the -property and derive a contradiction.
In this case there exists a group G which has a normal subgroup R with an g-projector
X not contained in any g-projector of G. Suppose that, among such groups, G has
minimal order, and let N be a minimal normal subgroup of G. Since XN/N is an
g-projector of the normal subgroup RN/N of G/N, the choice of G implies that
XN/N < HN/N for some H e Proj6(G). Now X is an g-projector of the normal
subgroup HN n R of HN since projectors persist in intermediate groups. If HN < G,
it follows from the minimal choice of G that X is contained in an g-projector of HN,
and hence X < H” e Proj6(G) for some n e N, contrary to supposition. Therefore
G = HN. If Corec(H) were non-trivial, we could choose an V inside it and derive a
contradiction. Thus G is primitive and belongs to b(g), and since R # 1, it follows
that Soc(G) = N < R.
Let J 6 Proj?(G), К = R n J < J, and Ye ProjjjRj. Since G satisfies (3.<5), it
follows from III. 4.18 that Ye Proj^TA). But clearly YN/N e Proj6(KA/N) =
Proj sfR/N). and so by III. 3.25(b) we have Y e Proj f,(R). Thus X is a conjugate of Y
and is therefore contained in some conjugate of J, which is an g-projector of G. This
final contradiction proves that g has the D -property. О
In Example 3.13 the Schunck class T is a D-class. and the boundary of g satisfies
(3.b); both g and T therefore have the /Г-property. But g л is not even idem-
potent, and so the set of Schunck classes with the D~-property does not form a
sublattice of .Yf, however, this set is closed under the join operation.
(3.16) Proposition (Forster [6]). If Schunck classes g and R have the D^-property,
then so does their join g v R.
460
VI. Further theory of Schunck classes
Proof. Let В e h(f> v Я), and let J be an § v Я-projector of B; further, let H be an
5 v Я-projector of a normal subgroup К of J, and let L (> H) be an § v Я-maximal
subgroup of HN, where N = Soc(B). In order to show that В satisfies Condition 3.<5
for § v Я, by III, 3.23(a) it will suffice to show that L = H.
By III, 5.8(a) we have H = (X, У> for some X e Proj8(K) and Y e Proj «(£') Since
§ is a D'"-class, an ^-maximal subgroup of XN containing X, which by III, 3.23(a)
and III, 3.25(b) is an ^-projector of KN < B, is contained in a conjugate of J and so
avoids N. Thus X is an ^-projector of KN and is therefore an ^-projector of L.
Similarly У is a Я-projector of L. Thus by definition of § vfl = <f>, Я>, it follows
that L = <X, У> = H, as desired. □
We have already observed that the Schunck class § of Example 3.13 has the D'rj-
property but not the D-property. We bring this section to a close with an example of
an idempotent Schunck class which does not have the -property.
(3.17) Example. Let £ = £(5/11), a primitive group of order 55, and let W =
£linal Alt(4). Then W has a unique normal subgroup of index 5, which we denote by
B, and it is straightforward to verify that В is primitive, that Soc(B), which we denote
by N, has order 114, and that a stabilizer J of В has a unique chief series 1 < M <
L < J with |M| = 53, [I.: M\ = 22, and |J: L| = 3. Set
® = (Z5, Zn,Dih(10),BX
Then ® is evidently a Schunck boundary, so that ® = b(fr>) for some Schunck class
Sj. The only normal subgroups К of J which belong to § are J and 1, and the
corresponding groups [Е](К/Ск(Е)) that arise in Condition З.у applied to В are
В itself and Zt,, both of which belong to b(§). It is clear that the remaining groups
of namely Z5, Zn, and Dih(10), also satisfy Condition З.у, and therefore § is
idempotent by (3.12). However, the ^-projector H of the normal subgroup Lof J has
order 22, and so Dih(22) arises as one of the groups [У] (H/CH(F)) when Condition
3.<5 is applied to B. Since Dih(22) ф b(Sj), it follows from (3.15) that § does not have
the №-property.
Exercises
1. Justify the assertions of (3.2)(c) that ©M is not closed under any of the operations
Sn, R-o. Еф.
2. If ЗЕ is a class of groups, a Fischer 3E-subgroup U of a group G has the defining
properties: (i) U e 3E and (ii) if [Ц G] < V e 3E, then V < U. If 3E is a Schunck class,
prove that
(a) 3E is idempotent if and only if the I-projectors of each group G are Fischer
3E-subgroups of G, and
(b) 3E is a D-class if and only if the I-projectors and Fischer 3E-subgroups coincide
in each group G.
3. Show that the minimal elements (atoms) of the lattice @ are precisely the classes
<SP and the maximal elements (dual atoms) the classes C₽ (p e P). Deduce that the
lattice is both atomic and dually atomic.
4. Schunck classes with normally embedded projectors
461
4. (Forster [6]). For n - 3, 4 let B„ be a primitive group with В /SoctB 1 = Svmim
Г Fl S/CB(F»5’gh’UPFLet ® den°te Ше C‘aSS C°nS1Stlng of a11 semidirect products
through the irreducible M'-modules and X = Alt(4)
or Dih(8). Show that (B3, B4) u 53 is the boundary of a Schunck class § which has
the following properties:
(i) § is not a D-class, and
(ii) if Я is a Schunck subclass of then Я «
(Cf. Remark 3.3(d).)
4. Schunck classes with normally embedded projectors
The theory developed in this section and the next is largely the work of Peter Forster
and grew out of an attempt to classify Schunck classes whose projectors always cover
or avoid chief factors. The corresponding problem for saturated formations had
already been settled by Doerk [1] and Doerk and Hawkes [1] (see Section 5).
Let P denote an embedding property of a subgroup in a group—for example, we
set P = NE to describe normally embedded subgroups, P = CAP to denote sub-
groups with the cover-avoidance property, etc. Then we shall denote by -#P the family
of all Schunck classes § with the property that, for all soluble groups G, the §-
projectors of G have the property P as subgroups of G. Thus in this notation Theorem
4.4 from Chapter III can be formulated thus:
= {O’:* £ P}.
It follows from I, 7.1(b) and I, 7.3(e) that
-wz5 z— т/f t— 'iff
^normal — ^NE — ^CAP-
At the time of writing no useful characterization of the Schunck classes in is
known. However, one of Forster’s achievements in [2] is to give an explicit descrip-
tion of the Schunck classes in the family This is stated :n (4.18), and its proof is
the main concern of this section.
We recall from I, 7.2(a) that a subgroup Я of a group G is said to be p-normally
embedded (written H p-ne G) if a Sylow p-subgroup P of H is simultaneously a Sylow
p-subgroup of its normal closure <PG>. By I, 7.3(d) such a subgroup satisfies
(4.a)
H r.N < G for all normal p-subgroups N of G.
It turns out (see Corollary (4.12) below) that Condition (4.a), when satisfied by aU
pairs (H G) with H e Proj^fG), is not only necessary for a Schunck class § to belong
to JT_NE but is also sufficient. This motivates the following weaker condition, which
will be helpful in the analysis of these Schunck classes:
(4jj) Hr, N <2 N for all normal p-subgroups N of G.
For convenience we incorporate this property in a formal definition.
462
VI. Further theory of Schunck classes
(4.1) Definition. A subgroup H is said to be p-stably embedded in G if it satisfies
Condition 4./1. (We denote this by H p-se G and the corresponding embedding
property P by p-SE.)
We shall see in Corollary 4.17 below that for any prime p
^p-SE n •tfp-CAF = -&P-NE-
(4.2) Terminology and Notation. Let p be a prime, let § be a Schunck class, and let
G be a group.
(a) A non-zero Fp G-module V is said to be ^-covered if the subgroup VH of the
semidirect product [ F] G belongs to § for some (and hence all) H e Proj6(G); more-
over, Vis said to be ^-avoided if H e Proj 8( VH) for some (and hence all) H e Proj 6(G).
(b) We write Mod'(G) for the non-zero ^-covered FpG-modules and Mod“(G) for
the ^-avoided ones. (We omit the suffix p when the prime in question is clear from
the context.)
(4.3) Remarks, (a) Modc(G) и (0) and Mod“(G) и (0) are <q, D„)-closed classes of
modules.
(b) If G e f>, then each simple FpG-module belongs to precisely one of Mod“(G)
and Modc(G).
(c) If G e and Modp(G) is empty, then G is a p'-group.
(d) Let GeSj. Then Modp(G) is empty if and only if Modp(G) is universal (which
by definition means that it contains all non-zero FpG-modules).
We briefly justify these remarks. In (a) the Q-closure follows obviously from the
fact that projectors for a Schunck class are preserved in quotient groups. The
Do-closure follows easily from III, 3.25(b). If V is a simple FpG-module, G e $j, and
[lz]G ф £>, then H e Proj6(G) by III, 3.23(a), and so Ve Mod"(G). Thus (b) holds.
To justify (c) suppose that p||G| and that Gefi. Then G has a p-chief factor H/K
such that G/H is a p'-group. If we write V for H/K regarded as an FpG-module, then
the semidirect product [FjG belongs to This is because H/K is complemented in
G/K (by a Hall p'-subgroup), and so the primitive epimorphic images of PG generate
the same class as the primitive epimorphic images of G. In this case Modp(G) 0.
We now consider Assertion (d). Since Modc(G) n Mod°(G) = 0, it is clear that
Modc(G) is empty when Mod“(G) is universal. On the other hand, if Modc(G) is empty,
Mod“(G) contains every simple Fp G-module by (b) and hence every semisimple
module by (a). But in this case G is a p'-group by (c) and by Maschke’s Theorem A,
11.4 every FpG-module is semisimple. Thus Mod“(G) is universal.
(4.4) Definition. A Schunck class § is said to have the universal IFp-module property
if the following condition holds:
(4.y) Whenever H e § and Mod“(H) contains a faithful module, then Mod"(H) is
universal.
Suppose that § satisfies (4.-)’). Let В e bp($), the p-boundary of §, and let
H e Proj8(B). Then Soc(B) is an FpH-module in Mod“(H) and is faithful for H.
4. Schunck classes with normally embedded projectors
Consequently Modcp(H) is empty, and
the following.
so H e 6p. by (4.3) (c). We have therefore shown
(4.5) Lemina. If § is a Schunck class with the universal
cp($>) s (The class cp(S5) is defined in (1.3)(c).)
fp-module property, then
(4.6) Lemma. Condition (4.y) is equivalent to the following weaker condition:
(4.y ) Whenever HeSyand Mod“(H) contains a faithful semisimple module, then
Mod“(H) is universal.
Proof. Suppose that the Schunck class satisfies Condition (4./). Let H e ft and let
L be a faithful H-module in Mod”(H); we must show that Mod“(H) is universal. Let
V* denote the direct sum of the factors of a composition series of V and note that
each such factor belongs to Mod”(к). Since Ker(H on V*) = Op(H) by B, 10.1, we
can regard V* as a faithful module for L = H/OP(H), and, so viewed, V* remains
semisimple; furthermore, V* e Mod£(L) by (4.3)(a). Since L e = ft, and since § is
supposed to satisfy (4.y'), we conclude that Modp(L) is universal and, in particular,
that L e Sp. by (4.3)(c). Suppose, for a contradiction, that OP(H) Ф 1, and let OP(H)/K
be a chief factor of H. Since OP(H)/K is complemented in H/K by a Hall p'-subgroup
of Н/K, we have [Op(H)/K] L = H/K e Qf> = ft, and consequently OP(H)/K, regarded
as an FpL-module, belongs to Modp(L), contradicting the fact that Modp(L) is
universal. Therefore we are forced to the conclusion that Op(H) = 1, whence H = L
and Modp(H) is universal. □
If the projectors of a Schunck class have the embedding property P in all soluble
groups, then we call it a “P” Schunck class.
(4.7) Proposition. A p-stably embedded Schunck class ft satisfies the universal Fp-
module property.
Proof. Let H e ft, and let V be a faithful H-module in Mod“(H). By (4.3)(d) it will be
enough to show that Modcp(H) is empty; we suppose not and obtain a contradiction.
Therefore let V *( / 0) be an FpH-module which is ft-covered. Let P denote the Hartley
group H(V, P*) constructed in B, 12.11 and recall that:
(1) P has class 2 and admits H as an operator group;
(2) there exist F„H-module isomorphisms P: P/P' and Ф: P' ;
(3) P has H-invariant subgroups S and S* which intersect P trivially such that
fliSP'/P'} = V fi(S*P'/P"} = V* and [S, 5*] = P'
In particular, the subgroup S*H of the semidirect product G = [P]H is isomorphic
with [F‘]Hand therefore belongs to ft. By III, 3.23(a)an
G containing S*H is an й-projector of G. Thus S*P - <S 02мод°(Н)
by hypothesis » _* »”fc.
U b-eModiOT Repealed appheab» rf®»
464 VI. Further theory of Schunck classes
argument shows that У1"1 ® V* e Mod'(H) for all where Vм denotes the n-fold
tensor power of V over Fp. Since we supposed that V is faithful for H, from the proof
of Theorem B, 10.13 we conclude that the regular module FpH is a direct summand of
(I © I')’"1 and hence of U © I')’”1 ® V* for sufficiently large values of n - here I
denotes the trivial FpH-module. Thus Modp(H), which is Q-closed, contains all simple
FpH-modules, in evident contradiction of the fact that Modp(H) contains a simple
quotient of V. Thus our initial supposition is false and Modp(H) is empty. Hence
Mod“(H) is universal and § satisfies (4.y). □
(4.8) Notation. It will be convenient in the sequel to ascribe special notation to three
formations, each associated with a prime p and a Schunck class § as follows:
(i)/p(S>) = QR„(ep.nCp(f>));
.. = f(D iffp(b) = 0, and
|(G: H e Proj8(G) => H e/p(§)) otherwise;
(iii) f>p = (G: H e Proj^G) => H e Gp-).
If Sp. n cp(£>) =£ 0, then/p(§) is a formation by II, 1.18(b). The other two classes are
formations by virtue of IV, 1.2. Evidently (5p. and hp($>) are subclasses of f>p. We also
remark that if /p(£>) # 0, then cp(§) n <5p. # 0, and there exists a group B( 1) in
bp(£>) n <5P<5P. Clearly В e hp(S>), and so hp(f>) = 1 if and only if fp(f>) = 0.
(4.9) Lemma. For all primes p and Schunck classes we have
f>p = <Zp.hp(§>).
Proof. It is clear from the definition that f>p = Sp.f>p and hence that Sp/ip(f>) c
Suppose, for a contradiction, that this inclusion is proper, and choose a group G of
minimal order in Sjp\(5p./ip(§). Evidently Op.(G) = 1, and therefore F(G) = Op(G).
Let H e Proj6(G) and note that § e (5p.. Let P = F(G). Since Cc(P) < P, we have
Op.(PH) = 1, and as PH e f>p, we obtain PH e hp($y) if PH < G. But then G e Iip(f>),
contrary to our choice of G. Therefore PH = G. The fact that F(G/®(G)) =
F(G)/<b(G) implies that 0р.(С/Ф(О) = 1. Therefore if Ф(С) # 1, we conclude that
С/Ф(С), and hence G, belongs to Iip(f>), another contradiction. Thus Ф(С) = 1,
and F(G) is semisimple by A, 10.6(c). Since both classes under consideration are
formations, by minimality G has a unique minimal normal subgroup, which there-
fore coincides with P. Therefore G e 6p(f>), and H e cp(f>) n <5p. S fp($>)- But then
G e A p(§>), a final contradiction which completes the proof. □
(4.10) Proposition. Let Sy be a Schunck class. Any two of the following conditions are
equivalent:
(a) 5 n <5„. has the universal Fp-module property;
(b) If hp(Sy) 1, then hp(Sy) is Enclosed;
(c) The formation Syp is saturated.
Proof, (a) => (b): Assume that (a) holds, and let L be a group in Sy n /p(§>). The first
step is to show that Modp(L) = 0. By definition of f(S>} we have L s G/K for
some G e R0(Sp. n cp(§)), and so G has normal subgroups Nlt .... Nr with trivial
4. Schunck classes with normally embedded projectors
465
intersection such that each quotient G/N, belongs to £ ,
GerS. = S „i c- , ®pncp(4>), m particular,
°. PI/ . ! ’I"’ r (Since G/N, e c (§), there exists a simple F (G/N}
module ^ which ,s faithful for G/N,. Let HeProjf,(G). Since G/H./ft
KertH on H ’T ° S T We СаП regard Vi “ M FrH‘modulc With
Ker(H on l')=-HnW„ moreover, V, is ^-avoided because ГККНЛНn H.n ~
[KJ(G/N|) e bp(£>). Therefore Modp(H) contains the FpH-module
и=
which is faithful for H because Q;=1 (H n H.) = 1. Therefore, since H e § n S . by
assumption Mod“p(H) contains all FpH-modules. Now G/K s L, which by definition
belongs to Hence G = HK, and L s H/(H n K). Since each H/(H r. K)-module
can be viewed as an H-module by inflation, we conclude that Mod“(L) is also
universal. p
We will now deduce that h₽(§) is Enclosed. Let G be a group with a minimal normal
p-subgroup N such that G/N e hp(§). IfH e Projg(G), we have HN/Nefp(f>) c 6
Let L e Hallp.(H). Then HN = LN, L n N = 1, and so L s HN/N e § r>fp($). V/e
conclude from the previous paragraph that N, viewed as an FpL-module, is ^-avoided
and hence that H = Le fp($>). Thus G e hp(f>), and it follows inductively that
/ip(§) = Gp/ip(f>), as desired.
(b) => (c): By (4.9) we have — Sp.hp(f>); hence §p is certainly saturated when
hp($>)= 1 On the other hand, if/ip(£>)# I, Assertion (b)implies that §p = Gp.Gp/ip(§),
and in this case is evidently locally defined by the formation function h given by
/1(9) =
hp(b)
if q = p, and
if « 14 p.
Since local formations are saturated by IV, 3.3, Assertion (c) holds.
(c) =>(a): Let H be a group in fin £p., and suppose that Modp(H) contains a
faithful H-module. Then the semidirect product [ F] H clearly belongs to §p. As we are
assuming that f)p is saturated, by Theorem IV, 4.6 it is locally defined, by a formation
function h say. Since V = Op.p(VH\ we have H s VH/Ve h(p). If W is any FpH-
module, the p-chief factors of [H']H are clearly h(p)-central, and it follows easily that
IVHesy. Hence H e Proj8(lVH) and Ii'eMod“p(H). Therefore §n£p. has the
desired universal Fp-module property. О
Remark If one of the three equivalent conditions of Proposition 4.10 holds, and
if * 0, then hp&) * 1 and it follows that Z„ e Лр(&), in other words, that
p Char(f>).
Forster’s next result is a good illustration of how the boundary of a Schunck class
ft influences the embedding of its projectors. (Of course the socks of groups tn the
p-boundary of & determine the simple modules in Modp(H) for H e &.)
(4.11) Theorem. Any two of the following statements about a Schunck class § are
equivalent:
466
VI. Further theory of Schunck classes
(a) § is p-stably embedded;
(b) 5 has the universal (rp-module property;
(c) S)p is saturated, and сp(F>) c <Sp.;
(d) Groups in Sp§ have p-normally embedded projectors.
Proof. We have already proved that "(a)=>(b)” in (4.7). If (b) holds, it is clear
that Condition (a) of (4.10) is satisfied, and hence by that result F>p is saturated.
Furthermore, Lemma 4.5 implies that cp(£») S 6p.; hence (b) =>(c).
(c) => (d): This step requires further work. Let G e <3pf>, set P = OP(G), and let
H e Projg(G). Then G/P e f>, and G = PH. Let g denote the formation /ip(f>). If
g = (1), we have cp(f>)n 6p. = 0, and therefore from the second assumption of
Condition (c) we deduce that bp(f>) = 0. But then Gpf> = Sj and Condition (d) is
certainly satisfied. Therefore suppose that g (1), and note that by (4.10) the assump-
tion that F>p is saturated implies that g = <5pg. Set L = Hs and R = OP(PL).
Then L is p-perfect, and consequently L < R. Since PH = G, the invariance of
residuals under epimorphisms (cf. II.2.4) implies that PL/P = (G/P)s; in particular,
R char PL < G, and so R < G.
Let Po = R n P, and observe that P0L = (Rn P)L = Rn PL = R. We assert that
Po < H. If this is not the case, there exists a maximal subgroup M of P0H containing
H. If Pi = Po n M, by A, 8.4 the section Ро/Л 's a chief factor of P0H. Since
H eProj8(P0H), this chief factor is ^-avoided, and therefore H/CH(P0/Pi)ecp(Sy) c g
because cp(£>) S <5p. by assumption. Consequently L centralizes P0/Pj, and it follows
that LPi is a normal subgroup of index |P0: l\ | in LP0 = R, against the fact that
R e Gp. Therefore we must have Po < H. But then R = P0L < H, and since H/R is
an ^-projector of G/R e <5pg = g £ Syp, we obtain H/R e <5p.. Therefore a Sylow
p-subgroup of H is a Sylow p-subgroup of R < G, and so H p-ne G. Hence Condition
(d) is fulfilled.
(d) => (a): Let N be a normal p-subgroup of a group G, and let H e Projg(G). Since
NH e Sp§, Condition (d) implies that H p-ne NH, and so by I, 7.3(d) we have
H n N < HN and a fortiori H nN < N. Therefore § is p-stably embedded, as
required. □
Let us now return to Condition 4.a. If В is a group in the p-boundary of a Schunck
class § and if H e Projg(B), then obviously (4.a) is satisfied by H in every epimorphic
image В of B, and yet, if p divides |H|, then H is certainly not p-normally embedded
in B. However, if (4.a) is satisfied by the ^-projectors in every soluble group, then the
^-projectors are indeed p-NE subgroups, as the next result shows.
(4.12) Corollary. The Schunck class f> is a p-NE class if and only if for all (soluble)
groups
HnN <G
whenever H e Proj6(G) and N is a normal p-subgroup of G.
Proof. The necessity of the condition is proved in I, 7.3(d). To prove the sufficiency,
we argue by contradiction: suppose that it is false, and among the groups which have
4. Schunck classes with normally embedded projectors
^-projectors whichi are: notp-normally embedded, let G be one of minimal order Let
H e Proj 6(G) and P e Syl„(H). If к denotes 0„.(G) and К * 1, the minimality of G as
a counterexample implies that KP/K is a Sylow p-subgroup of <(FK/K)™> -
( d Ж’ Bn^enJ £.SyM<PC>)’“C°MradictS ‘heCh““ ™
and F(G) = 0„(G). Next let R = H n 0„(G). Then R < G by hypothesis, and if R * 1
we reach a similar contradiction to the one above by using the fact that H/R is
p-normally embedded in the proper quotient group G/R. Therefore H n F(G\ = 1
and hence Mod;(H) contains the IF „Я-module F(G)/<t>(G), which is faithful for H by
A, 10.6(c). Since by hypothesis the subgroup H satisfies (10.a), it is certainly p-stably
embedded, and so has the universal IF „-property by (4.11). It follows that Mod“(H)
is universal, and therefore P = 1 by Remark 4.3(c). But then Я p-ne G and G is'not
a counterexample. q
From now on we prepare the way for Forster’s classification of NE Schunck classes.
A major step in the proof is to show that for such a class § there exists a set л of
primes (depending on p) such that hp(f>) = G„ for each prime p, and this is carried
out in two stages: the first is to show that hp(§) has a certain wreath product property;
the second is to show that this property characterizes classes G„. In Section 5 of this
chapter, we shall again derive this property for /ip($j), but under a set of hypotheses
sufficiently different to make a unified treatment unwieldy; however, the central idea
of the proof is the same in both cases, and it is facilitated by the following technical
lemma.
(4.13) Lemma. Let p be a prime, and let § be a Schunck class such that the associated
formation hp($>) is Enclosed. Let X be a primitive group such that Soc(A') and the
^-projectors of X are p'-groups. Also assume that the ^-projectors of groups in the
class product S„(X) either cover or avoid p-chief factors. Further, suppose that X
has a subgroup Y which contains an ^-projector Я of X and a normal subgroup К such
that
(i) Y/Kehp(b\
(ii) Soc(A') f K, and
(iii) Y/K has a faithful simple module over F„.
Then X e /ip(Sj).
Proof. Condition (iii) ensures the existence of a simple F„F-module Lsuch that
Ker(F on U) = K. In view of Condition (ii) there exists by B, 11.6 a faithful simple
A'-module V over F„ such that Pr has a quotient module P/Po isomorphic with U.
Since [17](F/K) e e„hp(f>) = Ap(5) by hypothesis, it follows that P/P„is; avoid у
an ^-projector of the subgroup VY of the semidireet product G == .P]Xb*H
an § projector of X contained in Y. Then an ^-maxima subgroup H of VH
confairing Я is an ^-projector of G by III, 3.23(a) and III 3.25 ЬХ Вu Ge<= M
and Я’ does not cover the minimal normal subgroup PofG because Я e P e g "
Therefore by hypothesis Я* avoids P, and we have Я= H* e ProHe 4
by hypothesis, we have G e §>p, andI hence G e h &) by ) Q
implies that G e /ip(f>) and hence that A'( = G/P) e Qh (x>)
468
VI. Further theory of Schunck classes
We now use this lemma to prove that if & is a p-CAP Schunck class, then h *'(§>) is
closed under forming certain wreath products. In view of Lemma I, 7.3(e), this result
will apply to p-NE Schunck classes.
(4.14) Proposition. Let p be a prime, and let 5) be a Schunck class whose projectors in
each group cover or avoid p-chief factors. Assume that the formation Syp is saturated.
If Ge hp(b) and if ge <r(G) и Char(/ip(§)), then
G e hp(^}.
Proof. Let g denote the formation hp(&). If g = (1), then <r(G) и Char(g) = 0, and
the result is true by default. Therefore suppose that g Ф (1). In view of the hypothesis
that is saturated, from (4.10) we have Zp rlj„8 G e g, and therefore we may suppose
that q # p. Let E = E(p/q), the primitive group defined in B, 12.5 with £/Soc(£) S Zp
and Soc(E) a g-group. Set Q = Soc(E), set
W'=Erb„gG
and let В = E1’, the base group of W Since Z(£) = 1, by A, 18.5(a) the Sylow
g-subgroup Q1’ of В is a minimal normal subgroup of W, and therefore W is primitive.
If I QI = it is clear that (A viewed as an F, G-module, is a direct sum of d copies
of the regular module and therefore has every simple module as a quotient (a fact
used in Cases (a) and (b) below). Let P e Sylp(E). Since g = Epg as we observed
above, it follows that P’G egs h" and hence that the ^-projectors of PbG avoid P.
Consequently Q“G contains an ^-projector, H say, of W and moreover H e <Zp.. We
now wish to apply Lemma 4.13 with W in the role of X and Q^G in the role of Y. We
distinguish 2 cases:
Case (a). If q e Char(g), let QVGo denote a trivial quotient module of the F, G-module
Q\ and take К = GQ0 < Q'G. Then Q^G/K = Z, e g, and it is clear that require-
ments (i) (iii) of (4.13) are fulfilled.
Case (b). If g| |G|, a group of minimal order in q(G)\<5,., say G/N, is primitive, and
its socle, M/N, is a g-group. Since M/N is a simple FeG-module, we can find a
submodule Qo of Q1’ such that QfQ0 is F4 G-isomorphic with M/N. Let К = Q0M.
Since M = Cc(M/N), it is clear that (fG/K s [Q’/Qo] (G/M) = G/N e g. Since Q'G/K
is primitive and has no non-trivial normal p-subgroups, by B, 10.7 it has a faithful
simple module over Fp, and again the requirements (i)-(iii) of (4.13) are fulfilled.
In both cases we can therefore apply Lemma 4.13 to conclude that We g and
hence that £ e f "(§) (defined in (4.8)(i)). Since H < QCG, it follows that Q'G e g, and
therefore, if R is a subgroup of Q of index q, we obtain Z„ Qj,„ G s (Q/R) Qj G s
Q'G/R» e Qg = g, as desired. ‘ Q
The next stage is to show that, among both formations and Schunck classes
ot soluble groups, the wreath product property described in Proposition 4.14
characterizes the classes of тг-groups.
4. Schunck classes with normally embedded projectors
469
(4.15) Theorem (Forster [2]). Let JE be a class of soluble groups of characteristic n
satisfying each of the following three conditions:
(i) ЭЕ = q3E;
(ii) Either JE = R(jJE or JE = гфЗЕ;
(iii) If Ge X and qenu <r(G), then Zt rlj,es G e X
Then JE = S„.
Proof. Let G e X let q e n и <r(G), and let 17 be a simple F, G-module. Since the base
group of Zq 1|ге8 G is the regular module, which has U as a quotient module, it follows
that the semidirect product [17] G is isomorphic with a quotient group of Z4rlreg G
and so belongs to JE. In particular, taking for 17 the trivial module, we obtain
Z, x Gel, and hence Z, e X Consequently <r(G) s n, and JE e
Now let qen. We aim to show that (5,JE S X and, to this end, let L be a group
with a minimal normal q-subgroup N such that L/N e X If N is complemented in
L, then L = [A'](L/N), and so LeJE by the previous paragraph. Now suppose
that N < <b(L). If JE = ecJE, then LeJE, as required. Therefore suppose that JE is a
formation. Since N is an elementary abelian q-group, it has normal subgroups
Nt, N2, N, with trivial intersection such that |Л':NJ = q for i = 1, ..., r. Let
W= N Qjreg (L/N). By A, 18.2(d) we have W/Nj = (N/N()Qj„, (L/N) e JE by hypoth-
esis, and as Nf,..., Л'/ have trivial intersection, we conclude that W e r0JE = X By
A, 18.9 the group L is isomorphic with a subgroup L* of W such that N^L* = W.
Since N( e 91, Theorem IV, 1.14 implies that L* e QR0(kV) с X Therefore in every
case we have shown that L e JE and can now conclude by induction that <5,JE c JE
for all qen. Thus <5„ S JE and equality holds. □
We are now ready to state and prove Forster’s characterization of p-NE Schunck
classes.
(4.16) Theorem (Forster [2]). Any two of the following assertions about a Schunck
class of soluble groups are equivalent:
(a) The ^-projectors of each group are p-normally embedded subgroups:
(b) The ^-projectors of each group are p-stably embedded and have the cover-
avoidance property for p-chief factors:
(c) Either (5P« Sj, or there exists a set n of primes containing p such that
D" « H « Q’6p..
Proof (a) => (b): If is a p-NE class, by the implication: (d) => (a) of Theorem 4.11 it
is also a p-SE class. Moreover, by I, 7.3(e) the ^-projectors are always p-CAP
subgroups.
(b)=>(c): Let § be a p-SE and a p-CAP Schunck class. By (4.11), (a)=>(c), we
have cp(&) e Sp-; therefore if h₽($j) = 1, we have bp(5) = 0 and hence Sp « § by
(1.6)(a). Suppose, from now on, that h₽(§) # 1. Then hp(f>) = G„ for some и S P by
(4.14) and (4.15), and since Ap(§>) is Enclosed by (4.11) and (4.9), we have pen and
p f Char(f>). Furthermore, £>₽ = Gp <5„ by (4.9).
We prove next that D" « and since л(С”) = S„ n 5J3, by (1.5) it will be enough
to show that b(Sj) s <S„. Let В e b(§), set N = Soc(B), and let H e Projs(B). If N is
470
VI. Further theory of Schunck classes
a p-group, then H e Qp. as remarked above; therefore В e ff, and consequently
В S B/O (B) e On the other hand, if N is a p'-group, let U denote the trivial
F Я-module, and’apply В, 11.6 to deduce the existence of a simple F„B-module V (a
quotient of UB) which is faithful for В and satisfies [F, Я] < V By III, 3.23(a) a
maximal ^-subgroup H* of VH containing Я is an §-projector of VH, and hence of
G = [И]В SinceZp e <з(ЕЯ) andp Char(§), thep-chieffactor FofGisnot covered
by H*. By hypothesis it is therefore avoided by Я*, and so by order considerations
Я = Я*. Let Л be a composition factor of VH. Since A is §-avoided, we have
| Л]/(Я/Ся(Л)) e />„(£>), and therefore H/CH(A) e £„ as above. Since by B, 10.1 the
intersection of the subgroups CH(A), as A runs over the composition factors of the
faithful Я-module Ц is Op(H), we conclude that Я e QpQ„ = and hence that
Be 5 -6, = fi1. It follows that G e whence G S G/Op(G) e <э„, and therefore
It remains to show that § « C'Sp.. For any group G and for Я e Proj{,(G), we
have O”(G) 5 Я by the preceding paragraph. Since the group G/O"(G) belongs
to ©„ - hp($) c its ^-projector H/On(G) belongs to Sp., and it follows that
§ « Q”SP-
(c) => (a): Let Я e Proj6(G). If Sp « §, a Sylow p-subgroup of Я is a Sylow p-
subgroup of G. On the other hand, if Q” « § « Q”SP-, then a Sylow p-subgroup of
Я is a Sylow p-subgroup of O"(G). In either case, we clearly have Я p-ne G. □
From the equivalence of Assertions (a) and (b) in the preceding theorem we have
the following:
(4.17) Corollary, (a) J^_SE -^-cap = J^-ne-
(b) ^SE Fl ^CKV ~ *T®P1E-
We have now arrived at our promised objective, namely Forster’s classification of
NE Schunck classes.
(4.18) Theorem (Forster [2]). A Schunck class § is an NE class if and only if there
exist sets n and a of primes with лэ a such that
= £>"<=„
Proof. It is dear from (4.16), (c) =>(a4 that each class of the stated form CS„ is an
NE Schunck class.
Now let § be an NE Schunck class of characteristic y. If p £ у, we have h ”(§) = S„(pl
for some set л(р) of primes containing p, as in the proof of Theorem 4.16, (b) => (c).
We show that the sets n(p) are independent ofp e P\y. Suppose, by way of contradic-
trnn that there exist a prune r in л(д)\л(р) for certain primes p, q e P\y. Then
E If r = 4’ then the ^-projector of Z, is trivial and it follows that
' £ . . ’ a conlracBction. Hence r p. Let E denote the primitive group E(r/q),
defined in B, 12.5, and let Я e Proj6(£). Since E e S,/i’(§) = h«(£>), we have
Wn boc(E)- 1, and the hypotheses of (4.13) are satisfied for the prime p with
e пр e (E, Я, Я) in the role of (X, Y, K). It then follows from that lemma that
5. Schunck classes with permutable and CAP projectors 471
E e hp($) = S„(p), which contradicts the assumption that Zr^h',(§). Therefore
n(q)\n(p) is empty, and by the symmetry of the argument we conclude that n(p) = 7t(q)
for all p,q£ y. Denote this common set by n.
It now follows, as in the proof of (4.16), that
(i) Q” « § « Q’6p. for all p e P\y, and
(ii) Sp « § for all pey,
for if p e Char(§), the formation h',(§) is not Enclosed and is therefore the identity
class. It is evident from (i) and (ii) that § = Q’S,, and on setting a = у n n, we have
§ = with ст s n. □
(4.19) Corollary. For a saturated formation g the following statements are equivalent:
(a) ^-projectors of every soluble group are normally embedded subgroups;
(b) g = for some a s P.
Proof, (a) => (b): Since g is a Schunck class, we can deduce from (4.18) that g has
the form Q"SO for some a S л s P. We claim that the class § = Q"SO is not a
formation if n с P. For, if 2 e P\n, the group SL(2, 3) x Z2 belongs to and,
if2 pe P\n, the group G x Zp e Ro§\§, where G is the semidirect product [E] <r>
of an extraspecial group E of order p3 and exponent p by an involutary automorphism
r which inverts E/Z(E) and centralizes Z(E). (Here take т = a/J in the notation of A,
20.12.) Hence n = P, and g = QPS„ = S„.
Since the ©„-projectors of a soluble group are its Hall <r-subgroups, the implication:
(b) => (a) is clear. □
Exercises
1. Show that Mod" (W) is closed under taking submodules.
2. (Forster, [2]) Let § be a p-SE Schunck class, let G e and let H be an
^-supplement to OP(G) in G. Let L be the /i,l(§)-residual of H. If P = Op(G}, show
that (OfPL) n P)H e Projg(G).
3. (Forster [2]) Let g be a p-SE saturated formation. Show that either Spg = g
or g C Qp._ If g is p-SE for all primes p, then g = S„.
4. Show that .^-Nt is properly contained in and in J^,_CAE.
5. Schunck classes with permutable and CAP projectors
The culmination of Section 4 was Forster’s determination of the Schunck classes
whose projectors are normally embedded subgroups. This could be regarded as a
special case of the problem of classifying JfcAE, namely the family of Schunck classes
whose projectors in every (finite, soluble) group cover or avoid chief factors. The main
result of this section, also due to Forster [3], is the characterization of permutable
Schunck classes, that is to say, Schunck classes whose projectors are always system
permutable (SP) subgroups. Although
472 VI. Further theory of Schunck classes
Forster’s description of the system permutable Schunck classes does not lead directly
to the explicit determination of the CAP Schunck classes, which is still open.
Although a subgroup with the cover-avoidance property is not in general system
permutable (see Exercise 1 below), in contrast we have the following.
(5.1) Proposition. A CAP Schunck class is system permutable.
Proof. We suppose that the statement is false and derive a contradiction. Let § be
a CAP Schunck class, and let G be a group of minimal order subject to having an
^-projector, H say, which is not system permutable. By I, 4.26 there exists a prime
p e <r(G) such that H does not permute with any Hall p'-subgroup of G.
Let N '< G. The minimality of G yields a Hall p'-subgroup S of G such that SN/N
permutes with HN/N, and in this case SHN is a subgroup of G. If SHN < G, then
the choice of G and the fact that H e Proj^(SHN) ensures that H permutes with some
Hall p'-subgroup S* of SHN; but then S* is a conjugate of S and hence S* e Hallp.(G),
a contradiction. Therefore SHN = G, and, in particular, HN has p'-index in G.
If A is a p'-group, then N < S and G = SH, a contradiction. Hence A is a p-group.
If N < H, we reach the same contradiction, and therefore H n N = 1 because by
hypothesis Я is a CAP subgroup of G. If P e Sylp(H), clearly PN e Sylp(HA), and so
PN e Sylp(G) since pj |G: HN\. Hence by A, 11.2 there is a complement, L say, to N
in G. By III, 3.23(c) we have Projg(L) £ Projg(G), and so, replacing L by a conjugate
if necessary, we can suppose that H < L. By the minimality of G, there exists a Hall
p'-subgroup T of L, which permutes with H and which is a Hall p'-subgroup of G
because |G: T|( = |L: T||A|) is a power of p. This final contradiction completes the
proof. □
(5.2) Lemma. Let § be a permutable Schunck class, let q be a prime, and let X denote
the class of groups whose ^-projectors have q-power index. Let p e q, and let G e SpX.
Then the ^-projectors of G are p-CAP subgroups.
Proof. The lemma is true if G = 1. Therefore suppose that G # 1 and by induction
that the lemma is true for groups of smaller order. Since X is obviously Q-closed, so
also is SpX, and if N < G, then we know by induction that an ^-projector Я of G
covers or avoids the p-chief factors above N. To complete the proof, it will suffice to
assume that A is a p-group and to show that Я either covers or avoids A. Set
? — and let Q be a Sylow q-subgroup of G which permutes with Я. Since
IG: PH\ is a power of q by hypothesis, it follows that G = PHQ. Let Ao = NnH,
and suppose that Ao 1. Then A = <A®> by definition of A, and since [A, P] = 1
by A, 13.8(b), we have A = <A™e> = <A"^> <; HQ. Consequently A <; Я because
\HQ: H\ is a power of q(J=p). [j
We remark that we have not used the full force of permutability in the proof of
Lemma 5.2. but only the fact that each ^-projector permutes with some Sylow
q-subgroup of G.
The next result, and its ingenious proof, are due to Forster; the result plays a crucial
part in his characterization of permutable Schunck classes.
5. Schunck classes with permutable and CAP projectors 473
(5.3) Proposition. Let 5 be a Schunck class, let p and q he distinct primes, and assume
that for each r e {p, <7}
(i) b,(§) 0, and
(ii) in each group G an ^-projector of G permutes with some R e Syl,(G).
Then § is r-stably embedded for each re{p, <7}.
Proof. Our eventual goal is to show that § satisfies Condition 4?/ of Lemma 4.6. We
can then deduce from that lemma that § has the universal Fr-module property for
r e {p, <7} and finally can appeal to Theorem 4.11, (b) => (a), for the desired result.
To this end, let H e $), let V = Ц © • © V„ be a semisimple F,,H-module, faithful
for H, and assume that V e Mod“(H). We may clearly suppose, without loss of
generality, that each simple Vt is not the trivial module and is a homogeneous
component (that is to say, has multiplicity one as a direct summand). Let G = [V]H,
and observe that Z(G) = 1 on account of our assumptions about V. Let Wt be a
simple FeG-module, faithful for G, such that [Wj, H] < H (the existence of such a
module is ensured by В, 11.7), and let L be a group in the non-empty class b,(§). Then
LhasthefonnL = W2EwithE e Proj ^(L) and IVj = Soc(L); W2 is simple and faithful
as an L-module over F,.
Next let W = Wt ® W2, regarded as an F4(G x E)-module according to B, 1.12.
First we assert that W is faithful for G x E: Let К = Ker(G x E on W). Since W is
clearly faithful for G and E, we have KnG = KnE = 1, and as K, G and E are
normal subgroups of G x E, it follows that К commutes with G and E. Therefore
К < Z(G x E) = Z(G) x Z(E) = Z(E), whence К < К n E = 1, and the assertion is
justified. Next, let M be a G x E-composition factor of W, and consider the action
on M of H x E, which by III, 6.3 is an ^-projector of G x E of p-power index. By
the Jordan-Holder theorem, the composition factors of Mo, like those of Wc, are
isomorphic with Wt; hence [M, H] < M. On the other hand, the composition factors
of ME are isomorphic with W2, and since [M, H] is an (H x E)-submodule of M, we
can find a submodule Mo of ME such that M/Mo s W2 as E-modules. But then the
semidirect product [M/M0](W x E) has a primitive epimorphic image isomorphic
with [W2]E e b,(§), and consequently M/Mo is avoided by the ^-projector of the
semidirect product R = [ И'] (G x E). Since the ^-projectors of G, and hence of
G x E, have p-power index, and since W e Qq, we can apply (5.2) (with the roles
of p and q reversed): the above reasoning implies that no chief factor of R below
W is ^-covered; therefore we conclude that W is ^-avoided and hence that
H x E e Projg(R).
Let {Mp I <ii m| denote the set of subgroups of index q in W. (Thus
m = (I Jkj - 1 )/(<7 - 1).) For each i = 1,..., m let W, denote a copy of the module W;
more specifically, let Op IT —* Hj be an Ffl(G x E)-isomorphism, and identify Л4, with
its image 0ДМ,) in Hj. Form the direct sum
w* = w, ©••© wm
and then the semidirect product S = [W*](H x E). If Z_= <Zj> © ••• © <zm> is a
vector space with basis {Z[,..., zm}, then the subspace Z = {£ 2,z(: 2( = 0} has
codimension 1 and satisfies Zn<z,> = 0 for i = 1, tn. Thus, taking
д7д VI. Further theory of Schunck classes
2 = S ®(IK/M), we can find a subgroup Hj, of index q in W* such
that W Let : be an element of G x E which leaves the subspace Wo
invariant. Then Af/z = (WB n WJz = M, for i = 1,..., m, and so z, in its action on W,
leaves invariant each subgroup of index q. From B, 8.7 we therefore deduce that
- has scalar action on IF and hence belongs to Z(G x E) = Z(E). Consequently
Л/(чЛ1*о) = Z* < Z(E). The group W*Z*/W„ is a metacyclic primitive group of
order q|Z*| and |Z*I divides q - 1, and by B, 11.7 this group has a faithful simple
module U over Fp such that [U, Z*] < V. By inflation regard U as an FpIF*Z*-
module with WB = Ker(IV*Z* on U), and set U* = Us. Because W*Z* = NS(IFO), it
follows from B, 7.4(b) and B, 7.6 that the FpS-module U* is simple. Then by Mackey’s
Theorem B. 6.20 we have (l/*)Hx£ S (U„,.z.^Hx£1)Hx£ = (l/z.)Hx£, and, writing
A' = (l'z.)£ we obtain (l/*)H x £ s XHx£. Since by construction C!z> has the trivial
simple module as a quotient, the E-module X has ((F,)z.)£ s F,(E/Z*). and hence
(F,)£, as a quotient. Therefore (U*)Hx£ has a quotient isomorphic with ((F,)£)Hx£,
which on restriction to H is isomorphic with the regular module F,W; in particular,
all simple F^H-modules. including the modules F,...., V„ defined earlier, appear as
a quotient of this restriction and hence as a quotient of (l/*/[G*, E])H.
Let T denote the semidirect product
T=[U‘]S= V*W*(H x E).
Since H x Ее Proj6(S), an ^-maximal subgroup of U*(H x E) containing H x E is
an ^-projector of T. Because (U*)H> E has a submodule V} containing [I/*, E]
such that = F,, for example, and because F, is ^-avoided, the quotient
U*(H x E)/Ut is not in §, and therefore V* is not covered by an ^-projector of T.
But S belongs to the class T of groups whose ^-projectors have q-power index and
T e Sp£. We can therefore apply (5.2) once again to deduce that U x E is an
^-projector of U*(H x E), and hence that H is an ^-projector of the semidirect
product [<.'*/[<.'*, E]]W. It follows that Mod“(W) contains all the simple FpW-
modules and is therefore universal by the argument used in justifying (4.3) (d). Con-
sequently Condition 4./ is satisfied for the prime p, and thus § is p-stably embedded.
Finally, by the symmetry of the hypotheses in p and q, we conclude that f> is also
q-stably embedded. q
We are now ready to prove Forster’s characterizations of permutable Schunck
classes.
(5.4) Theorem (Forster [3]). Any two of the following statements about a Schunck
class § are equivalent:
(a) § is permutable:
(b) For each group G with an ^-projector H of prime power index, and for each
prime p not dividing that index, every simple ^-„G-module is either fo-covered or Л-
avoided:
(c) Either § = <$p.§ for some prime p, or there exist sets cf primes n and a with
r. 2 a such that
5. Schunck classes with permutablc and CAP projectors
475
Proof. It follows at once from (5.2) that (a) => (b). We will show next that (b)^(c).
Therefore assume that Condition (b) is satisfied for and set
p= {reP:br(§)/0}.
Ifp = 0. then§ = S, and Condition (c) obviously holds. lf|p| = l,sayp= {p}, then
§ = Sp§ by (1.6)(a), and again Condition (c) is satisfied.
Therefore suppose that |p| > 2, in which case Proposition 5.3 can be applied to
conclude that § is r-stably embedded for all rep, and hence for all primes r because
<=, « § when br(§) = 0. Next we assert that <?„(&) = c,(£>) for all p.qe p. We suppose
that cp(§)\c,(§) # 0 and derive a contradiction. This supposition means that there
exists a group G in bp(Sj) with H 4 c,(&) for H e Projg(G). By В, 11.7 we can find a
faithful simple G-module V over F, such that VH contains the trivial module (FQ)„
as a quotient module, V/Vo say. Since q 6 p, we have bQ(H) 0, and therefore
Z, 6 S,h’(§) — hq($>) by Theorem 4.11, (a) => (c), and Proposition 4.10, (c) => (b) (here
we need to observe that 0 # c4(Sj) £ Sp. implies that /“(&) 0 and hence ihat
b’(§) 1). Hence Z, e b,(&), and since the semidirect product [И]Н has a quotient
VH/V0H = Z,, the module V is not ^-covered. Since § satisfies Condition (b) by
assumption, we conclude that V e Mod“(W), which is therefore universal by Theorem
4.11, (a) => (b). By Remarks 4.3 (d) and (c) it follows that H is a q'-group and, in
particular, that 0,(H) = 1. Since H has a faithful simple module over Fp (namely
Soc(G)), by B, 10.3 it also has a faithful simple module, W say, over Fg. Because
Mod°(H) is universal, W is §-avoided, and so [ IV] H e b,(§). But then Hec,(§),
which contradicts the choice of H. It follows that cp(§) = c,(§) and hence by defini-
tion (see (4.8)) that
b₽(§) = h’(§)
for all p, q e p. Let g denote the common value of the formation hp(ff) for pep.
The next stage of the proof is to show that g satisfies the following wreath product
property, which was previously encountered in Proposition 4.14:
(5.a) If G e (J and q e <t(G) Char(g), then Z, Qj„g G e g.
Since, for q e p, we have S,g = S,b’(§) = b’(§) = g, in proving that (5.a) holds we
may suppose that q$ p and hence that S, « § (because then bq(H} = 0). Next we
show that without loss of generality we may suppose that G e §. Let H e Projg(G).
If q 11G I, then q 11H | because S, « §. Now the base group В of Z, Tj„g G, considered
as an F,//-module by restriction, is the sum of |G : H| copies of the regular F,H-
module by A, 18.8 (a) and B, 3.16 (a). Therefore BH e H). If we knew that
Z, rLteg H belonged to g, we could deduce that BH e r0 g = g and hence that BG e g
by the definition of g and the fact that Projg(BW) £ Proj6(BG). It will therefore
suffice to prove that (5.a) holds when Cefing.
Let p and r be distinct primes in p; then p q # r because we have supposed that
q$ p. Let E = E(p/q), the unique non-nilpotent primitive group in to,(Zp) defined in
B, 12.5. The proof of (5.a) follows closely that of Proposition 4.14. Let
476 VI. Furiher theory of Schunck classes
IF=Erb„eG = e’P’G
where Q = Soc(E) and P e Sylp(E). By A, 18.5 (b) the group W is primitive with socle
Q- Because Ge§ng and because therefore P'G e r:pg = Е,ЬР(§) = hr(§), it follows
that G e Proj MP^G). Consequently QCG e Proj 6( IV) since S, « § by supposition.
Since the ^-projectors of g-groups are ('-groups^for all 16 p, we have g n f) E S„.,
and because q e p', it follows that Q"Ge Sp- E or- We will now apply Lemma 4.13
with r in the role of the prime p in its statement, and with W and Q*G in place of X
and V respectively. If Z, e g, then we take GQ0 for K, where Qe/Qx is a trivial simple
quotient module of (6’)c (which we know contains a regular quotient module F,G).
On the other hand, if q||G|, we take Q'CfjQfQft for where C’/Co is a quotient
module of Q" isomorphic with the first q-chief factor down some chief series of G. As
in the proof of (4.14), it is straightforward to verify that the hypotheses of (4.13) are
satisfied for these choices of X, Y and K; in particular, the requirement that §-
projectors of groups in S,(IV) cover or avoid the r-chief factors is guaranteed by the
assumption that § satisfies Condition (b) of this theorem. Therefore we deduce from
Lemma 4.13 that We b'(§) = g and hence that the ^-projector Q^G of W is in g.
Since G eQ(C’G) E Qg = g, we conclude finally that (5.a) is satisfied. But
then by (4.15) we have g = S„ for some л E P, and the proof that § = C'6„ now
follows, word for word, the final paragraph of the proof of (4.18).
(c) => (a): If § = ©,,§, then an ^-projector of a group G has p-power index; it
contains, and therefore permutes with, some Hall p'-subgroup of G. If q ± p and
Q e Halle (G), then H and Q have coprime index and therefore permute by A, 1.6 (b).
Hence H is a system permutable subgroup of G by I, 4.26.
On the other hand, if § = C'S„ with a E n, then an ^-projector H of a group G
contains the normal subgroup R = 0r‘(G). In fact, H/R is a Hall c-subgroup of G/R
and therefore certainly permutes with the subgroups of a Hall system of G/R to which
it belongs. Such a Hall system has the form Y.R/R for some Hall system Z of G, and
thus, for Le Z, we have
HL = (HR)L = H(RL) = (LR)H = LH,
in other words, H permutes with Z. Therefore Condition (a) follows from Condition
(c)- □
In view of Proposition 5.1 we now get the following.
(5.5) Corollary. Let § be a Schunck class whose projectors have the cover-avoidance
property. Then either
(i) § = Q"6S for some а с л c p, or
(ii) Sp. « § for some prime p.
On the quest for CAP Schunck classes, Forster has made the following observation.
(5.6) Lemma. Let § be a CAP Schunck class with is not an NE class. Then there exists
a set n of primes and a prime pen such that § = Q"fi, where Я is a Schunck class,
contained in to,, that satisfies
5. Schunck classes with permutable and CAP projectors 477
(5.P) For each G e S„, an й-projector of G is a CAP subgroup of p-power index.
Proof. By (5.5) and (4.18) we have Sp. « § for some prime p and therefore b(§) =
bp(§). Let л = a(b(§)). Then b(§) c S„ n %! = fl(E") by (1.4)(f), whence Q” « § by
(1.5). (In fact, Q” is the largest normal Schunck class contained in §.)
Let £ = § n S„. If Ge ft, then G/O”(G) e § n S, = fl, and so G e £!'£. On the
other hand, if G e Q”fl and H e Proj6(G), then H covers the ^-quotient G/O"(G) and
contains the O"-prejector 0"(6); thus G = H e §, and we have shown that § = £>”£.
The fulfilment of Condition 5.J3 is an obvious consequence of the fact that, in
©„-groups, the ^-projectors and the fl-projectors coincide. □
Thus a classification of CAP Schunck classes depends upon a classification of
Schunck classes fl of л-groups which satisfy Condition 5.J3. This has been done by
Forster for the case where fl is a local Schunck class (see Definition III, 5.2(c)); the
list he obtains includes all the CAP Schunck classes known at the time of writing.
(5.7) Theorem (Forster [3], Satz 5.12). Let p be a member of a set n of primes and set
7i* = л\{р}. Let fl be a Schunck class of n-groups satisfying Condition 5.Д of Lemma
5.6. If fl is a local class, then fl = S„., S„.Sp, or ©„.
We will not offer a proof of this theorem, which, like the proof of (5.4), is long.
However, in Section 7 of Chapter VII we give a proof of the classification of CAP
saturated formations using different, more elementary methods. To end this chap-
ter we deduce from (5.7) the full list of CAP Schunck classes which are locally
defined.
(5.8) Corollary. Let g be a local Schunck class whose projectors have the cover-
avoidance property. Let у = Char(g). Then either g = Sz, or у = P and g = Sp-Sp.
Proof. From (5.5), (5.6), and (5.7) we know that, for a suitable set л of primes, we have
g = C" fl, where C = S„ for some a £ л or fl = S„np S,,. It will therefore be sufficient
to prove that if
(5.y) Q" « g * 6,
then л — IP (and so Q" = (1)).
We suppose that (5.y) holds for the locally defined CAP Schunck class g with л IP
and derive a contradiction. By familiar reasoning. Condition (5.y) is equivalent to the
statement: 0 b(g) £ Let В e b(g), say В e b„(g), and let q e л'. Let E denote
an g-projector of B, and set U = Soc(B), regarded as a faithful simple E-module over
Fp. Further, let V be a faithful simple module over Fp for a cyclic group Q of order q.
Now consider a composition factor IVof U ®Гр F, regarded as a module for the direct
product D = E x Q according to B, 1.12. Since E is a л-group, we have (|E|, \Q\) = 1
and therefore W is faithful for D by B, 9.19 (b).
Let f denote the local function which defines the Schunck class g. Since В e bp(g),
wc have E e g\/(p); moreover, the fact that b(g) £ S„ implies that S, « g, and, in
particular, that Q e g. Hence D e Dog = g, but D <ff(p) because f(p) is Q-closed by
478
VI. Further theory of Schunck classes
definition of a local function. It follows that [IV]D e b„(g) £ G„ and hence that
q e <t( WO) £ n. This is the desired contradiction. □
Since saturated formations are examples of locally defined Schunck classes, we
obtain as a special case of this corollary Doerk’s classification of the saturated
formations whose projectors universally have the cover-avoidance property; these
are the classes Gz and Gp.Sp (see Theorem VII, 7.8 below). Doerk’s result was the
springboard for Forster’s attempt to determine the CAP Schunck classes. This proved
to be a difficult undertaking, which is still incomplete at the time of writing.
Open Question. It is a simple matter to verify that the classes Q"S„ and O’S„,P'Sp
are CAP Schunck classes. Are these the only ones?
Exercises
1. Let V be the natural module (of order 32) for the group S = SL(2, 3), and let
G = [F]S. Show that G has subgroups of order 3 which are CAP subgroups but
not system permutable.
2. If H permutes with some Hall system of G which reduces into H, show that H
permutes with every Hall system that reduces into H.
3. Let § = LF(H) be a local Schunck class. Show that § is permutable if and only
if either f> = G„ for some л £ IP or § = Sp W(p) for some prime p.
4. Let § be a Schunck class of the form ефg for some formation g. If § is permutable,
show that = g.
5. Let peP. Then the Gp. Gp-normalizers of a group G coincide with its Sp Sp-
projectors if and only if G e SpGp.Gp.
6. Let g be a saturated formation, and let G be a group with an g-projector U having
the cover-avoidance property. Let V be a subgroup of G which covers and avoids
the same chief factors of G as U. Then V e Proj ,(G).
Chapter VII
Further theory of formations
1. The formation generated by a single group
Most of this section is devoted to proving a result of Bryant, Bryce and Hartley [1].
This asserts that a formation generated by a finite soluble group contains only finitely
many subformations, and also that a similar statement holds for saturated forma-
tions. We shall keep closely to their original proof, which exploits a technical
result used by Oates and Powell [1] as one of the principal tools in the proof of
their celebrated theorem that the laws of a finite group are finitely based. A
detailed, self-contained proof of a version of this result is given in the Appendix a
(Theorem a.19). It is the following consequence of Theorem a.19 that we shall need
here.
(1.1) Theorem. Let к > 1, and let G be a finite group cf the form G = NlN2...NtL,
where L <. G and < G for i = 1,..., k. Let 0 = {1,..., k}, and for A.sQset
Ga = П
(Thus G0 = L and Gn = G.) Assume that for each a e Sym(k) we have
(l.a)
[lV..,JVb] = l.
Then, for each A £ 12, we have
Ga e QR„(Gr: A Г £ 12).
Proof. Let E denote the external direct product
and let nr denote the projection of E onto the component Gr. It follows from Theorem
a.19 of the Appendix a (see page 843) that E contains a subdirect subgroup R such
that R n Gr = 1 for all Г £ 12.
If A £ 12, then RG^/Gf, is obviously subdirect in Xr<_ri(GrCA/GA), and therefore
we have
480
VII. Further theory of formations
R S RGrJGb e R0(GrGA/GA: Г £ fl)
= R„(Gr: Л # Г £ fl).
Since nAR = Ga. we have GA e q(R) S QR0(Gr: Л # Г S fl). □
(1.2) Remark. Let G be a finite group and L a supplement to F(G) in G. If c is the
nilpotency class ofF(G), then Condition La is satisfied with к = c + 1 and N, = • =
дгк+1 = f (G) Since GA = G for 0 # A £ fl, we can deduce from Theorem 1.1 that
L = G0 e qr„(Ga: A # 0) = QRo(G).
This is another proof of Theorem IV, 1.14.
The following definition needs a more subtle formulation than the corresponding
concept in the theory of vaneties.
(1.3) Definition. A finite group G is called formation-critical if
G Ф QRo((QS(G)\(G)) n qr0(G)).
If p stands for the formation generated by G, this definition requires that G be not
in the formation generated by the proper Д-sections of G.
(1.4) Lemma. A formation of finite groups is generated by its formation-critical
groups.
Proof Let g be a formation, and set
& = QR0(H e g: H is formation-critical).
Clearly § £ g, and if § # g, we can select a group G of minimal order in g\f>. By
this choice each proper section of G lying in qr0(G) (and hence in g) must lie in f>. Thus
QR0((QS(G)\(G)) n QRo(G)) £ QRof) = f>.
Since the group G is not in f>, it is formation-critical and therefore by definition lies
in f>. This contradiction shows that 5 = g. □
(1.5) Proposition. Let G be a finite soluble group, and let g = qr0(G). For each d e N
there exists a number k(d) = kc(d) with the following property: If H eg and H has d
generators, then |H| < k(d).
Proof. Let e denote the exponent of G and I its derived length. Let H e g. The class
of groups with exponent at most e is a formation, as is the class of groups with derived
length at most I; therefore exp(H) < e and H*11 = 1. It is therefore sufficient to find
numbers kt(d) such that < k,.(d) for i = 1,.... I whenever fie g.
1. The formation generated by a single group
481
Set At H /Н , and let d(X) denote the minimal number of generators of a
group X. It will clearly suffice to prove the following statement by induction on i:
(1.0) There exist natural numbers kfd), d,(d) such that
Mil < k,(d) andd(Ai+l) < di+1(d).
Let i = 1. Since At = H/H' and H is generated by d elements, so also is Л,. Because
H, and hence Л,, have exponent at most e, it follows that |A11 < and we can take
k1(rf) = e‘'. Then by A, 1.8 it follows that H' is generated by 2d\H:H'\ elements.
Consequently d(A2) < 2d| A J, and so we can set d2(d) = 2dkfd) to establish (1.0) in
the case i = 1.
For the general induction step we assume that |AJ < k^d) and that d(A1+1) <
di+1(d). The above reasoning for the case i = 1 then yields:
|Л|+1| < and d(Ai+2) < 2dl+1 (d)|Ai+I|
and it is now clear how to define ki+i(d) and di+2(d) to complete the induction step.
□
Theorem 1.6 (Bryant, Bryce, and Hartley [1]). Let G be a finite soluble group, and let
g = QR0(C). Then
(a) g contains only finitely many formation-critical groups, and
(b) g contains only finitely many subformations.
Proof. Assertion (b) follows at once from (a) by (1.4). To prove Assertion (a), let H
be a formation-critical group in g. We aim to find an upper bound for the order of
H. From A, 10.6(c) we know that
Т(Я)/Ф(Н) = А,/Ф(Н) x • x Nk/®(H),
where each А,/Ф(//) is a chief factor of H.
Let c be the largest nilpotency class of a Sylow subgroup of C. Since soluble groups
whose Sylow subgroups have class at most c constitute a formation, it follows that
the nilpotency class of F(H) is at most c. We claim that
(Ly) к < c.
Suppose, for a contradiction, that к > c. If Ь/Ф(Н) is a complement to Т(Н)/Ф(Н) in
Н/Ф(Н), whose existence is guaranteed by A, 10.6(c), it follows that H = N1N2. . NkL
satisfies the hypotheses of Theorem 1.1, and we can then deduce that H(= Hn)e
QR„(Hr: Г <= Cl) in the notation of that theorem. Since L < Hr, it follows from 1.2 (or
from IV, 1.14) that Hr e qr0(H) for each Г S П. Moreover, Нг < H whenever Г <= П
because £/Ф(Н) complements the non-identity group Т(Н)/Ф(Н). Therefore H e
QR<>((QS(H)\(H)) n QR0(H)), contrary to the assumption that H is formation-critical.
This contradiction establishes (l.y).
f(p) = {
4g2 VII. Further theory of formations
Let r denote the largest order of a chief factor of G. Since the class of soluble groups
whose chief factors have order at most r is a formation, it follows that the chief factors
of H, in particular, the groups Л^/Ф(Н) have orders at most r. Consequently
\F(H)/®(H)\ <rc, =m say.
By A, 10.6(c) the group is isomorphic with a subgroup of Aut(F(H)/®(H)),
and therefore,in particular, |H/F(H)\ < m!; consequently |Н/Ф(Н)| < m(m!)andthen
certainly d(H) = <1(Н/Ф(Н)) < m(m’). FinaUy, by Proposition 1.5 we have |H| <
and therefore Statement (a) holds. □
Corollary 1.7 (Bryant, Bryce, and Hartley [1]). A saturated formation generated by
a finite soluble group G contains only finitely many saturated subformations.
Proof. Define
QR0(G/Op.„(G))forp e <r(G),
0 otherwise,
and let g = LF(J"). By IV, 3.2 and IV, 3.3 the class g is a saturated formation
containing G. It will therefore be enough to prove that g contains only finitely many
saturated subformations. If § is one such, then we can write § = LF(h), where by IV,
3.11 the formation function h is the unique smallest defining f>, and for which
therefore h(p) e /(p) for all p e IP. Thus h(p) = 0 for рф alG), and for p e <r(G) the
formation h(p) is a subformation of QRo(G/Op. p(G)). Consequently by Theorem 1.6
there are only finitely many choices for h and hence finitely many possible saturated
subformations f> of g. □
Open Question. Is Theorem 1.6 true for all finite group G?
Some progress has been made with this problem. Skiba [1] shows, as a special case
of a more general result, that form(G) contains only finitely many subformations when
the group G has no Frattini chief factors below Gs, the soluble residual. In another
direction, P.D. Foy [1] has shown that this is also true when Gs, the soluble radical,
is a maximal normal subgroup of G.
Exercises
1. Show that the formation generated by finitely many finite soluble groups contains
only finitely many subformations.
2. (Bryant, Bryce, and Hartley [1]). Let G be a finite group. Prove that the set of all
formations which are generated by subclasses of qd0(G) is finite.
3. (Bryant, Bryce, and Hartley [1]). Let g = qr0(SL(2,5)). Then g contains exactly
six subformations, including the empty formation.
2. Supersoluble groups and chief factor rank 483
2. Supersoluble groups and chief factor rank
All groups considered in this section are soluble.
The theme of this section is classes of groups defined by chief factor rank. The
archetype is the class U of supersoluble groups comprising all soluble groups with
chief factor rank 1. By IV, 3.4(f) the class U is the local formation LF(u) defined by
taking for u(p) the formation of abelian groups of exponent dividing p - 1 for all
primes p. We begin by summarising those properties of U which follow directly from
this local definition.
(2.1) Theorem. Let p denote the reverse natural ordering of P, and let Tp be the class
of Sylow tower groups corresponding to this ordering. Then
U = sll £ 9121 n
Proof Let и be the above local definition of U. Clearly u(p) = su(p) for all p e IP, and
therefore U = sU by IV, 3.14.
Next observe that 9121 = LF(f), where/(p) = 21 for all pe IP by IV, 3.4(b), and
that if g(p) = S„(p) with n(p) = {q e IP: q < p} for all p e IP, then X() = LF(g) by IV,
3.4(g). Since u(p) £ f(p) n g(p) for all p e IP, the rest of the theorem follows from IV,
3.5(a). □
(2.2) Theorem. Let G be a finite soluble group. Then any two of the following state-
ments are equivalent'.
(a) G e U;
(b) All chief factors of G are cyclic,
(c) (Huppert) [2]) All maximal subgroups of G have prime index;
(d) (Iwasawa [1]) All maximal chains of subgroups of G have the same length;
(e) (Venske) [2]) Every maximal subgroup U of G has a cyclic supplement which is
permutable with a Hall system of U.
[Remark. That G is supersoluble follows from each of Statements (c) and (d) without
the assumption that G is soluble.]
Proof. The equivalence of (a) and (b) is proved in IV, 3.4(f) and the equivalenceof
(b) and (c) follows at once from III, 4.14(d). Statement (d) follows from (c) because
U =sU.
Next, assume that all maximal subgroup chains of G have the same length. Because
G is soluble, this length is that of a composition series and therefore coincides with
the number of prime divisors of | G | (including multiplicity). In particular, all maximal
subgroups of G have prime index and we have proved that (d) implies (c). Finally we
note that (c) and (e) are equivalent by 1,4.31. □
The characterization of supersoluble groups as soluble groups with chief factors of
rank 1 naturally raises the question: which saturated formations can be described in
terms of chief factor rank?
484
VII. Further theory of formations
(2 3) Definitions, (a) A rank function R is a map which associates with each prime p
a set R(p) of natural numbers. With each rank function R we associate a class
g(R) = (G e <5: for all p e P each p-chief factor of G has rank in R(p))
of finite soluble groups and call R saturated if 8(R) = Еф<5(К).
(b) A rank function R is called minimal, if for each prime p and each n e R(p) there
exists a group G in Jv(R) which has a p-chief factor of rank n.
(c) A rank function R is said to have full characteristic if R(p) # 0 for all p e P.
(24) Remarks, (a) If R is a rank function, it is clear from IV, 1.3 that g(R) is always
a formation. Thus saturated rank functions are those that give rise to saturated
formations 8(R).
(b) If R is an arbitrary rank function, then the function R' defined by R'(p) =
{r(H/K): H/K is a p-chief factor of some G e 5(R)} is obviously the minimal rank
function with g(R') = 5(R).
(c) If R is a saturated rank function of full characteristic, by IV, 4.3 we have 1 e R(p)
for all ре P and therefore U £ g(R) by Theorem 2.2. Furthermore, hy the same
theorem, 11 = g(R) when R(p) = {1} for all p e P.
(d) If R is a saturated rank function of full characteristic, then R is already minimal
(Exercise 11 below shows that this need not be true when R does not have full
characteristic). To see this, let n e R(p) and H a cyclic group of order p" — 1. By B,
9.7 there exists a faithful H-module V over with Dim V = n and V is irreducible
by B, 9.8. Since by assumption 1 e R(q) for all primes q dividing p" — 1, it follows
that g(R) contains the semidirect product [VJH and hence that R is minimal by
Remark (b) above.
The question of the existence of saturated rank functions not of full characteristic
was first taken up by Kohler [1] and Huppert [4]. In [1] Heineken proved that
saturated rank functions of full characteristic satisfy a certain set of properties, and
subsequently Harman [1] showed that these properties characterize such rank func-
tions. In the same work Harman also characterized the saturated functions which
are not of full characteristic and also extended the theory to absolute ranks (obtained
by viewing chief factors as modules over a splitting field). In order to keep our account
reasonably elementary we restrict ourselves mainly to rank functions of full charac-
teristic and stay close to the treatment presented by Harman in [1J; Huppert’s are
the only results we will discuss from the case of non-full characteristic.
Our next observation shows that the saturated rank functions are characterized
by the property that their associated formations can be completely described in terms
of indices of maximal subgroups.
(2.5) Lemma. Let R be a rank function, and define
£>(R) = (G: ifM< G and |G: M| = p", then n e R(p)).
Then R is saturated if and only if g(R) = f>(R).
2. Supersoluble groups and chief factor rank 485
Proof. First suppose that R is saturated. Since R(R) is obviously a subclass of f>(R),
it will be sufficient to derive a contradiction from the assumption that G is a group
of minimal order in &(R)\g(R). Since &(R) is clearly Q-closed and g(R) is saturated,
G is a primitive group. Let N = Soc(G), and let M be a complement to N in G. Since
G e f>(R), we have |N| = |G: M| = p" with n e R(p). But G/N e g(R) by the choice
of G, and therefore G e 5(R). This contradiction proves that g(R) = f,(R).
Now suppose g(R) = &(R). Since f>(R) is a Schunck class by Remark (b) following
III, 4.21, it is saturated; hence the formation g(R) is saturated and therefore so is R.
□
(2.6) Hypothesis. The rank function R is saturated and of full characteristic (and is
therefore minimal), and <5(R) = LF(f), where f is an inclusive local definition.
As usual, Г(р") will denote the group of all semilinear transformations of an
n-dimensional vector space over (Fp (see B, 12.8 and B, 12.9). The group Г(р") is
primitive, has a socle of order pn, and a stabilizer, denoted by Г*(р"), which is
metacyclicand has order n(p" — 1); in particular,Г*(р")eUs R(R).
(2.7) Lemma. Assume Hypothesis 2.6. Let p e IP and n e R(p). Then
(а) Г*(р")е/(р),
(b) Zp„_, e/(p), and
(c) Z„e/(p).
Proof. Denote the primitive group Г(рп) by G, and set N = Soc(G) = OPiP(G). Since
G/N = Г*(р") e g(R), and since the minimal normal p-subgroup N has rank n e R(p),
we have G e g(R). Therefore Г*(р") = G/Op. p(G) e /(p), and furthermore Z„ e
О(Г*(р")) £ Qf(p) = /(p). Thus Assertions (a) and (c) hold. Assertion (b) follows by a
similar argument, using the fact that if Q denotes the cyclic normal subgroup of order
p” - 1 in Г*(р"), then NQ is primitive by B, 9.8(d). □
(2.8) Lemma. Assume Hypothesis 2.6. Then
RF1: If ne R(p) and m|n. then m e R(p);
RF2: If {m, n} s R(p), then mn e R(p).
Proof. RF1: Let G S Z^. By (2.7)(b) we have G e/(p). Since m|n, it follows that
(pm — 1)I(P" - О and therefore G has a cyclic factor group G e qf(p) = f(p) with
|G| = pm - I. By B, 10.7 and B, 9.8(d) the group G can be represented faithfully and
irreducibly on a module V of dimension m over (Fp. By IV, 3.5(c) it follows that
[V]G e S(R) and hence that »i e R(p).
RF2: Let M and N denote the natural modules over Fp for Г*(р") and Г*(р")
respectively. Then Dim(M) = m, Dim(A) ~ n, and by B, 12.10 both M and N are
absolutely irreducible. By (2.7)(a) the direct product D = Г*(рт) x Г*(р ) belongs
to D0/(p) = /(p) £ g(R) and by B, 5.23 acts irreducibly on the tensor product
T=M®lpN. Since O/CD(T)6Qf(p)=/(p), from IV, 3.5(c) we conclude that
[T] D e <5(R) and hence that mn = Dim(T) e R(p). □
486
VII. Further theory of formations
(2 9) Corollary. Assume Hypothesis 2.6.
(a) If p e IP and n(p) = IP n R(p), then R(p) is uniquely determined by tt(p).
(b) 5(R) = s,5(R).
Proof, (a) It is clear from RF1 and RF2 that R(p) consists of all n(p)-numbers
(products of powers of primes in л(р)).
(b) This follows at once from RF2 and Clifford’s theorem. □
(2.10) Lemma. Assume Hypothesis 2.6. Then
RF3: If p, q are distinct primes with q e R(p) and if me R(q), then q" - 1 e R(p).
Proof Let T s , and let V be a faithful, irreducible T-module over (Fe; by B,
9.8(d) we have Dim( F) = m. Let Q = Z,, and let U be a faithful, irreducible g-module
over By (2.7) we have T e f(q) and Q e f(p), and therefore if n = Dim(U), we have
n e R(p).
Now let W = ([C]g)Qjr([F]T). By A, 18.5(a) the normal subgroup 17’ of W is
minimal and has (Fp-dimension nqm, which belongs to R(p) by RF2. Since the q-chief
factors of IF are centralized by Oe-^W) = (UQ)" V, the groups induced on them by
W all belong to q(W7O, .,(W')) = q(T) s <tf(q) = /(q). It follows that W e g(R) and,
in particular, that Q“ VTe f(p\ whence VT eqf(p) = /(p). Because VT is primitive
and p # q, we can apply B, 10.10 to deduce that there exists ante N and a faithful,
irreducible [F x x F]T-module X over Fp such that |T| divides Dim(20. Since
[F x x VJTe Rof(P) = f(p), we conclude that Dim(A') e R(p). By RF1 therefore
gm - 1 = | T| e R(p). □
The following observations are helpful for calculating examples.
(2.11) Corollary. Assume Hypothesis 2.6.
(a) If q, p e IP and p # qe R(p), then 2 e R(p).
(b) If p,qeP and q e R(p), then R(q) c R(p).
(c) Let Pi,..., pk(k >2) be a sequence of distinct primes satisfying
(i) Pi e R(pi+1) for i = 1,..., к - 1, and
(ii) P*eR(Pi)-
Then R(pt) = M for i = 1,..., к.
(d) If 2 6 R(p) for all p 6 P, then R(p) = N for all odd p.
Procf. (a) If q = 2, we are done. Otherwise 2|(q - 1). Since 1 e R(p) by Hypothesis
2.6, we have q - 1 e R(p) by (2.10) and therefore 2 e R(p) by RF1 of (2.8).
(b) We may clearly suppose that p # q and that R(p) # 1. By (2.8) it will suffice to
show that R(q) n P s R(p). Let t e R(q) n P. If t = q, then t e R(p) by assumption.
Suppose that t q. Applying (2.10) twice, we obtain first that t — 1 e R(q) and then
* a* 9 — 1 e R(p). Since t divides q' 1 — 1 by Fermat’s theorem, we conclude from
RF1 of (2.8) that t e R(p), as required.
(c) By hypothesis and (2.10) the set R(pi+1) contains the even number р((р, — 1)
for i = 1,... ’ к (reading suffices mod к). Therefore 2 e R(p,) for i - 1,..., к by RF1.
Suppose that we have already shown, for all m < n, that m e R(pt) for i = 1,..., к. If
2. Supersoluble groups and chief factor rank
487
n is not a prime, from RF2 of (2.8) we conclude that n e R(pf) for i = I,..., k. On the
other hand, if n is prime, it divides the integer - 1). Because the set R(p(+1)
contains p, by hypothesis and contains pf 1 - 1 by (2.10), it also contains n by (2.8).
Therefore n belongs to each set R(p,), and it follows by induction that R(p) = N for
i=l,...,k.
(d) Since by hypothesis 2 e R(p), and since 4 e R(2) by RF2, from RF3 of (2.10) we
see that for odd primes p, the set R(p) contains 24 - 1 = 3.5 and hence contains 3
and 5 by RF1 of (2.8); in particular, 3 e R(5) and 5 6 R(3) and consequently R(3) =
R(5) = N by Part (c) above. If p > 3, Part (b) above yields N = R(3) c R(p), and
therefore R(p) = N for all odd primes p. □
(2.12) Lemma. Assume Hypothesis 2.6, let ре P, and let n e 14 Then the following
statements are equivalent:
(a) There is an me R(p) such that nj(p" — 1);
(b) Z„ e f(p) and p I n.
Proof. (a)=>(b): If meR(p), by (2.7)(b) the formation /(p) contains Zp„_t. Since
n|(p“ — 1), we conclude that Z„ e Qf(p) = /(p).
(b) => (a): If Z„ e /(p), then E(n/p) e %(Rf If the minimal normal subgroup of £(n/p)
has order pm, then m e R(p}, and by B, 9.8(d) we have n|pm — 1. □
We are now ready to state and prove the fourth, and final, necessary condition for
R to be a saturated rank function.
(2.13) Lemma. Assume Hypothesis 2.6. Then
RF4: If p,q e IP and r e N satisfy the following conditions:
(i) p\(qm — I) for some m e R(q),
(ii) qtfp" — 1) for some n e R(p),
(iii) r|(pk — l)ybr some к e R(p), and
(iv) p e R(p) and r e R(q),
then r e R(p).
Proof. By RF1 of (2.8) we can suppose that m, n, and к are the smallest natural
numbers such that p\(qm — 1), ql(p” — 1), and r|(p* — 1). Let V = Zr, P = Zp, and let
U be a faithful irreducible P-module over IF,. Then Dim(L) = m by A, 9.8(d). Applying
B, 10.12 in the case E = £(l/r), we deduce that the wreath product
W = UP%„8 V
is a primitive group which possesses a faithful irreducible module Y over IFp of
dimension npb, for some b # 0. Since n and p are in R(p), by RF2 of (2.8) we have
npb e R(p). The p-chief factors of W evidently have dimensions 1 and к e R(p) and the
unique q-chief factor of W has dimension rm e R(q). Hence it follows that W and
| Y] W both belong to g(R), and consequently W = YWIOP- P(YW) e f(p).
Now L" is the unique minimal normal subgroup of W and P" V is a complement
to in Ж Therefore by B, 10.10 there is an a e N such that the semidirect product
VII. Further theory of formations
S = [l/11 x x
has a faithful irreducible module X over (Fp whose dimension is divisible by |P* V|;
the action of P * V on each of the a copies of U * here is as in W. Since S e Ro( W) c f (p)
and | VI = r. we have r|Dim(X) e R(p), and hence r e R(p) by RF1 of (2.8). □
The following corollary will also be useful for working out examples.
(2.14) Corollary (Heineken [1]). Let p.qeP, and assume that p\(qm - 1) for some
m e R(q) and that qKp" - 1) for some n e R(p). Then
(a) if p e R(p) and q e R(q), then R(p) = R(q) = 14 and
(b) if R(p) = {p‘: i e N}, then R(q) = {1}.
Proof, (a) Setting r = q in (2.13) yields q e R(p); similarly, interchanging p and q, we
obtain p e R(q). Then by (2.1 l)(c) we have R(p) = R(q) = 14
(b) We suppose that R(q) # {1} and derive a contradiction. If q belonged to R(q),
we could conclude from Part (a) that R(p) = N, contrary to hypothesis. If t is a prime
in R(q), then t # q, and therefore 2 e R(q) by (2.1 l)(a). If p were odd, we could set
r = 2 in (2.13) and conclude that 2 e R(p), contradicting our hypothesis about R(p).
Hencep = 2. Since q|(p'1 — l)withn e R(p), it follows that q|22'— 1 for some suitable
s e 14 Since 2 e R(q) and 2s e R(p), we can deduce from RF3 of (2.10) that 22’ — 1 e
R(q). From RF1 of (2.8) we now conclude that q e R(q), a possibility which we have
already ruled out. This contradiction forces R(q) = {1}. □
Our next major objective is to prove that Conditions RF1-RF4 together charac-
terize saturated rank functions among all functions R: P -> S'W) which satisfy R(p) #
0 for all primes p. The derivation of these four necessary conditions is due to
Heinecken [1], and the proof of their sufficiency is due to Harman [1]. First we prove
a number-theoretical result.
(2.15) Lemma. Let qeP, and O^neZ with qin. Let a be the smallest natural
number with q\(n° — I) and b the largest natural number with qb\(n° — l).Ifb < m e (4
then q"^""" - 1).
Proof. We proceed by induction on m — b. By definition of b the conclusion is
certainly true if m = b. Suppose that it has already been proved for a given value of
m>b. Then there exists a d e Z such that
= dq" + 1.
Consequently n«r"1 = = (d<r + = + , where
d0 = d’q"4-"1'1 + Q^d’-'q"”-2-"-! + + Q 9 + d,
and the induction step is complete.
□
2. Supersoluble groups and chief factor rank 489 *
(2.16) Definition. Let R: IP -<• 0»(N)\0. For each prime p define ।
л(р) = R(p) n IP, and
e(p)= (p"~ l:meR(p)}.
Then Sf„(p)-(e(p)) is defined to be the class of all abelian n(p)'-groups whose exponents
divide an element of e(p). Evidently 9I„(p) (e(p)) is q- and s-closed, and with the help
of RF2 it is not difficult to see that this class is also o0-closed and hence a formation.
We also need to introduce another condition:
RF3': If p.qeP and p/qe R(p), then q — 1 e R(p).
Clearly RF3' is a consequence of RF3 when R(q) # 0.
(2.17) Lemma. Let R: IP -► £P(M)\0, and assume that R satisfies Conditions RF1,
RF2 and RF3'. If G e '2f„(p).(e(p))Grfp), and if V is an irreducible FpG-module, then
Dim(F) e R(p).
Proof. Write К = Fp and A = G^"1". Let Exp(G) = p°u, where p) u, and let f be the
smallest natural number such that u](pi — 1). By B, 5.21 the field L = GF(pf) is a
splitting field for all subgroups of G. Let W be an irreducible L[G]-submodule of VL.
By B, 7.14(a) we have Diml(W')| |G : Л|, and furthermore, in view of the definition of
n(p), the index |G : 4| belongs to R(p) by RF1 and RF2. Since Dim(f(I')|/ Dim,(H/)
by B, 5.18, it will be enough to show that f e R(p), again because of the hypothesis
that R satisfies RF1 and RF2. t
Let u = q*'... q^rf... rf be the factorization of и into distinct prime powers with 1
qt e zr(p)' (1 < i < s) and rf e rr(p) (1 < j < t). Since G e 5И„(р|. (e(p))S„(pt, the definition |
of e(p) guarantees the existence of an h e R(p) such that <7з'|(р/1 — 1)- For I
1 < i < t let и,- be the smallest natural number such that Г;|(р"‘ — 1) and dt the largest
integer such that r/,,|(p"' — 1). Furthermore, define
(0 ifCj < dj
(c, — dj if с, > dj.
Set ht = щгр. Then by (2.15) we have
rf'Kp*' - 1) for 1 < i < t.
Since г1|(рг'“1 - 1) by Fermat’s theorem, it follows that n,|(r( - 1), and since r, e R(p),
using the hypothesis that R satisfies RF1, RF2 and RF3’, we can conclude that
ntr‘‘ e R(p). But then the integer
к = h П n.tf'
i=l
490
VII. Further theory of formations
also belongs to Rip), and it is clear that u|(p‘ - 1). By definition the integer f divides
k, and so f e R(p) by RF1. О
We are now ready to prove the main result of this section.
(2 18) Theorem (Harman [1], Heineken [1]). Let R: P - 0W\0, and let f(p) de-
note the formation ^Ptpy(e(p))Znlp} defined in (2.16). Let g(R) denote the local forma-
tion defined by f. Then any two of the following statements are equivalent :
(a) R is a saturated rank junction:
(b) R satisfies Conditions RF1-RF4;
(c) 0f(R) = 8(R).
Proof. That (a) implies (b) follows from (2.8), (2.10) and (2.13). Also the implication:
(c) => (a) is immediate from the fact that a local formation is saturated. It remains to
prove that
(b) => (c): From (2.17) it is clear that g(R) £ g(R). With a contradiction in mind,
we now suppose that §(R) / ^(R) and choose a group G of minimal order in
g(R)\§(R). Since g(R) is a saturated formation of full characteristic, G is a non-
abelian primitive group; let V denote Soc(G), a p-group say, and let Я be a comple-
ment to V in G. Then evidently Я flp), and consequently Я is also non-abelian; for
otherwise we should have |Я||(р" — 1) and m = DimF (F)g R(p), and this would
imply that Я e f(p).
Let M be a proper normal subgroup of Я. Since g(R) is s„-closed by (2.9)(b), it
follows that VM e $(R), and therefore VM e §(R) by the choice of G; since V =
Op. p(MV), it follows that M e f(p). If Я had a non-trivial n(p)-quotient, we could
take M = Я®"” and conclude that M e 91я(р)(е(р)), which yields the contradiction
Я e flp). Therefore Я = Яе,,г1.
Now let M be a maximal normal subgroup of Я, of index s say, and note that
s e n(p)'. Set A = Then A < H and A e ^lPtpV(e(p)). Let S e Syl,(H) and Q e
На11„(я(М) £ Hall„w(H), and use the^ ‘bar’ notation to denote images under the
natural homomorphism Я -* H/A — H. Our final goal will be to show that
P-«) [ft S] = 1.
This achieved, it will follow that Q = 1, since H (and therefore Я) is 7t(p)-perfect; from
e ®«(pr we can then deduce that Q = 1, in other words that Я is a n(p)'-group. But
the supposition G e g(R) means that m e R(p) and hence that (Dim(F), |Я|) = 1, and
it then follows from B, 9.11 that H is cyclic, which yields the desired contradiction.
To this end we suppose that [ftS] 1 (in particular, that Q 1) and arrive at a
sequence of contradictions. Since |S| = sand Я is n(p)-perfect, it follows that M(= Q)
is the unique maximal normal subgroup of Я; moreover, S complements Q in Я. Let
Q/Qo be a chief factor of Я. Then |Q/Q0| = qr for some q e rt(p) and r e N; indeed,
re R(q) because Я e g(R). Suppose, for a contradiction, that q + p. Since RF3 holds
у hypothesis, we have qr — 1 e R(p), and since H/Qo is a primitive group of order
q's, we also have s|qr - 1 and therefore s e R(p) by RF1. But then s e R(p) n P =
n(P), contradicting the fact that Я is a(p)-perfect. Consequently q = p, and we have
2. Supersoluble groups and chief factor rank
491
(2./J)
p e n(p), and
(2.y)
s|pr — 1 for some r e R(p).
In particular, p||H|. Furthermore, the same argument shows that each n(p)-chief
factor X/Y ofH with \H/CH(X/Y)\ = .s is ap-chieffactor. We now distinguish 2 cases:
Case 1: Q is not a p-group. Set Ё = OP(Q) and D = ОР'(Ё),both normal subgroups
of H such that D < E < Q. Let E/Eo be a chief factor of H with D < Eo. If P is a
Sylow p-subgroup of H/Eo, it follows easily from the Frattini argument that the
normalizer of P complements Е/Ёо in Н/Ёо, and therefore since Й/Ёо is n(p)-perfect,
E/Eo is not central. It follows that s| |T|, where Г = Aut„(E/E0). If Г has order s,
then | E/Eo| would be a p-group, as remarked above. Since this is not the case, it
follows that F(F) is a non-trivial p-group with |T/F(T)| = s.
Let N be a minimal normal subgroup of Г. Since H e g(R), we have [Ё/Ё0]Г e
$(K), and^hence [E/E0]A'e = g(R) by (2.9)(b). If U is an N-composition
factor of E/Eo, it follows that Dim(L') e R(q), where q is the prime divisor of [Ё/Ёо|,
and that N/CK(U) s Zp. Thus
(2.<5)
p|(<7° ~ 1) for some a e R(q).
Since q e R(p), from RF3 we obtain q — 1 e R(p), and therefore by Fermat’s theorem
we have
(2.c)
<?|p’ 1 — 1 with q — 1 e R(p).
Since the group [E/E0]T has smaller order than G, it belongs to g(E), whence
Г e f(q) = 9l„(Q). S„(Q), and as every abelian normal subgroup of Г is contained in
F(T), we conclude that
(2.0
s e n{q).
But now, using (2./J), (2.y), (2.<5), (2.e) and (2.0, we deduce from RF4 that s e K(p), and
we have a contradiction in Case 1.
Case 2: Q is a p-group. Since G is primitive and Soc(G) = Op(G), we have OP(H) = 1.
Therefore Q n C„(A] = 1. Since (|2|, |Л|) = 1, by A, 12.3 there exists a chief factor
K/L of H below A such that Q £ CH(K/L\ Let t be the prime dividing | K/L\.
Evidently A < C„(K/L) because A is abelian, and since AQ is the unique max-
imal normal subgroup of H containing A, it follows_that s||K/CH(K/L)|. Let
H = H/CH(K/L). Then F(H) is a p-group and \H:F(H)\ = s. As in Case 1 we
obtain
(2.tl)
p|(tr - 1) for some c e R(t}.
Since t divides the order of A e 91„|р)(е(р)), we also have
492
VII. Further theory of formations
pg) t|(pe - 1) Гог some e e R(p).
Moreover, the fact that H e R(R) means that H e fit) = 9IMI).(e(t))6„,),and therefore
(2j) senW-
In view of (2.0), (2.y), (2.q), (2.0) and (2.t), Condition RF4 forces s e R(p), against
s e л(р)', and this is the final contradiction. □
(2.19) Corollary. If R is a saturated rank function of full characteristic, then g(R) is
subgroup-closed.
Proof. By (2.18) the formation g(R) has a subgroup-closed local definition f and is
therefore itself subgroup-closed by IV, 3.14. □
The following theorem is a direct consequence of (2.5) and (2.19).
(2.20) Theorem. If R is a saturated rank function of full characteristic, then any two
of the following statements are equivalent:
(a) Geg(K);
(b) If V is a maximal subgroup of G and if p 11G : V |, then | G : U | 6 K(p);
(c) If I ' < G and if V is a maximal subgroup of V, then | V: U | e R(p) when p 11V: U |.
Although Conditions RF1RF4 characterize the saturated rank functions of full
characteristic, an explicit description of all such rank functions is still lacking. We
therefore content ourselves with some examples.
(2.21) Examples, (a) (Heineken [1]). Let л and p be disjoint sets of primes. Suppose
that for each r e p a set л, £ л is given which contains all the prime divisors of t — 1
whenever t e л,. Let R: P -* ^*(ftJ) be defined thus:
R(p) =
'{1}
- the set of nf-num bers
ifp e л
ifpep
ifp e (л и p)'.
Then we assert that R is a saturated rank function. By (2.18) we must check that
Conditions RF1 RF4 are satisfied. Clearly RF1 and RF2 hold. In RF4 it is assumed
that p e R(p), which can only happen in this Example when p e (л и p)'. In this case
R(p) = N and RF4 is trivially fulfilled. It remains to verify RF3. Therefore suppose
that p and q are distinct primes with q in R(p), and let m e R(q). We must show that
qm — I e R(p), which is certainly true for p e (л и p)'. Since q e R(p), we cannot have
pen, and we may therefore suppose that pep. Then q e np £ л, whence R(q) = 1.
Thus m = 1, and since q e np, which by assumption contains the prime divisors of
9 ~ 1. it follows that q — 1 e R(p). as desired.
(b) The special case of (a) obtained by setting л = {pjandp = 0, yields R(p) = {1}
and R(q) = N for all q p, and thus defines the class of p-supersoluble groups (soluble
groups with cyclic p-chief factors).
2. Supersoluble groups and chief factor rank
493
(c) Another interesting special case of Example (a) is obtained by setting л = {2},
p = {2}'. and np = n for all pep. This yields R(2) = {1} and R(p) = powers of 2 for
odd primes p. Evidently g(R) c S2.S2, and from the local definition described in
(2.18) we obtain g(R) c <tM|2|.S2.
(d) If we require of a saturated rank function R of full characteristic that each R(p)
should contain only odd natural numbers, then it follows at once from (2.1 l)(a) that
for all p e P
(2.k)
R(p) =
{1} or
{p‘:i = O, 1,...}.
Conversely, suppose that R: Р-*ф(М) is a map which satisfies (2.k), and let л
denote the set of primes p for which R(p) {1). Since the map R obviously satisfies
Conditions RF1, RF2 and RF3, a criterion for R to be a saturated rank function is
given by the following observation.
(2.22) A map R satisfying (2.к) satisfies Condition RF4 if and only if
(2.2) for all pairs p, q e л
either
or
pp' f l(mod q]
q‘‘"' l(mod p)
for all 1 = 0, 1,...,
for all m = 0, 1,....
Proof. Suppose first that R satisfies RF4 and that there exist primes p and q in л and
non-negative integers I and m such that q[(pp‘ — 1) and pKg’” — 1). By definition of
л we have p e R(p) and q e R(q), and so q e R(p) by RF4. But then p = q, which is
impossible. Therefore л satisfies (2.2).
Now suppose that (2.2) holds; we must check that RF4 is satisfied. Suppose that
the primes p and q and the natural number r satisfy the hypotheses of RF4 (as laid
out in Lemma 2.13). If r = 1, thenr e R(p) because R satisfies (2.k), and RF4 certainly
holds. On the other hand, if r / 1, we may clearly suppose without loss of generality
that r e P. By RF4 (iv) we have r e R(q) and therefore r = q e л because of(2.s). Again
by RF4 (iv) we have p e R(p) and therefore pen. But (2.2) can only be satisfied when
RF4 (i) and (ii) hold if p = q, in which case r = q e R(p). Thus RF4 is fulfilled.
(e) Let p be a prime, and set
(1) for q,tp,
powers ofp if= p.
Then R is a saturated rank function by (2.22).
(f) We now give a general procedure for constructing sets л £ P which satisfy (2.2).
It depends on the following observation:
R(<7) =
(2.23) Let p and q be odd primes such that q = rp - 1 for somereN, and let m be
an odd natural number. Then qm — 1 = (rp — 1)" — 1 = — 2(mod p); in particular,
qi(<]"-V).
494
VII. Further theory of formations
Now let pi be an odd prime, let p2 be a larger prime of the form rp, - 1, and if p,,
., p, have already been chosen, let pr+I be a prime of the form spj p2... p, - 1. (By a
well-known theorem of Dirichlet’s, infinitely many such primes exist.) By (2.23) no
p,'+1 th power of pl+1 is congruent to 1 modulo p, for any i = 1, 2,..., t and therefore
the set Tt = {p,: i e N} (and indeed, any subset of it) satisfies Condition 2.2 of (2.22).
Consequently the map R given by
R(p) =
{1} for p<fn,
powers of p for p e л
is a saturated rank function, and therefore 5((R)) is a saturated formation, which
consists of groups whose chief factors all have odd rank, and which is not p-
supersoluble for infinitely many primes p.
Remark. It is clear from (2.x) that the formation T> of all finite soluble groups whose
chief factors all have odd rank cannot have the form g[R] for any rank function R,
and it is easy to see that D is not saturated. However, Schacher and Seitz [1] have
shown that surprisingly the class Ts (comprising all groups whose subgroups are all
in T>) is indeed a saturated formation, although again T>s is not defined by a rank
function. More details are given in Exercises 13, 14, and 15 below.
Harman [1] has also characterized the saturated rank functions which do not have
full characteristic. These are also described by a set of arithmetical conditions on the
primes in the support of R, but they are too unwieldy to justify the space a general
treatment would call for in this book. We consider just one special situation of
non-full characteristic.
(2.24) Theorem (Kohler [1], Huppert [4]). Let ne N, Zet it'= P: q|n}, and let
pen. Then the map R defined thus'.
/?(«)= <
{m:m|n} if q = p,
N if q e 7t\p,
0 if 6 it'
is a saturated rank function.
Proof. VJe must show that the formation 8(K) defined in (2.3) is saturated, and this
we do by showing that the formation function f defined thus:
W-l) if<7 = p
/(</) = - S if q e 7t\p,
0 for qen'
is a local definition for g(R). Let § = LF(f). First we show that g s S(R). Let H/K
be a p-chief factor of a group G in g, of rank k say, and set Г = G/Cg(H/K). Since Г
is a e ian, we deduce from B, 9.8 (a) and (d) that Г is cyclic and has order dividing
2. Supersoluble groups and chief factor rank
495
p‘ - 1; let к be the smallest such natural number with this property. On the other
hand, by supposition Г belongs to f(p) and so has exponent dividing p" - 1. It follows
that к|и and hence that к e R(p). Consequently G e g(R).
Now we prove that g(R) s g. Let H/K be a p-chief factor of a group G in g(R),
with \H/K\ = pr say, and let A = G/CO(H/K). Since R(q) = 0 for q e n', the group
G, and hence A, is a л-group. Since G e g(R), the rank r divides n, and therefore
(|A|, r) = 1. By B, 9.11 the group A is cyclic and |A||p' - 1. But г|и by definition of
R(p), and hence A e /(p). It now follows that g(R) eg. □
It is a consequence of this theorem that the class of groups of odd order whose chief
factors have rank at most 2 is a saturated formation.
If V is a chief factor of a group G and if К is a splitting field for G of characteristic
p, then V ®ffK is a direct sum of irreducible KG-modules of the same dimension by
B, 5.15(b). We denote this dimension by ro( L) and call it the absolute rank of И If we
substitute “absolute rank” for the usual rank in the Definition 2.3, we obtain the
concept of an absolute rank function R. Saturated absolute rank functions have been
fully characterized in arbitrary characteristic by Harman [1]. In this situation the
class 9191 takes over the role of supersoluble groups, and because the complicated
arithmetical conditions fall away, the results can be formulated in a much simpler
fashion. We refer the reader to Hannan’s University of Warwick doctoral thesis for
the details.
Exercises
1. Show that the class of supersoluble groups is not a Fitting class.
2. Let Я denote the class of all finite groups with the property that each maximal
subgroup has a cyclic supplement.
(a) (Kegel [3]) Show that G e St if and only if each non-Frattini chief factor of
G is either cyclic or has order 4. Verify that Sym(4) e Я\11 (whence the assumption
of system permutability in (2.2) (e) is indispensible).
(b) Show that Я is a Schunck class with basis (Ф nllju (Sym(4)), but that Я is
not a formation.
3. (Baer [1]) Let G = N]N2...Nt be a product of supersoluble normal subgroups
Nj. Show that G is supersoluble if and only if G e 9191.
4. (McLain [1]) G is supersoluble if and only if for each U < G and for each d\| U|
there exists a subgroup D of U with |D| = d. [Thus U is the largest s-closed
subclass of the class of groups which satisfy the converse of Lagrange s Theorem
A, I.4.]
5. (Huppert [1]) Let G = ZfZ2...Z, be a product of pairwise permutable cyclic
subgroups Z,-. Then G is supersoluble.
6. (Huppert) Let G = GIG2...GS be a product of pairwise permutable subgroups
G(. If each product GjGjG* is supersoluble, then so is G.
7. Show that each of the Statements (c) and (d) of (2.2) implies that G is supersoluble
even when the hypothesis that G is soluble is dropped.
8. (Venske [1], [2]). Let U be a subgroup of a soluble group G and a Hall system
of U. Then the subgroup
496
VII. Further theory of formations
/V*(1L.) = <x e G: <x>F = F<x> for all V e £„>
is called the weak normalizer of in G. The weak normalizer of U in G is then
defined as
N*(G) = UNg(tv).
(a) Any two of the following statements are equivalent:
(i) GeU;
(ii) G = A'*(T) for some Hall system Г of G;
(iii) If U < G, then N*(U) = G;
(iv) G has a complement basis К such that A'*(Q) = G for all Q e K.
(b) Part (a) fails in general when “Sylow basis” is substituted for “complement
basis” in Statement (iv).
9. A soluble group G is supersoluble if and only if each link in each maximal chain
of subgroups above each Sylow complement has prime index.
10. Show that a soluble group G is supersoluble if and only if for each maximal
subgroup M of G the following condition holds:
G = <x e G|<x‘)M = M{x'y for all i e
11. Let R(2) = 0, R(3) = {1, 2}, and R(p) = N for primes p > 3. By (2.24) this R is
a saturated rank function. Show that it is not minimal in the sense of (2.3)(b).
12. (Heineken [1]). Let R be a saturated rank function of full characteristic. Show
that
RF4': If p, q, r and se P and a, b, c e N are chosen such that
(i) s|(rc - 1), r|(<7b - 1), and q\(p“ - 1), and
(ii) ar e R(p), bs e R(q), and c e R(r),
then s e R(p).
Furthermore, show that RF4 follows from RF4'.
13. Let D denote the formation of all finite soluble groups with odd chief factor ranks.
Show that is not saturated.
14. (Schacher and Seitz [1]). Ifp e P, let n[p] = {q e P: q has odd order mod p}. Let
6 e ant' l£t V be an irreducible FpG-module. Show that V has odd Fp-
dimension.
15. (Schacher and Seitz [1]). Let be as in Exercise 13. Use Exercise 14 to show
that T)s is a saturated formation, whose local definition f is given by
f(p} =
ifp = 2
if p is odd.
0
Deduce that D! cannot have the form 8(R) for any rank function R.
3. Primitive saturated formations
497
3. Primitive saturated formations
All groups considered in this section are soluble.
The family & of soluble saturated formations referred to in the title of this section
may be characterized as follows:
(1) 0, 6e#;
(2) If /(p) e (P for all primes p, then LF(f) e
(3) If 8,- e 0> and g, s S2 s • •, then (JF=1 8. e
(4) & is the smallest family satisfying (1), (2) and (3).
The (non-empty) formations in & are closed under most commonly-met closure
operations; in particular, they are subgroup-closed Fitting classes and have been
characterized as such by Bryce and Cossey [8]. They often arise in the study of
special families of Fitting classes which satisfy additional closure conditions.
(3.1) Definition, (a) If 9C is a set of formations, possibly including additionally the
empty class 0, we define DST =SF и {LF(f): f(p) e 3C for all p e P}. Thus DSF\0F
consists of the saturated formations which have a local definition whose codomain
is SC.
(b) We set = {0, S}, 3\ = for i e and = (JS=o (Thus, for
example, = {SJrt £ P} и {0}, and 3*г contains, among others, the classes 91,
Sp.tSp, Sp.SpS,,., and the class of Sylow tower groups of a given complexion.) If
/(p) = 91'’ for all p e P, it is not difficult to see that l.F(f) ф 3^.
(c) For SC as in (a) above, we define
USF = |b: 8 = U Bi, where g, e SC and £ g2 c • •
Since the union of an ascending chain of (saturated) formations is again a (saturated)
formation, it follows that a non-empty class in VSC is a formation, and that the family
З’ defined thus:
& =
consists of saturated formations. We call the non-empty elements of 3 primitive
saturated fortnations.
(3.2) Theorem (Hawkes [6]). 3* = D3*.
Proof. Let 8 e D3>. Since 3> £ D3>, it will suffice to derive a contradiction from the
assumption that g e D3>\3>. By definition of the operator D and the family 3\ the
local definition f of 8 has the form
Яр) = U /l(p),
i = l
where f(p) e for all i e N. Let p, denote the ith prime, and for j e N define
498
VII. Further theory of formations
, „ Mp) if p g {pn .... P;}> and
0jP 1.0 ifp>Pj.
and set g. = LF(gj). For a given j e N, there exists a к = k(f) such that f(p) e &k
for all p e P Therefore fy e &k+l S Since g, £ g2 S •, it follows that the
formation 6 = U"=t 8. belongs to = & Hence it will be enough to prove that
g = ®.
If G e fy and p||G|, then G/OP-P(G} e ftp) s /(p), and so G e fy Conversely, let
G e g and let ps be the largest prime in <r(G). If p e o(G), we have
G/Op..p(G)e/(p)= ОЛР).
4 = 1
and so there is a natural number i(p) such that G/Op. p(G) e ftftp). Let и = n(G) =
s +maxp(,o|G| {i(p)}. Then
G/Op,p(G)e/;.(p)£/„(p) = 0„(p)
for all p e o(G). Hence G e g„(C) s ®, and we have shown that g £ ®. □
(3.3) Lemma. If g2e^ for all kA, then the formation g = Q2eAg2 belongs
to 3\.
Proof. We proceed by induction on k. If к = 0, each g2 is either 0 or 6 and the
conclusion is clear. If к > 0, then g2 = LF(fk) with fk(p) e i for all p g P and all
Ле A. By induction the formation f(p) = (~)keAfk(p) belongs to and by IV,
3.5(b) we have g = LF(f). □
(3.4) Lemma. A primitive saturated formation g can be locally defined by a formation
function f such that f(p) is a primitive saturated formation for all primes p.
Proof Let g = U»o fy with fy e and fy £ g.+1 for i = 0, 1, 2..........If ® g
then S = LF(g) with p(p) e for all p G P. Since © n g(p) e 3-k by (3.3), it is clear
that the formation G(p) = Sp(6 mg(p)) belongs to .^+1. Thus the canonical local
definition F, of g,- satisfies Ft(p) e for all p e P. Since gf — g,+i, we conclude from
IV, 3.7 that FAp) S Fj+1(p) for all p e P. If f(p) = (Jjl0 Л(р)> we conclude that
f(p) e 3* for all primes p, and evidently g = LF(f). □
(3.5) Lemma. If g is a primitive saturated formation, then g n W e ^r+1.
Proof. The statement certainly holds when r = 0. We prove the general statement
by induction on r. By (3.4) we can find a local definition f for g with f(p) e 0> for all
primes p. Let r > 1. By induction the formation g(p) = /(p) n W1 belongs to JF, and
by IV, 3.17 the formation g n 9Г is locally defined by g. Therefore gn'Jl'e D3^r =
3. Primitive saturated formations
499
(3.6) Lemma. If is a primitive saturated formation for each 2 e A then so is
8=0леЛ8л-
Proof Let r be a non-negative integer. Then Зл n 91' e Pr+1 by (3.5). Thus 3 n 91' =
ПлелС^л r' 91') e 8,+l by (3.3), and we conclude that
8= Uo(8n9l3eU^ = ^ a
(3.7) Theorem. P = UP.
Proof. If g e UP, then 3 = |j£0 with 3, e P and 3,- s 3j+1 for i = 0,1,.... But
then 3 = Ui=o (8, n 91'), where 3, n 91' e Pi+1 s Pa by (3.5). Therefore 3 e UP„ =
□
(3.8) Proposition (Bryce and Cossey [5]). Let X he a primitive saturated formation,
and assume that X S 91' for some r > 0. Then there exists a countable set of classes
X(, each a finite product of <o,’s (that is, of the form G^ for suitable neN and
tij £ P) such that
(3.«) X = П X,.
i=l
Furthermore, if p is a finite set of primes, then Gp is contained in all but a finite number
of the classes X(.
Proof. We justify both conclusions simultaneously by induction on г. И r = 0, then
X = (1), and we may take Xj = (1) and X,- = G for i > 1. Therefore suppose that r > 1.
By (3.4) we know that X has a local definition/for which f(p) is a primitive saturated
formation for all primes p. Since /(p)n9l'-1 is a primitive saturated formation by
(3.6), by IV, 3.17 we lose no generality in assuming that /(p) £ 9I'-1 for all primes p. ।
Then by induction there exist classes Xj(p)(j = 1, 2,...), each." product of G,’s, such
that /(p) = Q? ! X7(p). Since each X,(p) is Q-closed, we have Gp. Gp(Qjtj X;(p)) = *
Q/i Gp. GpX,(p), and so by definition off we have
x = n er.G„/(p)= n n e,.Gpx/P)= Q
peP pePj=l (j.p)
where X7P = Gp. GpXp(p), evidently a product of G„’s. Since the set N x P is counta-
ble, Assertion 3.a follows.
If p is finite, then Gp f &p- for only finitely many primes p, and for each such p we
may conclude by induction that Gp is contained in all but a finite number of the
classes X/p). The final assertion of the proposition now follows. □
(3.9) Lemma (Bryce and Cossey [5]). Let X he a primitive saturated formation with
X £ 91' for some r > 0. Then there exist (i) a set of primes p and (ii) primitive saturated
formations X„ for each n £ IP such that
500
VII. Further theory of formations
I = s А х.е,ея ).
\яеР /
Furthermore, if p is finite, then Sp £ i'„ for almost all n.
Proof By (3.8) every primitive saturated formation of restricted nilpotent length has
the form A^ £,, where each X, is a finite product of classes of the form 6„, and if
(p| < oo, almost all I, contain <sp. Since
A(s„n(n = epn( A (A
< = 1 \ \«L P // P V=1 / /
where p = A,® i Po by (3.6) it will suffice to prove the two assertions of the lemma for
classes I of the form
x = 6, e, ...e,.
*1 n2 nn
(Of course, such products no longer necessarily have restricted nilpotent length.) We
will prove both assertions simultaneously by induction on n, the length of such a
product.
If n = 1, we have i = n (Ans p and both assertions are clear.
For n = 2, let r = nt и л2, and let О; = г' и n, for i = 1, 2, so that и <r2 = P and
o'i £ o’]. Then obviously
1 = sn, ®пг = s. Sn2 = s. n eOi 6„. e„2
since G„; c GO|, and on taking = GO| andX„ = S for л Ф <r2, we evidently obtain
the two desired conclusions.
Finally, suppose that n > 3. Then, as in the case n = 2, we can find suitable sets v,
Pi, p2 — P such that
б^,е^ = е,пбЯ1е,2ей,
whence
£ = s„..,e„ e.л 2,...e, s s s .
Using the induction hypothesis for n - 1, we obtain
1=s₽ n Со. n
\П£Р
where £„2 — J л 6Ж1... S„n 2, and £„ = X'n for n p2. If p is finite, by induction
-=>p is contained in almost all and hence clearly in almost all □
3. Primitive saturated formations
501
(3.10) Theorem. Let a be a finitary closure operation with the following properties:
(i) G„ = a£, for all n £ P.
(ii) 91' = a91' for all r e f<J.
(iii) An л-closed formation is saturated.
(iv) A saturated formation g is А-closed if and only if it has a local definition f such
that f(p) = Af(p) for all p e P.
Then a formation is А-closed if and only if it is a primitive saturated formation.
Proof. Let JT be the family of А-closed formations. We must show that &> = u 0.
First let 0 # g e 0>. Then g = (J?=i g., where gj £ g2 £-- and g, e ,^ =
Ur=o By Property (i) the formations in are А-closed, and it follows from Prop-
erties (i) and (iv) by induction on i that the formations in are А-closed for all i e IM.
Thus each g; is А-closed, and consequently so is g by II, 1.7(b). Thus t X u0.
Conversely, let g be an А-closed formation. Set gr = g n 91'; then certainly g =
Ui=o g> ancl Agr = A(g n 9P) £ Ag n a9T = g n 9Г = g, by Property (ii). Thus
each formation gr is А-closed, and so it will suffice to prove the following assertion:
(3.a) An А-closed formation X £ 91' is a member of .'3%+1 for r = 0, 1,2,....
We prove (3.a) by induction on r. If r = 0, then I = (1) e 3\. Suppose that (3.a) holds
for r = 1,..., n and let X = <q, rc, a)X £ 9l”+1. By Properties (iii) and (iv) we have
X = LF(f) with Ajip) = f[p) for all primes p. By IV, 3.17 the formation function g
defined by g(p) = J(p) n 91" is also a local definition for X. and Aglp) = glp) for all
primes p by Property (ii). Thus glp) e &n+l by the inductive assumption, and con-
sequently X e D^„+i = ^„+2. This proves (З.сс) and shows that g = (J? 0 gf e &.
□
(3.11) Lemma. Hypothesis (i), (ii), and (iv) of Theorem 3.10 hold when A is any one of
the following closure operations: S, s„, Nc, <s, No>.
Proof. It is clear that S„ is closed under all the listed operations. If flp) = 91' 1 for
all primes p, then 91' = LFlf). It therefore follows from IV, 3.14 by induction on r
that 91' is also closed under s. s„ and No, and hence under <s. No).
To verify (iv) we observe that if ae {s,s„, n0}, then by IV, 3.14 and IV, 3.16 a
saturated formation g is А-closed if and only if its canonical local definition F satisfies
F(p) = aF(p) for all primes p. It follows that the same is true with a = <s, No)> and
Hypothesis (iv) is satisfied in each case. □
Remark. In Chapter XI, Theorem 1.2 we will prove that Hypothesis (iii) also holds
when a = <s, No) and thereby characterize primitive saturated formations as
subgroup-closed Fitting formations.
Exercises
1. If r is a non-negative integer and if g e .3^, then
g n 91' e 3*f+1 for i > 0.
502
VII. Further theory of formations
2. (Bryce and Cossey [8]). Let g be a primitive saturated formation contained in the
class f = where p,, p2, • ••, P„ are primes satisfying pf # pf+1
for i = 1,, n -’1. If ^"contains groups of nilpotent length n, prove that
5 = 1- ’
4. The saturation of a formation
This section has two main themes. The first is the proof of a theorem of Cossey
and Oates-Macdonald, which states that in the universe of finite soluble groups
<Q, Ro, E»> = (e»QRc)2; in other words, if X £ to, then EeQR(jE4QR(,X is the smallest
saturated formation containing I. The second subject under investigation is the
saturation of the formation § j g (introduced in IV, 1.2). Here g and § are formations
of soluble groups, § is saturated, and j g denotes the class of groups whose
^-projectors are in g. The culminating theorem states that only when = S„ for
some л £ IP is it true that j g is saturated for all saturated formations g.
(4.1) Lemma. If g is a saturated formation of soluble groups, then
g = EA>(gn‘₽),
where ф denotes the class of primitive groups. In particular, g = E^RoflJpe p F(p) Ф)
when F is the canonical local definition of g.
Proof. Let G e g and H = G/4>(G). Ifdenotes the set of maximal subgroups of
H, then Пл<6.^Соге„(М) < Ф(Н) = 1 by A, 9.2(e). But H/CoreH(M) e Qgn ф =
g n and therefore G e e»(H) £ EcR0(g n ф). Hence g £ F^Rjg r> ф), and since
the reverse inclusion is satisfied trivially, the first statement is clear.
If G e g n and Soc(G) is a p-group, then G e SpF(p) = F(p); therefore g n SP =
Up.PFlP)n^- □
(4.2) Lemma. Let X^&,let^= <q, r0, e„>X, and set
h(p) = QR0(G/Soc(G): G e qFo ф')
for all p e P. Then h = f, the smallest local definition of g.
Proof- If S’ = LF(h), we assert that qX £ f>. We suppose not and choose a group G
of minimal order in QX\f>. Then G e ф” for some p e IP and therefore G/Soc(G) e
Q3E n F £ h(p). But by the choice of G the group G/Soc(G) also belongs to §,
and it follows that G e f>, a contradiction, which proves that QX £ f>. In con-
sequence g £ f>, and therefore f(p) £ h(p) for all p e IP by IV, 3.11. But since h(p) £
QR„(G/Soc(G): G e Qg n F), we can conclude from IV, 3.10 that h(p) £ flp) and
hence h(p) = f(p). w-JW.
4. The saturation of a formation
503
(4.3) Proposition. Let £ £ S, and let 8 = <q, r„, Еф>£. if g = LF(F)> where F
notes the canonical local definition, then
F(p) £ QRoe4qro£ for all p e P.
Proof Let H e F(p) n and set N = Soc(H). By (4.2) we have f(p) £ qr^ and so
we may suppose that H f(p). Since Sp/(p) = F(p) by IV, 3.8(a), we have H/N e f(p)
and N is a p-group, Hence by (4.2) we see that H/N s S/T. where S is subdirect tn a
direct product X[=1 A, and At s D^SocfDJ with £>, e <j£ n for i = 1.r. Let
»:О = ХД-ХЛ
i=l 1=1
denote the natural homomorphism with Ker(v) = X'=l Soc(D;), and let S = v'*(S),
the inverse image of S under v. Since each is in ']!р, the group M = Ker(v) may be
viewed as an FpS-module faithful for S, and from this viewpoint we obtain
S =! [M]S.
Since S is subdirect in D e r0(q£ n ЧН £ qr0£, it follows that S e R(IQR„S = qr0£
by II, 1.18(b). Since H/N ~ S/T, we can view N as an irreducible FpS-module with
T in its kernel and can then apply IV, 4.1 to deduce that [IV] S e qroe*ro(S) £
qr„ecroqro;E = qrof^qr„£. But then we conclude that
H S [tV](S/T) ([N]S)/Te QQRoE»QRoi = QR0E4QR<,£. □
(4.4) Theorem (Cossey and Oates-Macdonald [1]). The saturated formation
<Q, Ro, еф>£ generated by a class £ of finite soluble groups is (e*qr0)2£.
Proof. Let 8 = <0, Ro, еф)£, and let F denote its canonical local definition. Then
from (4.1) and (4.3) we have
£ E»RoQRoE»QRo£ £ (e»QRc,)2£ £ <Q, Ro, Еф>£ = 8-
□
Remark. In [13] Forster shows that <Q, Ro, Еф> = (QR,T»)2 in the universe of ar-
bitrary finite groups and that <q, r0, еф> = QR0E»QR0 in the case of soluble groups.
The rest of this section is devoted to a method which allows us to describe the
‘saturation’ <Q, Ro, еф>£ of certain formations £, in particular the saturation of
formations of the form £ = [ 8< which are defined in IV, 1.1.
(4.5) Proposition (Doerk [5]). Let 8 be a formation, and let & = LF(H), where H is
the canonical local definition. Let p be a given prime, and assume that to each group
G e 8 there corresponds a unique conjugacy class r(G) of subgroups with the following
8 = E*R„( U F(p)n$)
\pe P /
504
VII. Further theory of formations
three properties:
(1) t(G) £& „ , ,
(2) If Hip) * 0, for each G e g there exists a group H e such that
(a) Soc(H) is a minimal normal subgroup of H of p'-order, and
(b) if T e r(G) and S = H x T, then Sefil x G) and SHW n H * 1;
(3) Let U = NL, the semidirect product of a normal elementary abelian p-group
N with an S-'group L, and let F e r(L). If F e Proj 6(NF), then Ue^.
Then the canonical local definition F of the saturation g = <Q, R„, E»>g satisfies
F(P) = 6pg.
[Remark. We shall see below that in favourable circumstances these hypotheses are
satisfied with r(G) = Proj&(G).]
Proof. By IV, 3.4(b) the class 'Jig is a local formation containing g, and therefore
F(p)<=6„gbyIV,3.11.
We now suppose that F(p) Z Spg and derive a contradiction. Let G be a group
of minimal order in the supposed non-empty class Spg\F(p). Since F(p) = SpF(p)
is a formation, G has a unique minimal normal subgroup M with G/M e F(p), and
M is an s-group for some prime s p; moreover, this forces Op(G) = 1, and hence
Geg.
Let T e r(G), and consider first the possibility that H(p) = 0. Let
W = ZpQjreg G,
andsetE = Z", the base group ofthe wreath product W\ then evidently E = Op- P(W).
Since Те § by Property (1), it follows from III, 3.23(a) that T is contained in an
fi-projector of ET: hence T e Projg(ET) because § C Sp- in this case. Then, because
of Property (3), we conclude that W e g and hence that G = W/Op. p(lV) e F(p), a
contradiction.
Thus we can suppose that H(p) 0, and by hypothesis there exists a group H in
S> n g satisfying Properties (2)(a) and (2)(b). Let D = H x G and S = H x T, where
again T e r(G). By B, 10.7 the group H has a faithful irreducible module A over Fp. If
R = S"w n H, then 1 R < H, and so СЛ(К) is an H-submodule of A. But A is
faithful and irreducible for H, and therefore СЛ(В) = 1. Let В denote the regular
IF,, G-module, and regard N = A ®Fr В as a D-module in the usual way (see B, 1.12).
Since СЛ(В) = 1, and Л/н = Лф-”фЛ, we have CN(B) = 1, and therefore S is
an ^-projector of [NJS by IV, 5.16(a). By Property (3) the semidirect product [/V]D
belongs to g. Since A is faithful for H, the kernel of D on the submodule A ® lc of
N is precisely 1 x G. But N1>r, SB® ® д which is faithful for 1 x G. There-
fore Ker(D on N) = 1, and consequently N = Op.p(ND). Because ND e g c g, we
have H x G S ND/N e F(p); in particular, G e F(p), which is the desired contra-
diction. Therefore F(p) = Spg. □
Next we describe conditions which guarantee that a function r exists satisfying at
least Properties (1) and (2) of (4.5).
4. The saturation of a formation
505
(4.6) Lemma. Let Я be a formation, and let H be the canonical local definition of a
saturated formation £>. For a given prime p set r(G) = Projg(G) for each Ge ft. If
fjnfiisa saturated formation and^cs^^ H(p), then t satisfies Properties (1) and
(2)0/(4.5).
Proof. Property (1) is immediate from the definition of an ^-projector, and if H(p) =
0, there is nothing further to prove. Hence suppose that II (p) ± 0.
We will first show that there exists a group H e (f> n g)\H(p) with the following
properties:
(a) Soc(H) is a minimal normal subgroup of H and has p'-order;
(b) Z(H) < Нн,,л.
Let L be a group of minimal order in (f> n g)\H(p). Then L has a unique minimal
normal subgroup, M say, and M is an r-group for some prime r p. If F is the
canonical local definition of £>ng, then L e Qr F(r), and therefore Le F(r) since
Or (L) = 1. Let Q be an r-complement of L, let l0 denote the trivial F,Q-module, and
let P = (1C)L, which is the principal indecomposable FrL-module by B, 6.16 and which
therefore has the trivial module 1L as its socle. Since Core,(Q) = O,.(L) = 1, the
module P is faithful for L. Let
H = [P]L.
Then H e 3 because L e F(r\ Since P is faithful for L, we have Soc(H) < P, and
since 1£ is the unique minimal submodule ofP, clearly 1L = Soc(H) = Z(H). Therefore
H has Property (a). Now H/P = Lf H(p) by choice of L, and so HHle> / P. Thus
1 Нн,р' < H, and since Z(H) is the unique minimal normal subgroup, H also has
Property (b).
Now let G e g, and let T e Projg(G). Since H e f>, the subgroup S = H x T is an
^-projector of// x G. If it were the case that n/f = 1. from A, 4.11(b) we could
conclude that
SH,P> < Z(H) x T,
and hence that H/Z(H) x T)/(Z(H) x T)e QH(p) = H(p), against Property (b).
Therefore S'llp' n H Z 1, and Property (2) of (4.5) is satisfied. □
We can now describe the saturation of the formation
£4 g = (G e 6: Projg(G) c g),
when £> and g are saturated formations. Since £> J g = £> | (£> g) by IV, 1 -2(b), we
can always assume that g £ §.
(4.7) Theorem (Doerk [5]). Let g and £> be saturated formations with canonical local
definitions F and H respectively, and assume that g E £>. Let T = 51 g, and define a
formation function E by
506 VII. Further theory of formations
f§|F(p) ifgsHlp).
£,p)-le(>3E if 8 <£H(P).
Then E is the canonical local definition of the saturation X = (Q, Ro, ЕФ)Х of X.
Proof Let r(G) = Proj6(G). Since 8 S &, it is clear that Property (3) of (4.5) holds
for this r with X in place of 8- Since f> n 1 = 8, it follows from (4.6) that Properties
(1) and (2) also hold if 8 S H(p). If X denotes the canonical local definition of I, the
conclusion of (4.5) shows that X(p) = E(p) whenever 8 £ H(p).
Now suppose that 8 S H(p) and, for a contradiction, that § J F(p) is not contained
in X'(p). Let G be a group of minimal order in (f> | F(p))\X(p). Then, as usual, G has
a unique minimal normal subgroup N, which is a p'-group because SpX(p) = X(p).
Let
И7 = Zp Qjreg G,
and let В denote the base group of W. If L e Projg(G), then L e F(p) £ H(p), and
therefore by IV, 5.16(a) the subgroup BL is an ^-projector of W. SinceBL e SpF(p) =
ftp), it follows that We 8 = £ and therefore G = W/B = W/Op.ДИ7) e A'(p),
against the choice of G. This contradiction proves that § | F(p) £ X(p).
Let G e qX n = X n let N = Soc(G), and let E e Proj6(G). Since E e 8 £
H(p) by supposition, we have NE e GpH(p) £ f>, and hence N < E. But N = C0(N)
because G is primitive, and therefore Op.(E) = 1. It follows that E e F(p) and hence
that G e F(p). From IV, 3.8(a) and (4.2) we now deduce that
X(p) = SpQR„(G/Soc(G): G e qX n ф")
£<Sp(&|F(p)) = 5|F(p).
Thus we have shown that X(p) = § | 8(P) when 8 £ H(p), as desired. □
(4.8) Theorem (Doerk [5]). Let 8 and be formations, assume that is saturated
and let H denote its canonical local definition. Then the formation § | 8 ls saturated
if and only if
(1) & n 8 is saturated, and
(2) for each prime p either £> n 8 £ hl(p) or H(p)r> 8 = G(p), where G is the
canonical local definition of £> n 8-
Proof. Set X = § 18- To prove the necessity of the conditions suppose that X is
saturated. Then n 8 = & <~ X is certainly saturated. Let p be a prime for which
&r, J J Hip}. Applying(4.7) with 5 n 8 in place of 8, we obtain S„X £ X and hence
= Evidently G(p) £ H(p) r> 8- Suppose this inclusion is proper, and let G be
a group of minimal order in (Hip) n 8)\Gf P). Then G has a unique minimal normal
subgroup N. and N is a p'-group because G(p) = <5pG(p). Let T = G, and let
В denote the base group of this wreath product. Then В = CT(B) and consequently
& ~ Since G e H(p) п8£&г>8£Х, we have T e &PX = X. On the other
4. The saturation of a formation
507
hand, T e topH(p) — H(p) £ f>. Therefore Те Jini = and we conclude that
G S T/B = T/Op. p(T) e G(p). This contradiction proves that G(p) = H(p) n ft.
To prove the sufficiency, assume that Conditions (1) and (2) are satisfied, let
_T = <Q, R„, ЕФ>Т, and let G be a group of minimal order in 3E\3E. As usual G has a
unique minimal normal subgroup, a p-group N say. Let H be an ^-projector of G
If 6 n 5 £ H(p), then HN/N e fir, j£ fir,ft £ H(p), and consequently N <, H.
Since G e T, it follows from (4.7) with f> n ft in place of ft that H/C„(N) e F(p) and
hence that H e ft. But then G e 3E, contradicting the choice of G. On the other hand,
if H(p) n ft = G(p), a similar argument gives H e LF(G) = 5 n ft and yields a final
contradiction. q
We can now give a precise description of those formations § for which § | ft is always
saturated for all saturated formations ft of full characteristic.
(4.9) Theorem (Doerk [5]). Let § be a saturated formation of characteristic n, and
let H be its canonical local definition. Set p = jpsP: Char(H(p)) n\. Then the
following statements are equivalent:
(a) Sift is saturated for all saturated formations ft containing 91;
(b) f> = G„n(QpepGp.Gp), the formation of p-nilpotent n-groups.
Proof, (a) => (b): If |n| = 0 or 1, then § clearly has the stated form. Therefore suppose
that |n\ > 2. First we prove the following claim:
(4.a) If Ge $>\H(p) and I = QRo(G/Op. p(G)), then H{p) £ Gpft.
Let ft = <эр. Gpl. Then clearly ft = LF( f) where /(p) = I and f(q) = G for all q p;
in particular, ft is a saturated formation containing 91. If F is the canonical local
definition of ft, then evidently F(p) = GpT. Let G be the canonical local definition of
f> n ft. Since § | ft is saturated by hypothesis, Theorem 4.8 implies that either
fl n ft C H(p) or Щр) n ft = G(p) £ F(p) = Gpl. Since G e £>\H(p), the first pos-
sibility is ruled out. Therefore H(p) n ft £ GPI. If Л is a g'oup of minimal order
in H(p)\GpI, clearly Op(A) = 1. But then A e H(p) n Gp.GpT = H(p) n ft £ GPI,
a contradiction. Thus H(p) £ GpI, as claimed in (4.a). We now distinguish two
cases:
Case 1: Let p e p. Then there exists a prime q in n\Char(H(p)) and Z, e £>\H(p)-
With G = Z, in (4.a) we deduce that H(p) £ Gp. Therefore H(p) = Gp in this case.
Case 2: Let pen\p. The hypothesis that 5191 is saturated implies that either
S>n91 £ H(p) or H(p)n91 = Gp by (4.8). Since Char(H(p)) = n and |n| > 2. the
possibility that H(p)n91 = Gp is ruled out. Therefore 91, = £>n91 £ H(p). We
suppose that f>\H(p) contains a group G and derive a contradiction. Let 1 =
QR„(G/Op. p(G))* so that H(p) £ GpX by (4.a). Then 91, £ GpT and consequently
6, £ Gpl for some q e n \ {p}. However, if c denotes the class of a Sylow «/-subgroup
of G, the class of every «/-group in 1, and hence in 6,1, is bounded above by c. This
contradiction shows that H(p) = fl for all p e n\p.
508
VII. Further theory of formations
These two cases together prove that f> is the formation described in Statement (b).
(b)=>(a): The canonical local definition H of the formation f) of p-nilpotent
л-groups is given by:
' for p e p
H(p)=^& forpen\p
0 for p £ л.
Let g be a saturated formation of full characteristic, and let G be the canonical
local definition off) g. If H(p) = Sp, then H(p) n g = = G(p); if H(p) = f>, then
£> n g £ £> = H(p); and finally, if H(p) = 0, then H(p) r> g = 0 = G(p). Thus Con-
ditions (1) and (2) of (4.8) are satisfied, and by that theorem the formation f) | g is
saturated.
In [2] Blessenohl has shown that the formation S„ J. g is saturated for all saturated
formations g. We now sharpen this result by showing that the classes S„ are
characterized among saturated formations by this property.
(4.10) Theorem (Doerk [5]). Let $ibea saturated formation of characteristic n. Then
the following statements are equivalent :
(a) f) | g is a saturated formation for all saturated formations g,
(b) 5 = s„.
Proof, (a) => (b): Let § = LF(H). By (4.9) it will be enough to show that the set
p = {pen: Char (H(p)) # n}
is empty. Suppose, if possible, that pep, and let q e л\СЬаг(Н(р)). By hypothesis
& | S, is a saturated formation, and therefore by (4.8) either 2q = fi n S, c l/(p) or
H(p) nS,; = G(p), where G is the canonical local definition of § n S, = <5,. Since
q ф Char(H(p)), the first possibility cannot arise, and since G(p) = 0 and H(p) # 0,
the second is also ruled out. Therefore p = 0.
(b) => (a): Iff, = S„, its canonical local definition H is given by H(p) = S„ for p e л
and H(p) = 0 for p ф n Let g be a saturated formation. If p e л, we have f) n g =
H(Ph and if рф n, we have H(p)n g = 0. But the canonical local definition G of
& n g(c gj satisfies G(p) = 0 for p ф n. Therefore Conditions (1) and (2) of (4.8)
hold and we conclude that | g is saturated.
Exercise
1. Let c be a product of the 3 operations q, r0, еф in some order (each used once).
Show that for each of the 6 choices for c there exists a class JE of soluble groups
such that cl is not a saturated formation.
5. Strong containment for saturated formations
509
5. Strong containment for saturated formations
All groups considered in this section are soluble.
In Section 1 ofChapter VI we introduced and studied the partial order « of strong
containment for the family of all Schunck classes in G; it was shown, in particular,
to induce the structure of a complete lattice on 3f. The first investigation of the
concept of strong containment, however, was carried out by Cline [1] in the context
of the family 5* of saturated formations in G. In this setting the theory is less
satisfactory: there is no inherited lattice structure from Xf the proofs are more
difficult, and results of general applicability are harder to find. Whereas the maximal
elements of (Jf, «) have been fully determined, those of (3-, «) are not yet well
understood. Cline was able to make progress only by restricting attention to satur-
ated formations of special type, for example those of the form Я = 91X for some
formation I. Cline’s work was subsequently simplified and extended by D’Arcy [1],
and it is his approach which we follow here.
The boundary criterion for strong inclusion between Schunck classes, proved in
VI, 1.5, is poorly suited to the study of (SF, «). In its place we shall use the following
criterion.
(5.1) Lemma (D’Arcy [1]). Let g = LF(F) £ = LF(H), where F and H are the
canonical local definitions. Then g « 1/ and only if for each H e § an ^-projector
E of H satisfies HH,p} < EF(p} for each p e Char(g).
Proof. First suppose that g « f>, let H e E e ProjR(H), and let p e Char(g). We
form the wreath product G = Zprljrce Hand denote its base group by B. By IV, 5.16(a)
the subgroup X = HCB(H"tp>] is an ^-projector of G and Y = ECB(EF,P>) is an
g-projector of G. Since g « f>, there exists an element g = bhe G = BH such that
Y9 < X. Since b e В e 91, we have
CB(EFtp,)e = CB(EF{p,)h <Xr.B= C„(HH,P>).
But because СВ(НИ,Р>) < G, we have CB(EF|'”) < CB(HH”’)), and it follows from B,
11.1(b) that HH,P> < EF,P>. Thus the condition is necessary.
We will prove the sufficiency by induction on |G|. Assume that the stated condition
holds. Let G e G, let M < G with F(G)M = G (see III, 6.4(b)), and let H e Proj6(M).
The induction hypothesis yields an g-projector E of M contained in H, and by
assumption HH,P> < EFlp}. It follows that, if P = O„(G), then
C„(EF'P>) < CP(HH,P>).
and the desired conclusion that an Sj-projector of G contains an g-projector of G
follows at once from IV. 5.16(a). □
We now apply this criterion to two examples of strong containment between satur-
ated formations.
510
VII. Further theory of formations
(5.2) Examples, (a) Let я S IP, let & and I be saturated formations with g s and
I £ G„-, and set
S = g ° X
Let F and H be the canonical local definitions of & and § respectively. By IV, 3.13
we have H(p) = F(p) о I when p e Char(g). If H e §, then IIх belongs to g and is a
normal Hall я-subgroup of H. Thus E = H1 is an g-projector of H, and, if p e
Char(g), we have En'” = (H1)"'” = HF,P'°1 = HHtp>- Consequently g « & by (5.1).
(b) Let я £ IP, let I be a formation, let g = G„GK. and § = Then the
formations g and § are saturated and have canonical local definitions, say F and
H respectively. By IV, 3.13 again we have F(p) = 5 and H(p) = § for p e я and
F(p) = and H(p) ~ for p <£ я. Let H e fi, let S be a Hall я-subgroup of H
and let E = N„(S). If E < V < H, then V = Nv(S}V3- = Nr(S)F» = EFB by the
Frattini argument. Therefore E is an g-covering subgroup and hence an g-projector
of H. If p e n, clearly HH,P> = 1 < EFtpp on the other hand, if p f я, then НЩр> <
O„(H) < S = EF(p\ Hence Hu,p> < EF,P> for all primes p, and therefore g « § by (5.1).
The next lemma gives more information about the situation g « § when the local
definitions have special properties.
(5.3) Lemma. Let LF(F) = g « & = LF(H) with F and H canonical.
(a) If F(p) = F{q) 0, then H(p) = H(q).
(b) If X is a formation and if F(p) = GPX for all p in a set it of primes, then there
exists a formation ф such that H(p) = Gp4) for all pen.
(c) If F(p) = g, then H(p) = fi.
Proof, (a) If possible, choose a group G in H(p)\H(q), let W = Zprb,eg G e <ZpH(p) £
f>, and note that OP(W) = 1. By (5.1) there exists an g-projector E of W such that
И7"”* < EF|,il. Since Ef<’1 = EFtp> by hypothesis and since EF,P> = Fs“/?<'11 is a p'-group,
it follows that If"'1” < OP(W) — 1. Thus W e H(q), and therefore G e Q( W) £ H(q).
This contradiction shows that H(p) £ H(q) and hence by the symmetry of the hypo-
theses that H(p) = H(q).
(b) Set 9) = P|,e„H(q). Let рея. Then ®рф £ topH(p) = H(p). By way of con-
tradiction, suppose that the class Н(р)\<5рф is non-empty; then it contains a group
G of minimal order, and Soc(G) is a minimal normal r-subgroup for some prime r # p.
By B, 10.7 the group G possesses a faithful irreducible module Л' over Fp, and the
semidirect product H = [NJG belongs to GpH(p) £ fi. Since G ф 'J, we have G ф
H(q) for some prime q, and therefore by (5.1) we have
(5.a) Soc(G) < G"*” < EF”> = E3«* < El
for some g-projector E of G. Because H H(q) and N = Soc(H), we have N <
H < E for any g-projector E of H, and it follows that EN is an g-projector of
H. Since N = CH(N), we have 0„.(EN) = 1, and therefore EN e Flp) = &pX. In par-
ticu ar, E is a p-group, and thus by (5.a) so also is Soc(G). This contradiction proves
that H(p) = S/2)-
5. Strong containment for saturated formations
511
(c) If E(p) = 8 and G e Sj, then G"1'” < Er'”' = Ей = 1 by (5.1). Thus & <= H(p)
(C§). and so H(p) = §.
Theorem 5.4 (Cline [1]). Let jy and § be saturated formations with 3 « S>-
(a) If 8 = 91X for some formation X, then § has the form f> = ВД) for some
formation ф.
(b) JfOiere exists a prime p such that jy = G„Sp.JE for some formation X, then
top. 4) for some formation ф.
Proof. Let F and H denote the canonical local definitions of jy and § respectively.
(a) If 8 = 91X, then F(p) = ZpX for all p e P. By (5.3)(b) there exists a formation
'Э such that H(p) = <эр'2) for all p e P. Thus § = 919). (We have appealed to Example
IV, 3.4(b), where formations with a “constant” local definition are described.)
(b) If 8 = then F(p) = 8 and E(q) = Sp.3E for all q e p‘. By (5.3)(a) the
formation H(q) has a constant value, 9) say, for all q e p', and therefore Sp.'2) = 9)
since each H(q) is full. By (5.3)(c) we have H(p) = f>, and therefore
S= A er.H(r) = §n A 6..? =§пб,?£&
re P \qep' )
Hence
S = to,?) = to, $,.?).
□
If F is the canonical local definition of a saturated formation 8, we recall from IV,
5.19(a) that the formation f* defined by
f*(p) = (G: the ^-projectors of G belong to F(p))
for all primes p is also a full local definition of jy If 8 « it seems that the
relationship between 8 and § is strongly influenced by f* and h*, in particular by
whether /*(p) and h*(p) are equal or not for a given prime p.
(5.5) Proposition. Let 8 = LF(F) « & = LF(H), and let p e Char(8). Then any two
of the following statements are equivalent:
(a) If E is an ^-projector of an fy-group H, then
yjHipi _ yjf'wi _ eF(p):
(b) We have h*(p) = f*(p);
(c) We have H(p) £ /*(p).
Proof, (a) =» (b): Let E and H be respectively 8- and Si-projectors of a group G with
E <H. From Statement (a) we deduce that He H(p) if and only if 1 = H"w = E w
if and only if E e F(p). Thus (b) holds.
(b) => (c): This is clear since H(p) £ h*(p).
512
VII. Further theory of formations
(c) => (a): Let H e ft and let £ e Proj B(H). Then by Statement (c) and (5.1) we have
£f(p) > > EFiP\
and therefore (a) holds. П
Remark. Example 5.2(a) satisfies h*(p) = f*(p) for all primes p e Char(g).
The condition that h*(p) = f*(p) for some prime p does not easily allow conclu-
sions about and 5 to be drawn. In contrast, the condition h*(p)#/*(p) in
conjunction with ft « § sets severe restrictions on the class §, as the following results
will show. By analogy with Schunck classes, this is only to be expected; for we saw
in VI, 1.5 that the condition ft « places greater limitations on the Schunck class
f> than on the Schunck class ft.
(5.6) Lemma. Let F and H be canonical local definitions of saturated formations ft
and § respectively. Let X be a formation, and assume that for some prime p we have
Hx < E whenever E is an ^-projector of a group H in H(p). If H(p) £ /*(p) # 0, then
H(p)<=X.
Proof. First we prove that
(S.jff) H(p)\/*(p) s X
Let H e H(p)\f*(p), let G = H, and let В denote the base group of G. Then
G e «pH(p) £ fj. If £ e ProjR(H), then by IV, 5.16(a) the subgroup D = ECB(EF(P>)
is an ft-projector of G. By hypothesis Gx < D, and by IV, 1.17(b) the residual Gx
contains Hx and hence its normal closure [B, HX]HX. Therefore
[B, < В n ECB(Ertp>) = (Bn E)CB(EF(pF)
= CB(EFIP>).
Therefore by B, 11.1(e) one of the following 3 cases arises:
(i) Hx = 1, (ii) EF,pi = 1, or (iii) |£f|<”| = p = 2.
If £ ,w = 1, then £ e F(p) and H e f*(p), contrary to the choice of H. In Case (iii) we
conclude that £ e S2F(2) = F(2), contradicting the fact that H £ /*(2). The only
remaining possibility is that Hx = 1 and H e X. Therefore (5.0) holds.
Let H e H(p)\f*(p) (non-empty by hypothesis), and let К be an arbitrary group
in H(p). Then certainly H x Ke H(p)\f*(p), and by (5.0) we have К e q(H x K) £
Qi = X Thus H(p) £ X □
(5.7) Theorem (D Arcy [1 J). Let ft and f> be saturated formations with ft « fj, and let
r and H be their respective canonical local definitions. Let p be a prime in Char(g) for
5. Strong containment for saturated formations
513
which f*(p} # h*(p). Then
(a) H(p) = Q (H(q): q e Char(g)), and
(b) If 'J(9, r) = Q(H: H e H(q) and 0r.(H) = 1), then
H(P) = П (9)(q, r): q e Char(g) and r e Char(F(q))).
Proof. We first observe that H(p) £ f*(p) by (5.5).
(a) By (5.1) we have < £f‘” < £ whenever H e H(p) and q e Char(g), and so
the hypotheses of (5.6) are fulfilled with JE = H(q). Therefore H(p) £ H(q) and State-
ment (a) holds.
(b) Let q e Char(g) and r e Char(£(q)). By IV, 1.10 the class 9)(q, r) is a formation.
Since 9)(p, p) e H(p), by (5.6) it will be enough to show that whenever £ is an
g-projector of a group H e H(p), then < £. Let G = Z/breg H, and let В be
the base group of G. Since r e Char(F(q)), evidently r e Char(g). Hence by Part (a)
we have H(p) £ H(r\ and consequently G e G,H(r) £ By IV, 5.16(a) the subgroup
D = £CB(£FW) is an g-projector of G, and by IV, 1.17(c) we have [B, HWg-rl] < GH,,);
furthermore G",9> < DFig' by (5.1). Since DB/£F,«’[B, £] s (£/£f‘”) x (B/[B, £]) be-
longs to F(q) because r e Char(F(q)), it follows that f/*” < £r<”[B, £]. Hence
[В, < [B, £], and we can conclude from B, 11.1(d) that < £. □
(5.8) Corollary. Let 91 £ g « § as in (5.7). Then for each prime p either h*(p) = f*(p)
or h*(p) = H(p).
Proof Suppose that h*(p) #/*(p)forsome prime p.ThenH(p) £ H(q) for all primes
q by (5.7)(a). Let G e h*(p) and let H e Proj6(G). Then H e H(q) for all primes q and
it follows from IV, 5.21 that G = H e H(p). □
The next result describes the saturated formations that can strongly contain one of
the form g = 91X.
(5.9) Theorem (Cline [1]). Let Xbea formation, and let g = °1X. Let Sjbea saturated
formation with g « § and g # § # S. Then
(i) JE = SP JE and g = epX for some prime p,
and there exists a formation 9) with the following properties:
(ii) 9) = S„.9) and § = Sp9);
(iii) If qe Char(JE), then 9) = q(G e 9): O,.(G) = 1).
Proof. Let F and H denote the canonical local definitions of g and § respectively.
By (5.4)(a) there exists a formation 9) with § = 919), and so we have
F(q) = S,X and H(r) = £,?)
for all primes q and r. If q # r, then F(q)n F(r) = X, and therefore we also have
/*(9) n/*(r) = X by IV, 5.22. If we had /*(р) = Л*(р) for two distinct values of p,
say p = q and p = r, we could conclude that
514
VII. Further theory of formations
9) = h*(q) r, h*(r) = = X
and hence that 8 = S, contrary to hypothesis. Therefore /*(p) = h*(p) for at most
one prime p. But if we had H(q) = Qr6 ₽ H(r) for all primes q, it would follow from
IV 5 20, (b)=>(c), that § = S, and this is also ruled out. Therefore by (5.71(a) there
exists just one prime p with/*(p) = h*(p), and for all primes q # p we have H(q) =
C)r6» H(r), = "D say. Thus £„.?) = 'J and f> = еД) = H(p) by IV, 5.20; in particular,
f*(p) = h*(p) = to, and therefore 5 = F(p) = <=pX by IV, 5.20 again. If q # p, then
S JE = F(q) s 8 = S„X, and consequently ®pX = X Hence we have shown that
Statements (i) and (ii) in the theorem hold, and finally we observe that (iii) follows
easily from (5.7)(b). □
The question of determining the maximal elements of the set SF of saturated forma-
tions partially ordered by « remains open. We end this section with D’Arcy’s
description of those maximal elements 8 of (^,«) which have the special form
8 = 'JtX for some formation X. Most of the work is contained in the following two
preparatory results.
(5.10) Lemma (D’Arcy [1]). Let Xbea formation, and let ЪЬеа formation of abelian
p-groups such that SBX = SPX. Then X = GpX.
Proof. We suppose that X # <2>PX and derive a contradiction. Let G e SpX\X, and
set H = G. Since H e GPX = SBX, the residual H1 is abelian. Denote the base
group of H by B; since BGl = BH1 by IV, 1.17(a), we have
GX[B, G1] < Hl < BG\
and therefore, since H* is abelian,
[B, G*J < Hs n В < CB(GX).
It follows from В, 11.1(e) that |G*| = p, and since H e GPX\X, the same argument
then yields that |HX| = p. Therefore G* = and so
[B,Gi] = [B,HI]<BnHI = BnGi = 1.
Hence we conclude that G1 = 1 and obtain the contradiction that Ge X. Therefore
X = topX. |-|
In the next result 8-normalizers are used to show that a certain formation 8 is not
maximal in (&,«).
‘Д' L? Proposition (D Arcy [1]). Let p be a prime, and let X be a formation satisfying
Set 5 = = ^pX, let SB # (1) be a formation of abelian p-groups.
5. Strong containment for saturated formations 515
= (G: the ft-normalizers of G are in SBX).
Then the class = ©р9) is a saturated formation satisfying g « SS and g / / g.
Proof. It is straightforward to verify that the class ф is a formation. Let G be a group
with a minimal normal p'-subgroup N such that G/N e 9), and further let D be an
g-normalizer of G. Since DN/N e SBX, and since D e g = gpX by V, 3.2(g), it follows
that D e SBX and therefore that Ge'l). Consequently ф = and so is a
saturated formation by IV, 1.9; in fact, = S)i9) and the canonical local definition H
of § is given by H(r) = S,9) for all primes r.
Since X e 9), we have g e §. If g = §, it follows easily that X = 'J and hence that
X = SBX because X g. As SB A (1), we conclude that X = <SPX which together with
the assumption X = 3pX implies that X = S, contrary to hypothesis. Therefore
g A 6. On the other hand, if § = S, then obviously '!) = (S, and therefore g = SBX
by definition of 9). In this case we conclude from (5.10) that again X = SpXand hence
that X = S, against hypothesis. Therefore § # S.
Next we note that H(p) = and H(q) = 'J for all primes q # p, and similarly that
the canonical local definition F of g is given by F(p) = g and F(q) = X for all primes
q # p. Thus, in order to prove that g « §, by (5.1) for each H e and E e ProjB(H)
it will suffice to show that Hv < Ex. This we do by induction on |G|. If H e g, then
since X £ '!). Therefore we can suppose that H ф g, in which case H
has an g-critical maximal subgroup U (see V, 3.5 and V, 3.6(b)), and by V, 3.7 the
g-normalizers of V are g-normalizers of H. Therefore, if F e ProjB(G), by V,4.1 there
exists a subgroup D, simultaneously an g-normalizer of both U and H, contained in
F. Note that U e § by IV, 1.14, and so by induction Uv < Fx. Since U is g-critical
in H. we have H = NU for some normal q-subgroup N of H. First suppose that q # p.
Since § = Gp'J, the residuals and W® are p-groups, and because NUV = NHV
by IV, 1.17(a), it follows that G® = Hv, for H® is the unique Sylow p-subgroup of
NHV. By IV, 5.16(a) the subgroup E = FCN(FFtq>) is an g-projector of H. By the
argument just used, we can also deduce that E* = Fx and can therefore conclude
that Ex > Uv = Hv, as desired. Finally, suppose that N is a p-group, so that E =
NF e ProjB(H) in this case. By definition of 9) we have G® = and H® =
<(nsi)No> and therefore HV = <(U®f> = G®J < F*[N, F*] < (NF)X by
IV, 1.17(b), which again yields the desired conclusion that H® < Ex. □
We are now ready to state and prove the promised description of maximal elements
of (^, «) of the form 91X.
(5.12) Theorem (D’Arcy [1]). Let X be a formation properly contained in g, and let
g = 91X. Then g is maximal with respect to strong containment if and only if g S„g
for all primes p.
Proof. First suppose that there exists a prime p such that g = S„g and let g =
LF(F) with F canonical. Then 3„X = F(p) = g by IV, 5.20, and therefore F(q) =
F(q) n g = G,X n GPX = X for all primes q # p. Consequently X = to,.X, and by
(5.11) the saturated formation g is not maximal in (SP, «).
516
VII. Further theory of formations
Conversely, if 5 is not maximal with respect to the partial order «, it follows from
(5.9)(i) that g = 5,5 for some p. □
Concluding Remarks, (a) By IV, 5.20 and (5.12) a saturated formation^ satisfying
the hypotheses of (5.12) fails to be maximal in (S. «) if and only if « ft for a
prime p, and so the non-maximal elements of this form are also not minimal. In
contrast, the class 9i'(r > 1) is both maximal and minimal in (.3^, «), maximal by
(5.12) and minimal even in the Schunck class lattice (JT, «) by Hawkes [9].
(b) It is not difficult to see that S is not a sublattice of (.#, «). For example, let
H denote the non-abelian metacyclic group of order 12, which has a cyclic normal
subgroup N of order 6. If V denotes the faithful, 1-dimensional F7N-module, then
у = uH is simple and faithful for H, and the semidirect product [l']H is a boundary
group for the Schunck class join 91 v U. Thus 91 v U # S. By (5.12), however, 91 is
maximal in (.^, «), and so S is the supremum of 91 and U in this poset. It is still
possible that the partial order « induces a lattice structure on S'. This seems unlikely,
but we know of no examples to rule out the possibility.
Exercises
In the following exercises ft and § denote saturated formations with canonical local
definitions F and H respectively.
1. If g « Sy and F(p) c F(q) for p,qe Char(g), deduce that H(p) £ H(q).
2. (D’Arcy [1]). If g « § and f*(p) = h*(p) for some p e Char(g), and if epg c §,
show that ft = F(p) and § = H(p).
3. Suppose that F(p) # ft for all primes p. Show that g « § if and only if the
formation § | g is saturated (cf. (4.7)).
6. Extreme classes
All groups considered in this section are soluble.
In order to describe the theme of this section we need some definitions, and these
make sense in the universe of all finite groups.
(6.1) Definitions, (a) For a class of groups JE we can define a unary closure operation
Si by
Si?) = (s(G)r,X: Ge?)).
Then, in keeping with II, 1.19(b), we can associate with any class ?) its ^-interior
?)S1 = (G: Si(G) <= ?)).
We note that ?)s« is evidently an s-closed class.
(b) A class ?) is called X-complete if ?)s« £ ?), in other words, if ?) contains those
groups all of whose 3E-subgroups belong to ?).
6. Extreme classes
517
(c) A group G is called s-critical for 9) if G is not in 9) but all proper subgroups
of G are in 9) Thus, in symbols, we define
Crits(9)) = (G ф 9): (s - 1 )(G) £ 9)).
In this section we shall be mainly concerned with the following two questions:
(1) How are the closure properties of 9)s* related to those of X and 9)?
(2) For which pairs of classes X and 9) can we conclude that 9) is X-complete?
In attempting to answer these questions, we shall restrict ourselves to the situation
where 9) is a saturated formation and X is an extreme class (in the sense of Definition
6.6 below). A more comprehensive treatment of this problem area is to be found in
Carter, Fischer, Hawkes [1]. At the end of the section we apply the results to a
description of the groups that are s-critical for certain saturated formations.
The connection between the critical groups for a class and its .^-completeness is
made clear by the following lemma.
(6.2) Lemma. Let X and 9) be classes of groups. Then 9) is X-complete if and only if
Crits('S) £ X.
Proof. First suppose that 9) is X-complete, and let G be an s-critical group for 9). If
G ф X, then s(G) n X £ (s — 1 )(G) £ 9), and so G e 9)s* £ 9), against the choice of G
as s-critical for 9). Thus G e X.
Conversely, suppose that Crits(9)) £ X. If 9) is not X-complete, we can find a
group G, of minimal order, in 9)s,\9). Clearly G ф X, and if H < G, then s(H) n X £
s(G) r .1 £ 'll, whence H e 9)4 Because of the minimal choice of G, it follows that
He'D and hence that G e Cri t s(9)). But then G e X by supposition, and consequently
G e 9) by definition of 9)4 This contradiction proves that 9)s‘ £ ?), namely that 9)
is X-complete. □
(6.3) Lemma. If X = <Q, еф>Х and ?) = Q9), then 9)s’ is Q-closed.
Proof. Let N < G e 9)4 If X/N is an X-subgroup of G/N, by A, 9.2(c) there exists an
X-subgroup X* of X with X = NX*. Since G e 9)4 we have X* e 9) and therefore
X/N = X*/(X* n N) e Q9) = 9). Hence G/N e 9)4 □
(6.4) Lemma. Let X = QX, and let 9) be a class of groups satisfying Ro9) r X £'!).
Then 9)S1 is R„-closed.
Proof. Let N2 and N2 be normal subgroups of G such that Nt N2 = 1 and G/N; 6
9)s* for i = 1, 2. It will suffice to show that G e 9)4 and to this end let X be an
X-subgroup of G. Then XNJN, s X/(X n IV,) e qX = X, and therefore X/(X n A,) e 9).
Consequently X s X/(X n N2) n (X n N2) e r0 ?) n X £ ?). and thus G e 9)4 □
(6.5) Corollary. IfX = <Q, ЕФ>Х and 9) is a formation, then 9_)S1 is a formation.
Proof. Lemma 6.3 ensures that 9)s‘ is Q-closed, and since r09)nX = 9)nX £ 9),
Lemma 6.4 shows that 9)Sl is R„-closed. □
518
VII. Further theory of formations
If 3E and ?) are saturated formations, the class 'J)s' is a formation by (6.5); but even
then it need not be saturated (see Exercise 2 below).
We now turn our attention to a special type of (Q, Еф ^-closed class X which will
be useful in establishing X-completeness.
(6.6) Definition. A class X is called extreme if
El: X = <Q, e„>X, and
E2: if a group G has a unique minimal normal subgroup N with G/N e X, then
GeX.
Obviously Conditions El and E2 are together equivalent to El and the following
condition:
E2’: If G e fp and G/Soc(G) e X, then Gel.
Before we give examples of extreme classes, we prove:
(6.7) Proposition Let X = <Q, еф>Х. Then any two of the following conditions are
equivalent:
(a) X is extreme;
(b) X contains all groups which possess an X-projector;
(с) X contains all groups which possess an X-covering subgroup.
Proof, (a) => (b): Suppose that the implication is false, let X be an extreme class, and
let G have minimal order among the groups in 3\X which have X-projectors.
Let X e Projj(G), and let N be a minimal normal subgroup of G. Since XN/N
is obviously an X-projector of G/N, we have G/N e X by choice of G, and therefore
G = NX. If CoreG(X) # 1, we can choose N < X, whence G = X e X. On the other
hand, if CoreG(Jf) = 1, then G is primitive and N is the unique minimal normal
subgroup of G; consequently G e X because X is extreme, and either way we have a
contradiction. Therefore Statement (b) follows.
(b)=>(c): By Definition III, 3.5(b) X-covering subgroups are X-projectors.
(c) =>(a): Assume that X contains all groups which have X-projectors. Let G e fp
with G/Soc(G) e X, and let X be a complement to Soc(G) in G. If G ф X, then evidently
X e Proj j(G). and so G e X by assumption. This contradiction shows that GeX.
Therefore Condition E2' holds and X is extreme. □
Next we present a varied selection of examples of extreme classes.
(6.8) Examples, (a) Let Щ = (G: Ф(С) = 1 and G has at least two minimal normal
subgroups) and let IB £
Then the class
X = (G:Q(G)n®= 0)
is an extreme class. For it is immediate from its definition that X is Q-closed, and the
fact that groups in 9B are Ф-free implies that X is E^-closed. Finally, if G is primitive,
an epimorphism from G onto a group with at least 2 minimal normal subgroups must
6. Extreme classes
519
have Soc(G) in its kernel and so, if G/Soc(G) e JE, then Gel. Therefore Conditions
El and E2' hold, and JE is extreme.
(b) The class
3E = (G: G/G' is cyclic)
is extreme. This is the special case of Example (a) with ® as the class of non-cyclic,
elementary abelian groups.
(c) Let r > 2, and let fbr denote the class of groups which can be generated by r
elements. Then <5, is extreme. For (5, is certainly Q-closed and by A, 9.2(a) also
Ew-closed. To see that Condition E2' of (6.6) is fulfilled, suppose that G is primitive
with N = Soc(G), and let X e ®, be a complement to N in G. Let
X = <Xj,...,Xr>
with X! # 1. (If all x, = 1, then G is cyclic.) Let s = |N|, and write N =
{n,(=l), n2, ...,ns}.
Now define
X, = <Xj, x2,..., xrn,>,
and suppose that X,< G for i = 1,2,.... s. Obviously NX, = G. and therefore either
{X(}f=1 is the conjugacy class of complements to N in G or Xt = X; for some
1 < i # j < s. But in the former case we get 1 # Xj e CoreG(X), and in the latter case
X, contains (xrn,)-1xrn,. = nf*n2 # 1, whence X^ . N 1. In either case we have a
contradiction, therefore G = X, for some i. Hence G e ®, and E2' is satisfied by ®r.
(d) The class JE of groups which can be generated by a single conjugacy class of
elements is extreme. Since it is clear that JE = <q, ЕФ)Х, we need only check that
Condition E2' holds. Let G e Як N = Soc(G), and let X e JE be a complement to N
in G. Let X = <АЛ: x e X). If X = 1, then G is cyclic, and so G e JE. If X 1, then X
is not normal in G, and therefore li'fX for some g e G. But then </ic> = G, whence
G e JE. Thus E2' holds, and JE is extreme.
(6.9) Definitions. We recall that the upper nilpotent series {Л(С)}|>0 of a group G is
defined recursively by FO(G) = 1 and F^Gl/E. ,(G) = F(G/Fri(G)) for i> 1. The
smallest integer I such that F,(G) = G is called the Fitting length (or nilpotent length}
of G and is denoted by 1(G).
We now define an associated series of characteristic subgroups {Ф,(С)},>1 by
Ф.(б)/Л ,(G) = O(G/F, JG))
for i > 1.
Since the intersection of a collection of extreme classes is obviously again extreme,
there exists a smallest extreme class. In order to characterize this class we need the
following lemma.
520
VII. Further theory of formations
(6.10) Lemma. Let G be a group with the property that F(G)/®(G) is a chief factor of
G. Then G possesses at most one complemented minimal normal subgroup, and if N is
such a subgroup, then ®(G/N) = <b,(G)/N.
Proof Let N be a complemented minimal normal subgroup of G. Since N < F(G)
and N Ф(С) = 1, our hypothesis implies that АФ(С) = F(G). From A, 9.2(e)
it follows first that F(G)/N < <D(G//V) and then that <D(G/N)/(F(G)/N) =
<b((G/N)/(F(G)/A)). Using the standard isomorphisms between (G/A)/(F(G)/N) and
G/F1GX we see that <b((G/A)/(F(G)/N)) = (<b2(G)/N)/(F(G)/N) by definition of Ф2(С).
Hence <t>(G/N) = <1>,(G)/N.
Now let M be a second complemented minimal normal subgroup of G. Since
МФ(С) = F(G), as G-module M is isomorphic with F(G)/<I'(G) = F(G/<I'(G)); there-
fore Cc(M) = F(G) by A, 10.6(c). However, MN/N < F(G/N) = F2(G)/N since
F(G)/N < <D(G/N), and therefore F2(G) < Cc(M) by A, 10.6(b). Consequently F(G) =
F,(G), and so G is nilpotent. But in this case G/d'(G) has prime order p, whence G is
a cyclic p-group and certainly has only one minimal normal subgroup. This contra-
diction proves that N is unique. □
(6.11) Theorem. The smallest extreme class Xo has the form
Xo = (G: F,(G)/<I>,(G) is a chief factor of G for i = 1,2,..., 1(G)).
Proof First we show that Xo is extreme. Let N <! G with /V < <1'(G). Then by A, 9.2(e)
and A, 9.3(c) we have
F,(G/N) = F,(G)/N, and
<J>,(G/A) = <t>i(G)/N
for i = 1,2,..., 1(G). Consequently F((G/N)/®,(G/N) is G-isomorphic with
Ff(G)/®((G), and clearly G/N e Xo if and only if G e Xo. Thus Xo = г:фХ0.
To show that Xo is Q-closed, let G e Xo and К <i G. In proving that G/K e Xo we
can suppose by induction that К is a minimal normal subgroup. If К <, Ф(С), the
argument of the previous paragraph shows that G/K e Xo. On the other hand, if К
is complemented, by (6.10) we have Ф(С/К) = Ф2(С)/К; in this case F;(G/K)/®,-(G/K)
is G-isomorphic with Fi+l(G)/<t>i+i(G) for i = 1,..., 1(G) - 1, and again G/K e Xo.
Thus 3E0 fulfils Condition EL
Finally, if G e *p, then Ф(С) = 1 and F(G) = Soc(G) is a minimal normal subgroup.
Hence, if G/Soc(G) e X(1, evidently G e Xo, and Xo also satisfies Condition E2'. The
class X„ is therefore extreme.
Let X be an extreme class. It remains to show that Xo <= X. If not, we can choose
a group G of minimal order in X0\X. If Ф(С) ± 1, then G/®(G) e qX0 = Xo, and so
/Ф(б) e X by the choice of G. But then G e ефХ = X and we have a contradiction.
Ф(С) — 1, then Soc(G) = F(G) is a minimal normal subgroup of G by definition of
Xo, and therefore G is primitive. Since G/F(G) e X by the choice of G, we deduce from
Property E2 for X that G e X. This contradiction proves that Xo s X. □
6. Extreme classes
521
Next we give some more characterizations of Xo
(6.12) Theorem. Any two of the following statements are equivalent:
(a) G e .Eo, the smallest extreme class:
(b) G has exactly 1(G) conjugacy classes cf maximal subgroups:
(c) A given chief series of G contains exactly 1(G) complemented chief factors:
(d) If H e Q(G), then H has at most one complemented minimal normal subgroup.
Proof, (a) => (b): Let G e Io, and consider the following normal series of G
(6.a) 1 < «DJG) < Ft(G) sc < Ф,(С) < l\(G) < < Ф1(С) < F,(C)(G) = G.
By (6.11) each quotient С/Ф,(С) is primitive, and therefore by A, 15.6(c) each of the
complemented chief factors F,(G)/®,(G) of G has a unique conjucacy class of comple-
ments. Since each maximal subgroup complements some chief factor in a given chief
series and since the ТДС)/Ф;(С)(| = 1,2,..., 1(G)) are the only complemented factors
in a chief series refinement of (6.a), Statement (b) clearly holds.
(b) => (c): We begin by recalling A, 9.13, the sharpened form of the Jordan-Holder
theorem, which ensures, in particular, that the number k(G) of complemented chief
factors in a given chief series of G is independent of the series chosen.
Assuming that statement (b) holds, we now proceed by induction on |G| to show
that (c) follows. Let p(X) denote the number of conjugacy classes of maximal sub-
groups in a group X. Clearly p(G/<I>(G)) = p(G) and к(С/Ф(С)) = k(G) by the above
remark. Since l(G/<I>(G)) = 1(G), it follows by induction that p(G) = k(G) = 1(G) if
<b(G) 4 1. Therefore suppose that Ф(С) = 1. In this case F(G) is the product of r(> 1)
complemented minimal normal subgroups by A, 10.6(c). Therefore
1(G) = p(G) > p(G/F(G)) + r > l(G/F(G)) + r = 1(G) — 1 + r,
and consequently r = 1. Hence by induction we have
x(G) = 1 + k(G/F(G)) = 1 + l(G/F(G)) = 1(G),
and Statement (c) holds.
(c)=>(a): This implication follows at once from (6.11) and the fact that for each
i = 1,2,..., 1(G) between Ф,(С) and F;(G) there is always at least one complemented
chief factor.
(a)«>(d): If G g Io and H e q(G), then H e q!0 = and by (611> the factor
F(H)/<t>(H) is a chief factor of H. But in this case H has at most one complemented
minimal normal subgroup by (6.10), and Statement (d) holds.
Suppose conversely that Statement (d) holds, and let H = G/<1,,(G) g q(G). Since
fj(G)/®,(G) is a direct product of r(> 1) complemented minimal normal subgroups
of С/Ф,(С), statement (d) implies that r = 1, namely that F,(G)/®,(G) is a chief factor
of G. By (6.11) we then have G G Io. □
522
VII. Further theory of formations
(6 13) Theorem (Carter, Fischer and Hawkes [1]). Let f be a full local definition of
a saturated formation 8. and let I be an extreme class. If h(p) = f(p)'1 for all primes
p, then 8'* = LF(h).
Proof. By (6.5) the map h is certainly a formation function. Let ft = LF(h). First we
show that:
(6./3) S 8”
Let Ge ft, and let X be an I-subgroup of G. Then XOp. p(G)/Op..p(G) S
X/(XrOp,P(G)) gq3E = I, and since G/Op. p(G) e/(p)s’, it follows that A7(AftOp. p(G)) e
f(p). But X r-, Op.p(G) < OP,P(X), and so X/Opp(X) e of(p) = f (p). As this holds for
all primes p, we conclude that X e 5 and hence that G e 8s’- Thus (6/i) holds.
In order to prove that ft = 8s’, we choose a group G of minimal order in the class
Rs’\ft and derive a contradiction. Since 8s' is Q-closed by (6.5), it follows from II,
2.5(a) that G is primitive. Let N = Soc(G) be a p-group, and let A < A' < G with
X/N e 3E. Certainly X e s8s' = 8s’. and if X < G, we have X e ft by the choice
of G But then X/Op- p(X) e li(p), and since X/Opp(X) e q(X/N) £ I, it follows that
X/0p- p(X)e f(p). However, as the subgroup Op,,P(A ) contains the self-centralizing
p-subgroup N, it is itself a p-group, and therefore X/N e Gp/(p) = f(p). If, on the
other hand, X = G, then G e X because I satisfies Condition E2' of (6.6) by hypo-
thesis, and we again conclude that X/N = G/Op. P(G) e f(p). Thus G/N e flp)31 =
h(p), so N is an (i-central chief factor of G. However, because G/N e ft, all chief factors
of G are h-central; therefore G e ft and we have the desired contradiction. □
In order to show that the property of being an X-complete formation is inherited
from a local definition, we will also need the following result.
(6.14) Lemma. Let X be an extreme class and 9) an X-complete formation. Then Sp^)
is an X-complete formation.
Proof. The class 3B = top'() is certainly a formation by IV, 1.8(a). We have to show
that SB5’ c 2B. Suppose not, and let G be a group of minimal order in 3BS’\2B. Since
!№’ is Q-closed and 3B is a formation, G has a unique minimal normal subgroup N
with G/N e SB, and since G f 2B, this A is a q-group for some prime q # p. Moreover,
the s-closure of 5BS* and the choice of G means that every proper subgroup of G is in
Let X be an X-subgroup of G. We show that X e 'J and conclude that G e ф5’ s
'8 £ SB. which will give the desired contradiction. Let W = XFiG), and first suppose
that
(1) W / G. Then We SB, and since F(G) is a q-group with Cc(F(G)) < F(G), it
follows that 0p(W) = 1 and therefore that W e -2). Consequently, X/(F(G) г, X) S
W/FiG) e q'J) = '(). But X e SB because G e ВД"’, and so X/OP(X) e f). It therefore
follows that X s ХЦ0р1 X) n (F(G) n X)) e R0'J) = 'I), as desired.
We consider next the case where
(2) W = G and N s <!>(G). Let V = XN. If V = G, then X = G by A, 9.2(b);
6. Extreme classes
523
consequently. G e X and therefore G e SB, against the choice of G. Thus V / G Since
f (G/W) = F(G)/N is a g-group, it follows that O„(G/N) = 1 in this case, and hence
G/N e 'll because G/N e 9B. Thus
(E/N)/((E n F(G))/N) V/(V r, F(G)) s VF(G)/F(G) = G/F(G) etf)) = 'I|.
Since V/N is an X-subgroup of G/N e q(2Bs*) = We have V/N e SB and hence
(V/N)/O„(V/N) e 7). Thus V/N = (V/N)/(OP(V/N) n (V n F(G))/N) e Rfl't) = ?), and
therefore X/(X n N) e 7). But X/Op(X) e ф because X e 'IB, and we deduce once more
that A'e Ml = ф.
Finally, suppose that
(3) W = G and N <b(G). Then G is primitive by A, 15.8(a) and N = F(G). Since
XN = G, we have G/N e qX = X, and therefore G e X because X is extreme by
hypothesis. But then G e lr, '2BSl s 'IB, a final contradiction. Therefore 2BSl s SB.
□
We can now prove the main theorem of this section.
(6.15) Theorem (Carter, Fischer, and Hawkes [1]). Let X be an extreme class, let f
be a formation function, and assume that ftp) is X-complete for all primes p. Then the
local formation g = LF(f) is likewise X-complete.
Proof. Since g is also locally defined by g, where g(p) = Sp/(p) for all primes p, by
(6.14) we may suppose without loss of generality that f is full. But then by (6.13) we
have gs* = LF(h) with h(p) = ftp)*1 S ftp) for all primes p, and so by IV, 3.5(a) it
follows that gs> E LF( f) = g, in other words, g is X-complete. □
(6.16) Corollary. Let g be a primitive saturated formation.
(a) If X is an extreme class, then g is X-complete.
(b) Each s-critical group belongs to the smallest extreme class Xo, defined in (6.11).
Proof. In the notation of (3.1) it follows easily from (6.15) by induction on n that all
the formations in are X-complete, because 0 and S in 3,, are clearly so; thus all
formations in are X-complete. Let {g,},*i be formations in 3f with g, £
g, £ ••, and let g = IJ”] g,. If G e gSl, then there exists an integer n such that
G e g*« since G has finitely many X-subgroups, each in some g,. But then G e g? £
g„ £ g. Hence g'> £ g, and g is therefore X-complete.
In view of (6.2) Statement (b) follows at once from (a) with X = Xo. □
Next we describe an example which shows that (6.16) fails for general saturated
formations.
(6.17) Example. We aim to show that the class U of supersoluble groups is not
X0-complete. The quaternion group Q of order 8 has a faithful irreducible module
M of dimension 2 over the field F5. [This may be seen in many ways: for example, M
can be the natural module for SL(2, 5) restricted to Q e Syl,(SL(2, 5)).] It is easy to
524
VII. Further theory of formations
verify that the semidirect product G = fM]Q is s-critical for U: for every maximal
subgroup of G which is not a 2-group is an extension of M by a cyclic group. However,
whereas G has 3 complemented chief factors in a chief series, it has 4 conjugacy classes
of maximal subgroups, namely the class Syl2(G) together with 3 singleton classes of
normal subgroups of index 2. Therefore from (6.12) we can deduce that G £ Io, the
smallest extreme class.
For the remainder of this section we will be concerned with descriptions of
g-critical groups for certain saturated formations g. Before turning to the special
case g = 91r, we first consider what can be said for a general g.
(6.18) Theorem. Let F be the canonical local definition of a saturated formation g,
and let Gbe a group not in g which has all its maximal subgroups in g. Then G has a
normal p-subgroup P satisfying the following properties:
(i) P=G*:
(ii) Р/Ф(Р) is an F-eccentric chief factor of G;
(iii) F(G) = P<b(G) = 0p. p(G), G/<b(G) e b(g) S Ф, Soc(G/®(G)) = F(G)/<D(G), and
Ф(С) < Cc(P);
(iv) Ф(С) = Zr(G), the Q-hypercentre of G;
(v) P has exponent p when p is odd and exponent 2 or 4 when p = 2;
(vi) P' = Ф(Р) = Pn Ф(С), and either P is elementary abelian or P' = Z(P): in
particular, P is special:
(vii) If F(G) < W < G, then W/F(G) e f(p), and if G e 91,+1 and F(p) £ брЭГ*1,
then W/F(G) e 91' *.
Proof. Since E^g = g, we have G/<b(G) f g. Let 1У/Ф(С) be a minimal normal sub-
group of С/Ф(С). Then G has a maximal subgroup U such that UN = G, and therefore
G/N s U/(U r^N)e Qg = g. Consequently С/Ф(С) e 6(g), and so С/Ф(С) is primi-
tive by III, 3.9(a); in particular, Soc(G/®(G)) = F(G)/®(G) is a minimal normal
p-subgroup of G for some prime p. It follows that Op.(G) < Cc(F(G)/®(G)) < F(G),
and since Op(G/F(G)) = 1 by A, 15.6(b), we conclude that F(G) = Op,p(G). Since
F(G)/®(G) is a p-group, the Sylow p-subgroup OP(G) of F(G) is not contained in Ф(С),
and there exists an M < G with MOp(G) = G. As before, G/OP(G) = M/(M r-, Op(G)) e
gS = g. and so the residual Gs is a p-group, which we shall denote by P for Statement
(*)
We show next that
l6T) P/(P r> Ф(О) is an F-eccentric chief factor of G.
Since G g = Eeg, we have P £ Ф(С) and therefore P n Ф(С) < P. Thus Ф(С) <
mJ? ~ F(G) and so P®(G) = F(G). It follows that Р/(РпФ(С)) is G-isomorphic with
( )/Ф(С)— F(G)/<I»(O), which we have shown to be an F-eccentric chief factor of
G. Thus (6.y) holds.
We show next that
,M) d>(G) = Z8(G) < Cc(P).
6. Extreme classes
525
Let E be an g-projector of G. Then £ is a maximal subgroup of G by the hypothesis
that maximal subgroups of G are in g, and so <b(G) < E. Since G g 9tg, it follows
from V, 4.2 that £ is an Й-normalizer of G, and so certainly <!>(G) < Corec(£) =
ZS(G). But F(G)/®(G) is the unique minimal normal subgroup of G/<I>(G) and is
F-eccentric, and so clearly Ф(С) = ZK(G). From IV, 6.10 we can then conclude that
Ф(б) — Cc(P). Thus (6.r5) holds, and Statements (iii) and (iv) are now clear.
Next we deal with the exponent of P. Since P< G, we have Ф(Р) < Ф(С), and
therefore Ф(Р) < Z(P) by (6.Й). ft follows that P has class 2 and hence that [x, y]p =
[x₽, y] g [Ф(Р), PJ = 1. Thus, if p is odd, we deduce from A, 7.3(b) that
(xy)’ = x₽y₽[x, y]^ = X1’)'’.
Consequently the map x->x₽ is a G-homomorphism from P onto U,(P)= {xp: xg P}
with kernel Г2, (P), and we obtain U,(P) s Р/ПДР). But L5,(P) < Ф(Р) = Zr,(G), and
we conclude that the chief factors of G between fij(P) and P are F-central. Hence
G/flJP) g g, which forces the conclusion that P = £2,(P). If p = 2, we note that
(xy)4 = x4y4[x, y]6 = x4y4 and use the same argument to deduce that the exponent
of P divides 4. Statement (v) is therefore justified.
Since an g-projector £ of G contains Ф(С) and satisfies PE = Gs£ = G, we
have P n Ф(С) < Pri £< G by A, 8.4. Therefore P r> Ф(С) = Pn £ by (6.y) and
P' < P n £. On the other hand, by IV, 5.18 the abelian g-residual P/P' of G/P' is
complemented by the g-projector E/P’ and so Pn £ < P'. Hence P r>Ф(С) = P' <
Ф(Р):й G, and again from (6.y) we conclude that P' = Ф(Р) = РпФ(С) and, in
particular, that Statement (ii) holds. If P' = 1, then Ф(Р) = 1 and P is elementary
abelian. If P' ф 1, then P n Ф(С) = P' < Z(P) < P since P has class 2, and therefore
P' = Z(P) by (6.y). This proves Statement (vi).
Finally, let F(G) < W < G. Since by hypothesis W e g, we have W/OP (W) g F(p).
But Op.ffV) centralizes the self-centralizing p-chief factor 7'(G)/<I’(G); hence Op (W) <
F(G) and therefore W/F(G) e <jF(p) = F(p). Moreover, because F(G)/®(G) =
Ор(С/Ф(С)), the second Fitting factor F2(G)/F(G) is a p'-group, and consequent-
ly Op(lV/F(G))nF2(G)/F(G) = 1. If GeW+1 and F(p) S <?p9li-1, it follows that
W/F(G) g r091'-1 = 91' ’. This proves Statement (vii). □
We now turn our attention to critical groups for the classes 9lr, r — 1,2,....
(6.19) Lemma. A group G is s-critical for the saturated formation 9F(r > 0) if and
only if
(i) G e Io, the smallest extreme class,
(ii) G g 91r+1 \9ir, and
(iii) for к = 0, 1,..., r and for all Frattini chief factors H/K of G above Fk(G) the
Fitting length l(AutG(H/K)) is at most r - к - 1.
Proof. First suppose that G is s-critical for 9Г. Then Condition (i) follows from
(6.16), and Condition (ii) follows from (6.18)(i) and the definition of s-critical. Since
G/F(G) is critical for 9Г 1 by (6.18) (vii), by induction on r it is sufficient to prove
Condition (iii) for к = 0. However, Ф(С) = Z№(G) by (6.18)(iv), which implies that
VII. Funher theory of formations
l(Autc(H/K)) < r - 1 for all chief factors H/K of G below <D(G). Since this inequality
holds^trivially for chief factors H/K above F(G) because G/F(G) belongs to 'Ji', which
has a focal definition f satisfying /(p) = 9Г'1 for all primes p. we conclude that
Condition (iii) is also satisfied.
Conversely, suppose that G is a group of Fitting length r + 1 satisfying Conditions
(i), (ii), and (iii). We proceed by induction on r to show that G is critical for 9ir. If
r = 0, then G is nilpotent; in this case Condition (i) implies that G is cyclic, and (iii)
that Ф(С) = 1, whence G has prime order and is certainly critical for the class
91° = (1). Therefore suppose that r>0 and that Conditions (i)—(iii) characterize
s-critical groups for 9Г, 0 < i < r. Then certainly G/F(G) is s-critical for 9T-1. Let M
be a maximal subgroup of G. If M > F(G), then M/F(G) e УГ‘, and so M e 9lr. On
the other hand, if G = MF(G), then M n F(G)o G by A, 8.4, and soAf n F(G) = Ф(С)
by Condition (i) and (6.11). By III, 6.5 the chief factors H/K of G below Ф(С) are chief
factors of M and satisfy Aut„(W/K) s Autc(H/K); therefore by Condition (iii) they
satisfy
(6.e) /(AutM(H/K)) < r - 1.
But ЛТ/Ф(С) s G/F(G), and so the chief factors H/K of M above Ф(О also satisfy
(6.6). It follows once more that M e 91' and hence, since 9lr is s-closed, that G is
s-critical for 9ir. □
In order to formulate our final characterization of groups s-critical for 9T we need
to establish some special notation.
(6.20) Notation and Terminology. Let G be a group in the smallest extreme class Io.
By (6.11) the section Ft(G)/®k(G) is a chief factor of G; we will call the prime divisor
Pr of | РДС)/Ф,(С)| the relevant prime for F^G)/Fj.^G).
Now let
G = Lr+1 > L, >
> Lt > Lo = 1
denote the tower nilpotent series of G, defined inductively by Lf = (Ь(+1)я for i = 0,
1,..., r = 1(G) — 1. Let Pj e Sylpi(F(G)). Since all chief factors of G between P1 and
F(G) are Frattini by the sharpened form of the Jordan-Holder theorem (A, 9.13) we
haveF(G)/P, < Ф(6/Р,), and so G/Pt e r^W = 9T; consequently Lt = G91' < P, and
L, is a p,-group. Since LJL^ = (G/L^' and G/L^ e QI0 = JE0, we can
likewise conclude that LJLi x is a p,-group for i = 1, 2,..., r + 1.
Now write F; = F)(G) for i = 0, 1,..., r + 1, and denote certain sections of G as
follows:
L? = LiFi_l/Fl.
p* = Р.Л-1/Fi-i for p,. G SylP((Ff(G)),
ф* =(®l(G)nL(F1_1)/F,_1.
6. Extreme classes
527
Because L; < Fh L^ < Fl^i and LJL;^ is a prgroup, we have L* < P* Further-
more, Р,*/Ф* s PiF,_1AOi(G)nPiF;._1)F(/O.(G), and so Р^/Ф* is a Pi-chief factor
of G. Since G/®,(G) f 91r+1 the quotient 0* is not contained in Ф*, and therefore
1*Ф* = P*.
(6.21) Theorem (Carter, Fischer, and Hawkes flj). f.ei r > 0. A group G is s-critical
for 91 if and only if, in the notation of (6.20), the following conditions are satisfied'
(i) G e .Eon(9ir ,l\9ir);
(ii) Lf is a special prgroup for i = 1, 2,..., r + I;
(iii) L* n Ф(* = (Lf)' fori= 1, 2,..., r + 1;
(>v) If Qi e Sylp.(L|), then Ф* = CP;((),,,) for i = 1,..., r.
Proof. First suppose that G is s-critical for 9ir. By (6.19) Condition (i) holds and the
quotient G/F(G) is s-critical for 9ir therefore by induction on r it will suffice to
show that Conditions (ii), (iii), and (iv) hold when i = 1, in which case Lf = Lt = G9|r,
F* = P, e Sylpi(F(G)), and Ф* = Ф(С)г> Рр Now by (6.18)(vi) the group is a
special P1-group and L\ = Ф(С1) = L1 r> Ф(С). Thus Condition (ii) holds, and since
Lt < P, (as we pointed out in the discussion in (6.20)), it follows that Condition (iii)
also holds when i = 1. By (6.19)(iii) the 9ir-1-resicjuai /2 of g centralizes all chief
factors of G below Ф*, and so Q2 induces a p2-group of automorphisms on the
Pj-group Ф* stabilizing a normal series. Since L2 = Q2LY and [P,, L2] = Л by
(6.18)(iii), it follows that p2 # p2 and that Q2 acts fixed-point-freely on Р,/Ф?; thus
CP1((>2) Ф*- Then by A, 12.3 the series-stabilizer Q2 now centralizes Ф*, and so
Condition (iv) also holds when i = 1.
Conversely, suppose that G satisfies Conditions (i)-(iv). To prove that G is s-critical
for 9T, by (6.19) it will be enough to show that
(6-0 f(Autc(H/K)) < r - i + 1
for each Frattini chief factor H/K of G/F^JG) for i = 1. ..., r + 1. Since
FfGj/Fi.fiGjP, < ®(G/F,_1(G)P1) and since /(G/F;(G)) = r - i + 1, we have
((G/Fi-JG)/^) = r - i + 1. Thus it follows that all chief factors of G between (G)
and F;(G) which do not belong to the relevant prime for that section satisfy (6.0.
But all Frattini p,-chief factors H/K of G between Fj_l(G) and F((G) are operator-
isomorphic with a chief factor of Ф* and are therefore centralized by Q. uFjfG) >
£>i+1L; = LI+1; for such H/K we therefore have L,+1 < Cc(H/K) and consequently
l(Autc(H/K)) = l(G/Cc(H/K)) < 1(G/L,+1) = r - i.
This proves that (6.0 holds for all required H/K, and so G is s-critical for 91. О
In [1] Carter, Fischer and Hawkes investigate the structure of s-critical groups for
other primitive saturated formations. In particular, they show that the critical groups
for the class l?p( n) for soluble groups of p-length at most n are just the critical groups
for 912" whose relevant primes satisfy pt = p3 = — P2n+i ~ P-
Among the extreme classes ,E = <Q,Ee>I are those which satisfy the following
condition:
528
VII. Further theory of formations
(6.t/) Whenever a group G has a minimal normal subgroup N such that
(a) G/N e X, and
(b) N has a unique conjugacy class of complements in G,
then G e X.
Such classes are called skeletal and have been investigated by Hawkes in [6]. Among
other things he proves that for certain skeletal classes X, among them the class 6,
of groups generated by at most 2 elements, a union of two primitive saturated
formations is X-complete.
Exercises
1. Let g = LFff j and X = LF(x) for formation functions f and л. Set h(p) = /(p)s'“
for all primes p. Show that:
(a) If X is a primitive saturated formation, then л may be chosen so that ft is a
formation function;
(b) If h is a formation function, then gSl = LF(h).
2. Let § denote the formation of all groups whose chief factors are all complemented,
letX = <53.<53&,andlet g = G3.to39I. Show that gSlis not a saturated formation.
[Suppose gs‘ = LF(g). Let V be the natural module for SL(2, 3) over F3. Then
f F]SL(2, 3) e gs', and so Alt(4) e p(3). But the semidirect product with Alt(4) of
a faithful irreducible Alt(4)-module over F3 is not in gs,J
3. Define a formation function h by
li(p) = (G: (|G|, p — 1) = 1).
Then LF(h) = 9is“, the class of groups whose supersoluble subgroups are nil-
potent.
4. (Carter, Fischer, Hawkes [1]). Let X be an extreme class, and let F be the canonical
local definition of an X-complete saturated formation LF(F). Then F(p) is X-
complete for all primes p.
5. Show that U is 62-complete, where 62 is the class described in (6.8)(c).
6. Let § be a Schunck class containing 91, and let X = (G: G = <Proj6(G)>). Show
that X is an extreme class.
7. By showing that a skeletal class (see (6.1/)) contains all cyclic groups, deduce that
the smallest extreme class Xo is not skeletal.
7. Saturated formations with the cover-avoidance property
All groups considered in this section are soluble.
We saw in Chapter V, Section 3 that for a saturated formation g the g-normalizers
of a group either cover or avoid its chief factors. If g = S„ then the g-projectors,
being Hall n-subgroups, also have this cover-avoidance property. Our central objec-
tive in this section is Doerk’s description of saturated formations for which this holds.
7. Saturated formations with the cover-avoidance property 529
It turns outjhat the g-projectors are САР-subgroups in every group if and only if
either g = for some n e p <jr g = Zp.Qp for some prime p.
(7.1) Definition. Let p be a prime, let 8 be a saturated formation, and let ® be a
subclass of G. If for all G e <B the g-projectors of G either cover or avoid each p-chief
factor of G, we say that 8 has the p-cover-avoidance property in ®; and if this happens
for all primes p, we say simply that g has the cover-avoidance property in ®. [We
shall mainly be concerned with the special case ® = G„.]
Remarks. Let F be the canonical local definition of 8- We recall that another local
definition/* is obtained by setting/*(p) = 8 [ F(p) = (G e G: Projs(G) s F(p)) and
that by IV, 5.19(c) the /’-central chief factors are precisely those covered by an
g-projector. Therefore g has the cover-avoidance property in a group G if and only
if an g-projector of G avoids the/’-eccentric chief factors of G; furthermore, it follows
from IV, 5.19(c) and the Jordan-Holder theorem that if an g-projector either covers
or avoids the chief factors of just one chief series, then it does so for all chief factors
of a group.
If g has the cover-avoidance property, we show next that g-projectors are char-
acterized by this property; thus by V, 2.3(c) the g-projectors coincide with the
/’-normalizers in this case.
(7.2) Lemma. Let g be a saturated formation, and let G be a group in which the
t^-projectors have the cover-avoidance property. Let V be a subgroup of G which covers
or avoids the same factors in a given chief series as an ^-projector of G. Then U is an
^-projector of G.
Proof. Let N be the minimal normal subgroup in the given chief series of G. By
induction on the group order we may suppose that UN/N e Projg(G//V) and hence
that UN = FN for some F e Proj8(G). If F covers N, certainly U = UN = FN = F.
If F avoids N, then U = UN/N = FN/N stFefr Therefore by III, 3.23(a) the
subgroup U is contained in some conjugate Fe of F e Proj S(FN). Since F avoids N,
we have U = Fe e Projg(G). О
Since G„-projcctors are Hall л-subgroups, the saturated formations G„ have the
cover-avoidance property in G. We now present another family of examples.
(7.3) Lemma. The saturated formation Gp.G„ has the cover-avoidance property in G.
Proof. Let g = Gp. Gp; then the canonical local definition F of g is given by
~ . (8 if«#p, and
9 t<=P if 9 = P-
Therefore /*(9) = G for all 9 * P. and by IV, 5.19(c) an g-projector of G covers all
the p'-chief factors of G. Consequently if £ e Projg(G), we have Op (£) 6 Hallp.(G),
530
VII. Further theory of formations
and it follows from the g-maximality of E that E = NC(S), where S = O„(E). Thus
Er'r> = S, and it follows at once from IV, 5.16(a) and B, 11.4 that g also has the
p-cover-avoidance property. □
We now characterize p-cover-avoidance in terms of local definitions in favourable
circumstances.
(7.4) Proposition. Let g and § be saturated formations with canonical local definitions
F and H respectively. Let pe Char(g) n Char(§). Then g has the p-cover-avoidance
property in Ft if and only if:
fl.a) If Ge H(p) and E e Projs(G), then EF<P' is a p-complement of a normal
subgroup cf G.
Procf. First we assume that g has the p-cover-avoidance property in 5, and let
G e H(p) be a group for which (7.a) fails to hold. Since for E e Projs(G) we have
E e Sp F(p), it follows that fi'7*'’' is a p'-subgroup which is not a Hall p'-subgroup of
any normal subgroup of G, and so by B, 11.5 there exists a simple FpG-module V
such that 0 # CV(EFW) # V. Let H be the semidirect product [t']G. Since G e H(p),
we have H e f>. But by IV, 5.16(a) the g-projectors of H do not have the cover-
avoidance property, contrary to assumption. Thus we conclude that (7.a) holds.
Now assume that (7.a) holds. Let G e § and let E 6 ProjB(G). We proceed by
induction on |G| to show that E has the p-cover-avoidance property in G. If N < G,
then E either covers or avoids the p-chief factors of G/N by induction. If N is a
p'-group. or if E avoids N, then E has the p-cover-avoidance property in G. Therefore
we can suppose that OP(G) = 1 and that N is a minimal normal p-subgroup of G not
avoided by E. Then by IV, 5.16(a) we have CN(£F<₽)) # 0. Since Ge § g Sp./7(p), we
have G e H(p) and so £FI₽) is a Hall p'-subgroup of some normal subgroup of G by
(7.a). But then Ск(£л₽>) = N by B, 11.5, and £ covers N. Thus £ either covers or
avoids the p-chief factors of G, and so g has the p-cover avoidance property in fj.
□
For arbitrary saturated formations g and we know of no criterion beyond (7.4)
for g to have the cover-avoidance property in §. For two special cases, we have the
following consequences of (7.4).
(7.5) Corollary. Let g0 and F>0 be formations, and let g = 91 g0, § = 91fj0. Then g
has the cover-avoidance property in § if and only if f) £ 91g.
Proof. First we observe that for an arbitrary saturated formation g, an g-projector
of a group G in 9lg is an g-normalizer of G by V, 4.2, and has the cover-avoidance
property by V, 1.5. This proves the sufficiency of the condition.
To prove the necessity, suppose that g has the cover-avoidance property in fj, and
note first that the canonical local definition F of g is given by F(p) = (Spg0 for all
primes p, likewise H(p) = defines the canonical formation function H for Sy
We aim to show that Jj0 £ g. Suppose not, choose a group G of minimal order in
7. Saturated formations with the cover-avoidance property 531
§0\g, and note that G is primitive because g is a saturated formation. Let N =
Soc(G), a p-group say, and let E be a complement to N in G. Since G/N e g by the
choice of G, we have E e Projg(G). Since c Q,ePH(q), we can apply (7.4) to
conclude that E = ES»R° is a ^-complement of a normal subgroup of G for all
primes q. But if q # p and E««> # 1, this can certainly never happen, for N is then a
q'-group contained in every non-trivial normal subgroup of G. Consequently E e F(q)
for all q # p, and so E 6 pF(q) = g0) which implies that G e 91g0 = g, a contra-
diction. Hence f)0 <= g, and § = 91f>0 s 91g, as desired. □
Our next main objective will be the classification of all saturated formations which
have the cover-avoidance property in ©„. Since the g-projectors of a group in S„
are also g r> ©„-projectors, without loss of generality we suppose in this undertaking
that g £ ©„.
(7.6) Lemma. Let n £ P, and let g = LE(E) s ©„, where the local definition F is
canonical. If g has the p-cover-avoidance property and if q e Char(g), then either
F(p) S F(q), or F(q) S E(p).
Proof. If p = q or p ф Char(g), the conclusion obviously holds. Therefore suppose
that qfpe Char(g). We suppose that F(q)\F(p) 0 # F(p)\F(q) and derive a
contradiction; this will prove the lemma.
First, let At be a group of minimal order in F(p)\F(q); then certainly At e Sn, and
Socl-djis a minimal normal q'-subgroup of Л, since F is full. By В, 11.7 there exists
a simple F^z^-module Ut, faithful for At, and we can form the semidirect product
[4i = ^2- Once again there exists a faithful simple A2-module U2, this time over
and we form the semidirect product A = [l/2]A2. Then A 6 S„ r> and since
the subgroup At is not in F(q), it is an g-projector of A2, and consequently the
subgroup E = U2At is an g-projector of A. Note here that E 6 SpF(p) = F(p).
Next let E be a group of minimal order in F(q)\F(p). Then E 6 S„, and Soc(F) is
a minimal normal p'-subgroup of E. Again by B, 11.7 there exists a faithful simple
E-module W over Fp, and the semidirect product В = [IV]F belongs to S„r>43₽.
Since E ф F(p), we conclude that E e Projg(B).
Finally set D = A x B, and note that E x F e Proj g(D), that (E x F)f(,) =
L/2Soc(X,), and also that (E x F)F(p) = Soc(F). Since A and В are non-abelian primi-
tive groups, D has exactly two minimal normal subgroups, namely V2 x 1 and 1 x W.
Since O„(D) = 1, by B, 11.7 we can find a faithful simple D-module V over F,. Let
G = [FJD e ©„ n Since U2 < D, it follows that CV((E x F)fi”) = 1 and hence
that E x F e Projg(G) by IV, 5.16(a). However, (E x F)f,p> = Soc(F) is not a Hall
p'-subgroup of a normal subgroup of G because V is a q-group and the unique
minimal normal subgroup of G. But this contradicts Assertion 7.a of (7.4) applied to
g with — S„ = H(p) for all pen. C
Our next result shows that, apart from some trivial exceptions, the p-cover-avoidance
property in S„ for a single prime p already implies this property for all primes.
(7.7) Theorem (Doerk [1]). Letg = LF(F) £ S„, where F is the canonical local defini-
tion. For p e Char(g) the following conditions are equivalent:
532
VII. Further theory of formations
(a) g has the p-cover-avoidance property in S„;
(b) Either Spg = g, or g = S,4plF(p) and g has the cover-avoidance property
in <S„.
Proof. Since the implication: (b)=>(a) is trivial, we need only show that
(a)=>(b): If Char(g) = {pj, then g = <SP and Condition (b) is certainly satisfied.
Hence assume that |Char(g)| > 1. Then by (7.6) we have F(p) £ F(q) or F(q) £ F(p)
for all q 6 Char(g). We first prove the following assertion:
(7./?)
g = F(p) и F(q) for all q e Char(g)\{p}.
Suppose, by way of contradiction, that (7./?) fails, and choose a group G in
g\(F(p) и F(q)) of minimal order. If G possesses two minimal normal subgroups, Nt
and N2 say, then G/N, e F(p) и F(q)for i= 1,2. If F(p) £ F(q), we obtain G 6 r0F(<j) =
/•'((/), a contradiction; likewise if F(q) £ F(p). Hence G has a unique minimal normal
subgroup, N say. If N is a (/-group, then G e &gF(q) = F(q) £ g, against the choice
of G. Therefore N is a (/-group, and by B, 11.7 we can find a simple F,G-module M,
faithful for G. Let H = [M] G. Then H e S„ r> Ф’, and G is an g-projector of Since
G $ F(p) and M is a p'-group, GF(p} is not a p-complement of a normal subgroup of
H, and we conclude from (7.4) that g fails to have the p-cover-avoidance property.
This contradiction proves (7./J).
If F(q) e F(p) for some prime q e Char(g), then g = F(p) by (7.fl), and then
Spg — g by IV, 5.20. Therefore we can suppose henceforth that
(7-}’)
F(p) <= F(q) for all q 6 Char(g)\{p}.
We suppose now that л\{р} contains a prime r such that F(r) = 0 and derive a
contradiction. Since |Char(g)l > 1, there is a prime q e Char(g)\{p). Let H be a
group of minimal order in F(<?)\F(p) (which is non-empty by (7,y)). Since F(r) = 0
and G e F(q) £ g, the socle of H is not an r-group, and so we can find a faithful simple
module N for H over F,. Let L = [N] Ц 6 <5n r> фг. Then H is an g-projector of L,
and HF,P' is not a Hall p'-subgroup of a normal subgroup of L. Since this contradicts
(7.4), we must have Char(g)u {p} = tt, so that (7.y) holds for all q e л\{р). From
(7.P) we know that g = F(p) и F(q) = F(q) for all q e n\{ p}, and so from IV, 5.20 we
obtain g = F(p); in particular, g has the (/-cover-avoidance property in for
all qen\{p}. Since g by assumption also has the p-cover-avoidance property,
Statement (b) of the theorem is now clear. □
We now come to the promised description of saturated formations which have the
cover-avoidance property in S„.
<38) Theorem (Doerk and Hawkes [1]). Let ‘ft be a saturated formation contained in
for some n £ P. The following statements are equivalent:
(a) g has the cover-avoidance property in <S„;
(b) Either g = Sp for some p c n or g = S„\(p)Sp for some prime p.
7. Saturated formations with the cover-avoidance property 533
Proof, (a) (b): Let F denote the canonical local definition of g, and let p=Char(g).
If = 8 for all pep, it follows easily from IV, 5.20 that g = (Sp. Therefore
suppose that Spg g for some pep. Then we conclude from (7.7) that g =
ап^ from IV, 5.20 that F(p) yt g. We aim to reach a contradiction by
assuming that F(p)\Gp 0. Let A be a group of minimal order in F(p)\Sp and В
a group of minimal order in g\F(p). Then A is primitive with M = Soc(A) a q-group
for some q e it\ {p] and N — Soc(B) is a minimal normal r-subgroup of В for some
r6 7t\{p}.
By В, 11.7 the group В has a faithful simple module V over F , and since N = Br,p>,
it follows from IV, 5.16(a) that В is an g-projector of the semidirect product C =
[F]B. Again, C has a faithful simple module W over F, and since (S,g = g, the
subgroup WB is an g-projector of the semidirect product D = [W]C.
Set H = A x D(eS„) and £ = A x WB e Proj^(H). Since M x 1 and 1 x W have
different centralizers in H, they are non-isomorphic H-modules and are therefore the
only minimal normal subgroups of H. Since OP(H) = 1, we know by B, 11.7 that H
has a faithful simple module X over Fp, and the semidirect product L = [X] H belongs
to S„ r> Since £n₽) = 1 x WN and [X, 1 x W] = X, it follows that E e Projs(L)
by IV, 5.16(a).
The minimal normal subgroup M of A may be viewed as an Fq£-module on which
E has kernel M x WB; call this inflated version M*. Then by B, 11.7 there exists a
simple FeL-module Y, faithful for L, such that YE has a submodule Z such that
Y/Z = M*. Form the semidirect product.
E
G = [Y]L.
Since g = S,g, it follows that YE e ProjB(G). Since Y/Z, as a chief factor of YE,
is isomorphic with (M x WB)Y/WBY, we have YE/WBZ eR0(A) s F(p), and so
(YE)r<p> r.Y<Y Since (YE)F,P> contains EF,pl = WN and WN does not centralize Y
because Y is faithful for L, it follows that
1 * (YF.)r<p,r. Y # Y.
Since Y < G, we conclude that (Y£)rw is not a Hall p'-subgroup of a normal subgroup
of G, and hence from (7.4) that g does not have the p-cover-avoidance property. This
contradiction proves that F(p) £ Sp, and since pe p, we have F(p) =/ 0 and therefore
(b) => (a): Clearly ©p has the cover-avoidance property in S„, and by (7.3) so also
does Sp. Sp n S, = ©„x(p) <=₽ °
We end this section with an application of Theorem 7.8 leading to a characteriza-
tion of the classes ©„ among saturated formations. It rests on the fact that an
©„-projector (viz. a Hall-subgroup) has a unique conjugacy class of complementary
subgroups, namely Hall„.(G).
(7.9) Theorem (Chambers and Makan [1]). Lot g be a saturated formation. Then the
following statements are equivalent:
534
VII. Further theory of formations
(a) For each group G each ^-projector E of G has exactly one conjugacy class of
complements.
(b) g = for some tt £ P.
Proof. Since it is clear that (b)=>(a), we have only to prove that (a) =>(b): We show
first that 8 has the cover-avoidance property. Let G be a group, E e ProjB(G), and
let N be a minimal normal subgroup of G. It will suffice to show that E either covers
or avoids N. But E e Proj B(EN) and since N is abelian, (ETV)® is a complement to E
in EN by IV, 5.18. But any complement to E r> TV in TV is a complement to E in EN,
and so by hypothesis and the fact that (ETV)® < EN, we conclude that (EN)® is the
unique complement to E r, TV in TV. But the only subgroups of an elementary abelian
group N that have a unique complementary subgroup are N and 1, and so E either
covers or avoids TV. Therefore by (7.8) we have g = S„ or Sp- <5p. We rule out Sp. Sp
by showing that Condition (a) fails to hold if g = <3p.Sp. Let G = Zprljrel! Zg. If В
denotes the base group of G, then D = CB(Zq) is the diagonal subgroup of В and
E = D x Zq is an g-projector of G. By Maschke’s theorem (A, 11.4) D has a comple-
ment in В which is normal in G. But D has other complements in B, and all such
complements are complements to E in G. Thus Condition (a) fails to hold for this
group G and therefore g # Sp.Gp. □
Chapter VIII
Injectors and Fitting sets
1. Historical introduction
In an attempt to dualize the theory of projectors—and this is the starting point for
the theory of Fitting classes—it would be natural to replace the concept “quotient
group by normal subgroup” wherever possible; in terms of closure operations this
would mean replacing q by s„. In the same spirit, the natural candidate for the dual
of the operation Ro is N„, and so from this point of view an <s„, N„>-closed class
becomes the dual of a <q, r0 ^-closed class; in other words, a Fitting class may be
regarded as the dual of a formation. However, by pursuing the analogy with projec-
tors and projective classes instead of closure operations, we shall see below that a
Fitting class may equally well be viewed as the dual of a Schunck class. Because of
this ambiguity, we cannot therefore expect the duality to be exact, certainly not to
the extent of having a well-defined procedure for translating each true statement
about projectors and Schunck classes into a true dual statement about injectors and
Fitting classes.
The two fundamental papers in the theory of projectors, that of Gaschiitz [8] in
1963 on saturated formations and that of Schunck [1] on saturated homomorphs
(Schunck classes) four years later, both use the concept of “covering subgroup” rather
than “projector”. In fact, the formal definition of a projector as we now know it did
not appear until 1969 in Gaschiitz’s Canberra notes [10], and perhaps one reason for
its late emergence was the fact that, as we saw in III, 3.21, the two concepts coincide
in the universe of finite soluble groups; this was proved for saturated formations by
Hawkes [3] and for general Schunck classes by Gaschiitz, loc. cit. Thus, when Fischer
came to lay the foundations of the theory of Fitting classes in his Habilitationschrift
[1] in 1966, he naturally chose to dualize the concept of an X-covering subgroup.
The reader will recall that this means an X-subgroup which covers every X-quotient
of each intermediate group; its dual should therefore be an X-subgroup which
contains all the normal X-subgroups of each intermediate group, or, equivalently,
which contains every X-subgroup which is normalizes. Such subgroups are today
known as Fischer X-subgroups (see IX, 3.1 below). For an arbitrary Fitting class g
Fischer was able to show the existence of Fischer g-subgroups in every finite soluble
group. But he was only able to prove their conjugacy under an additional hypothesis
(which represents a slight strengthening of the requirement of s„-closure for the Fitting
class in question). These ‘better-behaved* Fitting classes are the so-called Fischer
classes (see IX, 3.3(a) below). In 1972 Dark [2] gave a cleverly-constructed and
complicated example to show that indeed Fischer g-subgroups need not be conjugate
536
VIII. Injectors and Fitting sets
for an arbitrary Fitting class g in the universe of finite soluble groups, and so some
extra condition like Fischer s is really necessary.
In 1967 Fischer, Gaschiitz, and Hartley [1] published a short and elegant paper,
which was the result of collaboration during the Spring of that year at a Group
Theory Symposium in the University of Warwick. With incisive and economical
arguments they show that the concept of an g-injector, whose defining properties
(see IX, 1.2 below) mirror those of a projector rather than a covering subgroup, is
precisely what is needed to consummate the dualization. Thus they obtain:
A class g of finite soluble groups is a Fitting class if and only if each finite soluble
group G possesses an ^-injector. Furthermore, the %-injectors of G then form a single
conjugacy class.
(As a surprising historical footnote, we were interested to learn from Gaschiitz that
the definition of an g-injector arose independently and was not, as one might easily
suspect, inspired by analogy with the definition of a projector. In fact, the truth is
quite the reverse. For although by 1967 the word “projector” had been adopted, the
concept had not; at that stage “projector” still meant “covering subgroup”. It was only
later, and in imitation of the injector concept, that the definition of a projector
assumed its present, well-established formulation.) When g is the Fitting class of
л-groups, the g-injectors of a finite soluble group, like its g-projectors, turn out to
be the Hall тг-subgroups. However, as we shall see below, this is the only situation
in which the injectors and projectors universally coincide, and so the two theories
are quite independent generalizations of the classical theory of Sylow and Hall
subgroups.
In 1973 Anderson* [1] (see also [2]) observed and exploited the fact that the proofs
of the main results of Fischer, Gaschiitz, and Hartley about injectors do not make
full use of the properties of a Fitting class. He showed that the requirement that a
class be ‘closed under isomorphisms’ is unnecessarily restrictive and that within a
fixed group it can be replaced by invariance under conjugation. This observation
leads to the theory of Fitting sets, which may be viewed as a local theory of Fitting
classes inside the subgroup lattice of a single group. (Various successful attempts
similarly to localize the theory of projectors for saturated formations had already
been carried out by Prentice [1] and [2], Wielandt [6], and Wright [3], [4]). The
injectors for a Fitting set of a group G retain all the important properties which were
established by Fischer, Gaschiitz, and Hartley for the injectors of a Fitting class g,
and these can be read off by specializing to the Fitting set consisting of all g-
subgroups of G.
The approach is not just an empty exercise in generalization. It has the following
distinct advantages:
(a) Theorems about injectors apply to a larger portion of the subgroup lattice.
(Thus, for example, all normally embedded subgroups are injectors in this sense.)
In his Diplomarbeil Michel [I] independenlly explores lhe same idea of a Fitting set, but his investigation
does not cover as much ground as Anderson’s.
2. Injectors and Fitting sets
537
(b) One gets information about injectors in quotient groups, which was not avail-
able in the case of Fitting classes.
(c) None of the proofs is made more difficult by the greater generality. In fact,
because of (a) and (b) one has more scope for the use of induction arguments, with
the result that some proofs are actually shorter; this is true in the above-mentioned
theorem of Fischer, for example.
For these reasons we have decided to expound the theory of injectors initially in
the framework of Fitting sets, and this is the task to which the rest of this chapter is
devoted. The far-reaching developments in the theory of Fitting classes are dealt with
in Chapters IX and X.
2. Injectors and Fitting sets
The main sources for the basic ideas and results of this section are Fischer, Gaschiitz,
and Hartley [1] and Anderson [1] and [2]. The universe up to and including Lemma
2.7 is G and thereafter S.
(2.1) Definition. A non-empty set S' of subgroups of a group G is called a Fitting
set of G if the following three conditions are satisfied:
FS1: If Tsn S 6 S', then T e S';
FS2: If S, T e & and S, T < ST, then ST e S>\
FS3: If S e S' and x e G, then S* e S-.
Clearly a Fitting set of a group G always contains the identity subgroup of G, and
the intersection of an arbitrary collection of Fitting sets of G is again a Fitting set of
G. If U < G and U e S', we call U an -subgroup of G.
(2.2) Examples, (a) Let g be a Fitting class (see II, 2.8) and G a group. Define the
trace of g in G thus:
Tr^(G) = {H < G: H e g).
Since g = s, g, Condition FS1 is fulfilled. The N0-closure of g ensures that FS2 holds,
and the fact that g is a class means that FS3 is also satisfied. Thus Tr^(G) is a Fitting
set of G. and consequently the many examples of Fitting classes described in Chapter
IX below yield a rich variety of Fitting sets.
(b) If N < G, the set of all subnormal subgroups of N is a Fitting set of G; this
follows easily from A, 14.4.
(c) If G is a p-group, it follows from IX, 1.9 below that the only Fitting sets of G
which are Fitting class traces are {1} and {17: U < G}. In view of Example (b), not
every Fitting set of G is the trace of a Fitting class if | G | = p with n > 1.
(d) In the group G = Sym(4) the set consisting of the four ‘point stabilizers’
together with all 3-subgroups of G is a Fitting set; it follows from IX, 1.9 that this is
also not the trace of a Fitting class.
538
VIII. Injectors and Fitting sets
(2.3) Definitions. Let .?’ be a set of subgroups of a group G.
(a) If H < G, define
S„ = {S<H:ScS}.
If is a Fitting set of G, clearly is a Fitting set of H. Where there is no danger of
confusion we shall usually denote simply by Я.
(b) The join Gf of all normal .Я-subgroups of G is called the S'-radical of G. Clearly
G, < G, and if Я' satisfies Condition FS2, then Gj e Я'. Note that by (2.2)(b) each
normal subgroup is the radical of a Fitting set; therefore, unlike a Fitting class radical,
the radical of a Fitting set need not be a characteristic subgroup.
(c) For g e G, we shall use the obvious notation Я'9 = {Xе: X e S}.
(2.4) Proposition. Let S be a Fitting set of a group G.
(a) If N <G, then = (Я\)9 for all g e G.
(b) If N <G, then < G (strictly, we should write here). In particular, if
N < G, then either N e S' or h'f = 1.
(c) G>- is the join of all subnormal S'-subgroups of G.
(d) If N sn G, then Nf =NrtGf.
Proof, (a) Clearly (SK)e = (Se)NC = SN since Se = S by Condition FS3.
(b) If g e G, we have (N = (N")^e = N.r by Part (a).
(c) Let К be a subnormal .^-subgroup of G, and let
К = Ko < Kt < < Kr = G.
By Part (b) we have (K,)^ < Ki+1 for i = 0, ..., r — I; therefore (K,) ^ < (Ki+1)^.
Consequently К = (K0)f < (Kr)f = Gf. Since Gf is a subnormal J5--subgroup of
G, Assertion (c) now follows.
(d) If N sn G, we have N < N nG# by Part (c). Since N n Gy is normal in N and
belongs to S by Condition FS1, we have N c\G? < therefore = N n G:f.
(2.5) Definitions. Let S be a set of subgroups of a group G.
(a) A subgroup V of G is called S-maximal if
(i) V e S', and
(ii) if V < U < G and U e 9C, then U = V.
(b) An S-injector of G is a subgroup V of G with the property that V n К is an
Я-maximal subgroup of К for every subnormal subgroup К of G. We shall denote
the (possibly empty) set of ЭС-injectors of G by Inj ,(G).
The next observation is a direct consequence of the definition.
(2.6) If KsnG and V e Inj f(G), then V r> К is an S -injector (strictly speaking, an
SK-injector) of K.
The following elementary fact is also important.
2. Injectors and Fitting sets
539
(2.7) Lemma. Let & be a Fitting set of a group G, let N < G, and let V e Ini ,(N)
Then Vе 6 Inj for all g e G.
Proof. Evidently Vе is an .^«-injector of № Since = & by (2.4)(a) and N" = N
the result is clear.
The proof of the existence and conjugacy of injectors in soluble groups given by
Fischer, Gaschiitz, and Hartley [1] for the case of Fitting classes carries over without
difficulty to Fitting sets. Their elegant treatment now follows. For the rest of this
section all groups under consideration will be assumed to be not only finite but also
soluble.
(2.8) Lemma (Hartley). Let & be a Fitting set of a group G. Let К be a normal
subgroup of G containing the nilpotent residual G91, let W be an P-maximal subgroup
of K, and let V and Vt be ^-maximal subgroups of G which contain W.
(a) If W < K, then V = (WCjf, where C is a suitable Carter subgroup of G.
(b) In any case V and are conjugate in <И Vt >.
Proof, (a) Clearly W = Kf when W < K, and therefore W < G by (2.4)(b). Let
N/W = Ne/№(V/W). Since N/(N nK)^ NK/K < G/K e 91, we have [IV, N,.'.,Nj<
К for a suitably large value of r, and because V< N, it follows that
[F, N,Z.,N] < EnK.
Now V n К < V e J5-, whence V r> К is an .^-subgroup of К containing W. Therefore
Vn К = W by choice of W, and so V/W < Za(N/W).
Let C*/W be a Carter subgroup of N/W. By 1, 5.9(b) and V, 4.1 we have V/W <
C*/W, and therefore V sn C*. It then follows from the ^-maximality of V that
V = (C*) ^. Let TV* = /Vc(C*). By (2.4)(b) we have F < N* and hence N* <NC(V) =
N. Consequently N* < Nn(C*) = C* because C*/W is self-normalizing in N/W by
definition of C*. Thus N* = C*, and we have shown that C*/W is a Carter subgroup
of G/W. By III, 4.6 there is a Carter subgroup C of G such that C* = WC, and (a) is
proved.
(b) Let G* = <F, F,> and К* = KnG* < G*. Note that G*/K* e 91; that
К* V = К r-, V = Was we showed in Part (a) above; similarly that K* n Vt = W;
and hence that W < <_V, fj> = G*. By Part (a) there exist Carter subgroups C and
C, of G* such that V = (WC)f and Vt = (WCfi^. Now the conjugacy of Carter
subgroups implies that (WCl )x = WC for some x e G*. Therefore Vf is a normal
.^-subgroup of WC by Condition FS3, and consequently Vf < (WC)? = V. But
because Pj is ^-maximal, so is Pjx; hence we conclude that Vf = V. □
(2.9) Theorem (Fischer, Gaschiitz, and Hartley [1]). If & is a Fitting set of a finite
soluble group G, then G possesses exactly one conjugacy class of ^-injectors.
Proof. The theorem is true for groups of order 1. Proceeding by induction on the
group order, we assume that G is a non-identity group all of whose proper subgroups
540
VIII. Injectors and Fitting sets
possess a unique conjugacy class of S -injectors. Let К — G51. Since G is soluble, К
is a proper normal subgroup of G. Let W be one of the ^-injectors of К whose
existence and conjugacy are guaranteed by induction, and let V be an .F-maximal
subgroup of G containing W. We aim to prove that V is an ^-injector of G, and this
will follow if we can show that Vn M is an .'^-injector of M for each maximal normal
subgroup M of G. Let Fo be an .>-injector of M (here we again make use of the
induction hypothesis). Since К < M, we have l'o n К e Inj ,(K) by (2.6), and hence
W = (I'fl n K)‘ = Vg n К for some к e K. Replacing Vo by F0‘ if necessary, we may
therefore suppose that Vo n К = W. Let tj be an .F-maximal subgroup of G contain-
ing Vo. Then by (2.81(b) we have Vf = V for some geG, and consequently
|Z«= F09nM< VfnM = VnMeS.
By (2.7) the subgroup Fo9 is an ,'F-injector of M and is, in particular, .^-maximal in
M. Hence If = IV;M, and therefore VnM is an -injector of M, as desired.
Having dealt with the existence of .^-injectors in G, we now complete the induction
step by proving their conjugacy. Let V* e Injy(G). Then V* n К e Inj >(K), and
therefore (F*)‘ n К = W for some к e K. Since (F*)‘ is .^-maximal in G, by (2.8)(b)
we have V = (F*)‘9 for some geG. □
The next observation is an immediate consequence of the above proof.
(2.10) Corollary. Let S' be a Fitting set of G, let G51 < К < G, and let W be an
S'-injector of K. Then an S’ -maximal subgroup of G containing W is an S’ -injector
of G.
(2.11) Lemma (Dark [2]). Let S’ be a Fitting set of G, and let N be an arbitrary normal
subgroup of G. Assume that N is supplemented in G by a subgroup Le S' and that
LnN is an S' -injector of N. Then L is an S' -injector of G.
Proof. We proceed by induction on |G|, and note that the result is clear if N = G
and, in particular, if G = 1. Thus we may suppose that there exists a maximal normal
subgroup M of G containing N. Since (i) M = LN n M = (L n M)N, (ii) L n M e Sr,
and (iii) (LnM)nN = Lr.N is an ,'F-injector of N, by induction L n M is an
^-injector of M. Let L <, H e SF. Then L n N < H n N e S, and hence L n N =
H nN because L n N is .^-maximal in N. It follows that
H = H n LN = L(H n N) = L(L n N) = L,
and therefore that L is .^-maximal in G. Since G/M e 91, we can now apply (2.10)
with M in place of К and conclude that L is an .^-injector of G. □
The next result provides the key to proving that injectors are persistent in inter-
mediate groups. Its analogue for Schunck classes is also true (see Chapter III, Section
3, Exercise 10(b)) and is used by Gaschiitz in his Canberra notes to prove the
persistence of projectors.
2. Injeciors and Fitting sets
541
(2.12) Proposition. Let & be a Fitting set of a group G, and let
1 =G0<G1 <-<Gn = G
be a series in which each factor GJG,^ is nilpotent. Then a subgroup V is an SF -injector
of G ij and only if V n G, is &-maximal in G, for i = 0,..., n.
Proof The necessity of the condition is part of the definition of an injector. We prove
its sufficiency by induction on n, the length of the series. If n = 0, there is nothing to
prove. If n > 1, the induction hypothesis ensures that V n G„_1 is an .^-injector of
G„_j.Then by (2.10) with G„_] in place of K, we conclude that Vis an .^-injector ofG.
□
Remark. This result shows, incidentally, that in the definition of an injector for a
Fitting set in (2.5) (b) the word “subnormal” can be replaced by “normal”.
(2.13) Theorem (Fischer, Gaschiitz, and Hartley [1]). Let V be a Fitting set of a
group G, and let V be an-injector of G. If V < H <G, then V is an d?-injector of H.
Proof. Let {GJ be a series of G with nilpotent factors as in (2.12) above (for example,
the upper nilpotent series), and let If = H n G;, i = 0,..., n. Then
1 =H0<Ht < - =
and H,+,///,s(HnG,+1)G,/G,es9i = 9i. Since f'rJl, = l'rHnG,= l'7iG„ the
subgroup V r: Hi is 3--maximal in H, for i = 0,..., n. Therefore by the sufficiency of
the condition in (2.12) we have V e Inj >(H). □
(2.14) Proposition. Let be a Fitting set and V an ^-injector of a group G. Let
К <G and N = NG(Vn K). Then:
(a) VdK is a pronormal subgroup of G;
(b) NK = G;
(с) V is a CAP-subgroup of G.
Proof, (a) Write W = VdK. Then W and W9 = Vs d К are .^-injectors of К and
consequently also of H = < W, W9) by (2.13). But then W is conjugate in H to W9 by
(2.9). Hence W pr G, as required,
(b) This follows at once from Part (a) by the Frattini argument.
(c) Let R/S be a chief factor of G, and let N = NG(V d R). From Part (b) we have
NR = G. Since the subgroup (V d R)S is normalized by N and also by R because R/S
is abelian, it is therefore a normal subgroup of G between R and S. Consequently
(V d R)S = R or S, and hence V has the cover-avoidance property. □
We now turn to an investigation of the behaviour of Fitting sets and injectors m
quotient groups. The next result shows that the injectors of a group map to injectors
of an epimorphic image.
542
VIII. Injectors and Fitting sets
(2.15) Proposition. Let be a Fitting set of G, and let N < G.
(a) The set .FG/fl = {SN/N: S is an .F-injector of SN} is a Fitting set of G/N.
(b) If V is an -injector of G, then VN/N is an &G/K-injector of G/N.
Proof, (a) Let K/N < SN/N, where S e Inj ,(SN). Since К < SN, we have
SnKe Inj.x(K). Furthermore, К = SN о К = (Sn K}N, and therefore K/N =
(SoK)N/N e SfG/N. It follows that .FG,K fulfils Condition FS1 of (2.1). Next let
SiN/Ne^n with S, e InjjdS.N) and S,N < (S, N)(S2N) for i=l, 2. Set T =
(StN)(S2N), and let W be an .^-injector of T. Let ie {1, 2} and put R, = WoS,N.
Since S,N < T, we have R, e Inj>(S,N), whence is conjugate to S; by (2.9). Hence,
in particular, R,N = \N, and it follows that
T = SjNS2N = R,NR2N = R,R2N < WN < T.
Consequently T/N = WN/N e .'FGIK, and therefore .'Л./к satisfies Condition FS2.
Since = Л for g e G, it is clear that S9 e Inj ^(S’N) and hence that J^/JV also
satisfies the third and last requirement of a Fitting set in G/N.
(b) Let K/N sn G/N. Then Ко К e Inj> (K')by(2.6), and so VoK e Inj ^(( V n K}N)
by (2.13). Hence the group (Vо K)N/N belongs to ^G/N: we claim that it is actually
.FG;f>.-maximal in K/N. To see this, let SN/N be an ,F(,;f>.-subgroup of K/N containing
(I'r K}N/N, with S e Inj,F(SN). Again by (2.13) the subgroup Vo К is an .F-injector
of SN; it is therefore conjugate to S in SN, and in consequence we have (V о K)N =
SN. This proves that (k'n K)N/N = VN/N о K/N is .F(,;f>.-maximal in K/N. There-
fore VN/N is an .‘/6;f,--injector of G/N. □
The set {SN/N: S e .//} presents itself as a natural candidate for the image of &
in G/N, instead of the above-defined S*CIN. However, the following example shows
that it is not in general a Fitting set of G/N.
(2.16) Example. Let P denote an extraspecial group of order pJ and exponent p
(where p is an odd prime). According to A. 20.12 it has the following presentation:
P = <x, y: x" = y” = 1; [x, у J 6 Z(P)>.
Furthermore, it admits a faithful group of operators A = (a) x <b> of order four,
where
(i) x“ = x1, y“ = y, a1 = 1, and
(ii ) xb = x,yb = y‘,b2 = 1.
Let G denote the semidirect product
G = [Р]Л,
and let .s denote the set consisting of the following subgroups of G:
(a) The conjugates of U = a>;
(b) The conjugates of V = < y, ft);
(c) The subgroups of P.
2. Injectors and Fitting sets
543
It is clear that the set {SP/P: S e SP} = {1, <flp>, <bp>} is not a Fitting set of G/p
We shall show that, nevertheless, ,'F is a Fitting set of G.
Since G = F s Dih(2p), the proper subnormal subgroups of conjugates of U and
V are p-groups and are therefore subgroups of P. Thus & evidently fulfils Condition
FS1 of (2.1). Since it clearly also satisfies FS3, it remains to verify FS2. Let R, S < RS
with R, S e SP. We must show that RS e .P. This obviously holds when R and S are
subgroups of P, and so it will be sufficient to consider the case where R is conjugate
to <x, a>. for the case where R is conjugate to <y, b> can be argued symmetrically.
Furthermore, by conjugating if necessary, we may suppose without loss of generality
that R = <x, a>. Let z = [x, у]. It is easy to verify that NP«x>) = <x> x <z>, that
z“ = z-1, and hence that W,,(R) = <х>Л. Therefore S is an &-subgroup of <х>Л. Now
any conjugate of <x, a> is contained in P<a>. On the other hand, the Sylow p-
subgroup of any conjugate of <b, y> lies in the normal closure of <y> in G, which is
<y, z>. Thus no conjugate of (b, y> lies in <х>Л. It follows that the .Л-subgroups of
<х>Л are all contained in <х>Л n P(a> = <x, a> = R. Therefore S < R and we have
RS = R e ,'F, as desired. Before leaving this example, it is worth mentioning the
obvious fact that the set consisting of all subgroups of P is a Fitting set of G and
that P is the ^-injector of G. It turns out that P is also the injector for the above
Fitting set Sp (see Exercise 8 below) and for several other Fitting sets as well (see
Exercise 9 below).
Next we show that the injectors of an epimorphic image of a group G are the images
of injectors of G.
(2.17) Proposition. Let N <3 G and let SP be a Fitting set of G/N.
(a) The set :F0 = [S < G: SN/N e & and S sn SN} is a Fitting set of G.
(b) If V/N is an SP -injector of G/N, then V is an SP,-injector of G.
Proof, (a) Let К < S e Po. Then KN/N < SN/N e &, and so KN/N e JP. Since
R S3 S sn SN, we have К sn SN, and therefore К sn KN by A, 14.1(a). Therefore
К e SP,}, and Condition FS1 is satisfied by SP,}. Next let R Se :P„ with R, S < RS.
Then (RS}N/N = (RN/N}(SN/N} e SP by Condition FS2. Also R sn RN < (RS)N,
therefore R, and similarly S, are subnormal in (RS)N, and consequently RS sn (RS)N
by A, 14.4. Hence RS e Л. and ’’is now cIear that is a Fittin8 set-
(b) Let KsnG. First we shall show that VrKt.f. Since KN/N sn G/N, it
follows from the definition of V that (V/N} n (KN/N} = (Fn K}N/N is an .^-injector
of KN/N, and therefore we certainly have (F n K)N/N e sp. Since К sn KN, for some
n e M we have
[N, K,..., K} < N c K
by A, 14.8. Then [N, FnK, < N rsK = N rs(VrsK}, and therefore
(V n K) sn (V c\ K) N by A, 14.8 once more. Hence V n К e &0. To complete the proo
we have to show that V n К is ^-maximal in K. Let V c\ К < H < К with H
544
VIII. Injectors and Fitting sets
Then HN/N e and by the ,^-maximality of (F n K)N/N in K/N we have HN =
(VoK)N. Thus H = (Vn K)N о H = (Vо K)(N C\ H). But N n H < N n К <
V n K, and so finally H = V n K. □
From Propositions 2.15 and 2.17, using their notation, we can now deduce the
following:
(2.18) Corollary. Let S be a Fitting set and V an -injector of G. If N S G, then
VN is an -injector of G.
In general there seems to be little connection between and (&G/K)0. For example,
let G = Sym(4), let N be the normal four-group of G, and set
S' = {3-subgroups of G}.
Then (3FGlf,)0 = {subnormal subgroups of Alt(4)}, and so S n(^/JV)0 = {1} in this
case.
We saw in (2.14)(c) that injectors are САР-subgroups. To end the section we touch
briefly on the question of the extent to which they are characterized as subgroups by
the chief factors which they cover and avoid. We begin with an example, which shows,
inter alia, that in general injectors are not so characterized.
(2.19) Example. Let T = Г(23) be the group of semilinear transformations of F8 (see
B. 12.9). We recall that T = [W]BA, where | rij = 23, |B| = 7, |Л| = 3 and В < BA.
Then T is a primitive group with W as its socle. Since С^(А) is the fixed field of the
Galois group A of FB, we have |Си,(Л)| = 2. Hence by A, 12.5 we have | W: [IF, Л]| =
2, and therefore К = [IV, AJA is a normal subgroup of WA of index 2.
Let X be an irreducible F3 IE4-moduIe with Кег(И'Л on A') = K, and let Y = XT.
By B, 9.8(d) the module X has dimension 1, and therefore Y has F3-dimension
|T: IF4| = 7. It follows from B, 7.7 that
(2.«) Y is an irreducible T-module.
Also, because W acts non-trivially on X, it cannot be in the kernel of T on Y;
consequently, since W is the unique minimal normal subgroup of T, we have
(2-0> Y is faithful for T.
Form the semidirect product
G = [У]Т
From (2.a) and (2./?) it follows that G is primitive, and therefore G has the following
unique chief series
I < Y < YW < YWB < G
2. Injectors and Fitting sets
545
with chief factors of order З7, 23, 7, and 3. Next we work out WG(B) Since Flf is a
normal complement to В in YWB, we clearly have NC(B) = Crlr(B)BA- moreover
CYW(B) - Cy(B) because В acts fixed-point-freely on W. Since В is a transversal to
И64 in T, by B, 6.20 we have
Yb )b = (XwArJ3)B — (Xt)B s (F3B)B.
It follows from В, 11.1(a) that |Cr(B)| = 3. Let Cr(B) = Z, = <2> say. Then NC(B) =
ZB4, and ZA is a Sylow 3-subgroup of NG(B) of order |Z| | A| = 9. Hence ZA is
elementary abelian, and it follows that
(2-7) W<,(B) = Z x S,
where S denotes the subgroup AB.
Next, let
S = {H < G: H sn S9, g 6 G}.
We assert that S is a Fitting set of G; since it clearly satisfies Conditions FS1 and
FS3, we have only to check FS2. Let Nit N2 e S with Nt, N2 < NjN2. In showing
that Nj N2 e S\ we may clearly assume without loss of generality that N, 1 N2,
and that Nt sn S and N2 sn S9 for some geG. Since S is a primitive group of order
21, its only non-trivial subnormal subgroups are S and 0,(5). Let R, = CL(Л/,). Then
1 R.char Nj < NjN2fori = 1,2, and therefore R,,R2 < RtR2; in particular, RjR2
is a 7-subgroup of G. Consequently R2 = RjR2 = Rt = O2(Nj) = O7(S) = B, and
В = R2 = O-,(N2) = O2(S9) = O1(S)9 = B9. It follows that g e NG(B) and hence that
S = S9 by (2.y). We therefore conclude that NtN2 sn S by A, 14.4 and that Condition
FS2 is therefore satisfied by S.
Let V be an .^-injector of G. Since V 6 S, we have V sn S9 for some geG, and
therefore V = S9 by .F-maximality. Hence S = Vs 1 is an J^-injector of G. Let
A = (a>, and let S* = B<az>. Then clearly
(2.6) S, S* < SS* = Zx. S.
An argument similar to that just given shows that the set
S* = {H < G: H sn (S*)9, g e G]
is also a Fitting set of G, and that S* 6 Inj^.(G). If S* 6 S, then Z x Se S by (2.6),
and this is clearly not the case. Therefore S* £ S, and in particular, S* is not an
^-injector of G. However, both S and S* cover the two chief factors G/YWB and
УИ'В/У If' of G and avoid the rest. We have therefore found the desired example of
a subgroup S* which covers and avoids the same chief factors of G as the ^-injector
S and which is not itself an S-injector.
Before leaving this example, we make two further observations. First we remark
that the subgroup Z x S cannot be an injector of G because it neither covers nor
546
VIII. Injectors and Fitting sets
avoids the minimal normal subgroup Y of G (see (2.14))(c)). Thus by (2.<5) we have
also established the following fact:
(2.20) The normal product of two injectors is not in general an injector.
Our second observation concerns the structure of primitive groups. It is tempting
to assume that a subgroup U of a primitive group G satisfying U n Soc(G) = 1 must
be contained in one of the stabilizers of G (as is the case when Soc(G) is a Sylow
subgroup). However, the group G described above provides a counterexample. For
let V = (azj. Since У is a faithful simple Т-module, evidently the group G is primi-
tive and its stabilizers are the conjugates of T. Furthermore, Y = Soc(G) and so
U nSoc(G) = 1. Suppose, by way of contradiction, that U is contained in some
conjugate of T. Then <az>" 6 Syl3( T) for some geG, and, writing g = yt with у 6 Y
and 16 T, we see that (azf = a’zeT. It follows that WB = O3(T) is normalized by
the elements a and a’z and hence by a~1 ayz = [a,y]ze Y. But Nr(WB) = Cr(IYB) <
С,(И') = 1, and we conclude that
(2.e) z = [а, у]-1 6 [Y, 4].
Since YB is isomorphic with the regular F3B, from Maschke’s theorem and B, 11.1(a)
we conclude that
YB = Cr(B) Ф [ Y, B] = <z> ® [ У, B],
andsincefY, B] is/-invariant, it follows that У/[У, В] is an (F,/-module of dimension
1 and that it is therefore a trivial module. Consequently [У, A] < [У, B], and hence
[Y, 4] n (z) < [ У, B] n <z> = 0, in contradiction of (2.e). Therefore U is contained
in no stabilizer of G.
To conclude this section, we describe two situations where injectors are indeed
characterized by their cover-avoidance properties.
(2.21) Theorem. Let G be a finite soluble group, and let C be a set of chief factors qf
G. Let t? be a Fitting set and V an ^-injector of G. Let V be a subgroup of G which
covers those members of C that V covers and avoids those that U avoids. Assume further
that either
(a) Vet? and C consists of the chief factors of a fixed chief series, or
(b) V is locally pronormal in G, and C is the set of all chief factors of G.
Then V e Inj^(G).
(In Case (a) this is due to Anderson [1] and in Case (b) to Chambers [1])
Proof. Case (a): We proceed by induction on |G|, noting that the conclusion clearly
holds when G = 1. Let K/l e C. We first show that Ve Inj B(FK). If К < U, this is
certainly true, for then by assumption VK = V e On the other hand, if U avoids
K, we have K f = 1 by (2.4)(b), and hence VnK = 1 e lnj^(K). Since V e SF, it
2. Injectors and Fitting sets
547
follows from (2.11) that I' e Inj.,( VK). Therefore VK/K e Fc/K. If we put
Cc/k = {(R/K)/(S/K): К < S and R/S e C),
then VK/K clearly satisfies the hypotheses of Case (a) for the Fitting set &C.K dG/К
and the chief factors Cc/K. Hence by induction we have VK = UK (after possibly
replacing U by a conjugate). But as we have just shown, V e Inj ,-(FK). Since
U 6 Injjr(FK), V is conjugate to U, and the desired conclusion follows.
Case (b): If U is locally pronormal in G, by (2.14)(c) and I, 7.12 it is normally
embedded, and then from I, 7.5(b) it follows that V is conjugate to U. □
Exercises
1. (Anderson [1]) (a) Let i/'(G) denote the set of all subgroups of G. A function
a: .C/'(G) -» if{G) is called a radical function if
(i) a(H) < H for all H e •'/'(G),
(ii) a(N) = No a(H) for all H e Sf(G) and all N <11, and
(iii) a(Hs) = a(H)? for all g e G.
Prove that there is a bijection between Fitting sets and radical functions of G.
(b) Let a, be the radical function associated with the Fitting set .'if of G, i = 1, 2.
Show that the composition at oa2 is a radical function of G and describe its
associated Fitting set.
2. (Anderson [1]) Prove that a set 3 of subgroups of G is a Fitting set if and only
if Sf is a union of conjugacy classes and every subgroup of G has an .F-injector.
3. (Anderson [1]) Let F- be a Fitting set of a group G,. i = 1, 2. Prove that
& = {S < G, x G2: n,(S) e Fj, i = 1, 2}
is a Fitting set of G, x G2. [Here a, denotes the projection onto the i-th co-
ordinate of the direct product.] Prove also that 3G. = 3^,i = 1,2.
4. Let .F be a Fitting set of G, V < G, and N < G. Show that V is an .'F-injector of
G if and only if V о N is an .F-injector of N and V is an .F-injector of NC(F n N).
Show that this no longer holds if ".F-injector” is replaced by “injector”.
5. Show that the conclusions of (2.17) hold when 30 is replaced by
= {S < G: SN/N e .Fj.
6. Let .F be a Fitting set of G, and let N <! G and N < H < G. Show that (3H)H/N
7.
8.
9.
Let U < Gand К < G. If U pr NC(UK) and UK pr G, we: know from I, 6.4 that
U pr G. Show that this is false when “pr" is replaced by “is an injector of ’.
Let G denote the group in Example 2.16. and let be the Fitting set of G
constructed there. Describe the .F-injectors of G and the Fitting set ,FC/P.
Let G again denote the group in Example 2.16, and define subgroups H, of G as
follows:
548
VIII. Injectors and Fitting sets
Hj = (ab, y> x <z>, and
H2 = (a, z> x <y>.
For i 6 {1, 2} let S; denote the set consisting of the subnormal subgroups of the
G-conjugates of H, and of the subgroup P. Prove that is a Fitting set of G and
that P is an .'^-injector of G, i = 1, 2. Deduce that if Л is any Fitting set of G
containing and S2, then P is not an «^-injector of G. (Thus the set of Fitting
sets giving rise to a fixed injector need not have a unique maximal element.)
10. Let U be an ,'F-injector of G, and let g denote the smallest Fitting class con-
taining (S. Show that U need not be a Tr(g)-injector of G. (Hint: Use Exercise 9.)
11. Let g be a Fitting class, and let U e Injs(G). Show that Tr(g) need not be
maximal among those Fitting sets S of G such that U 6 Inj^ (G).
12. Let H be a fixed injector of a group G, and let S(H) = {S( S is a Fitting set
and H e Inj^(G)}. Then S -»SClf, is not necessarily a surjective mapping from
S(H} to S(HN/N).
13. Let H be an injector of a group G. Call H dominant in G, if H is dominant in the
sense of IX, 4.1 for all S e S(H). Show that if H is dominant in G and /V < G,
then HN/N is not necessarily dominant in G/N.
14. The normalizer of an injector is not necessarily an injector.
15. If H is an injector of HN and if HN/N is an injector of G/N, then H is not
necessarily an injector of G.
3. Normally embedded subgroups are injectors
All groups considered in this section are finite and soluble.
“Which subgroups are injectors?” is the motivating question in this section. Our
main objective will be to prove that normally embedded subgroups are injectors
for subgroup-closed Fitting sets. In Section 4 we shall then show that the converse
is true, namely that injectors for subgroup-closed Fitting sets are normally embedded
subgroups.
(3.1) Definitions, (a) A subgroup H of a group G is called an injector of G if H
is an S -injector of G for some Fitting set S of G. The set of injectors of G will
be denoted by Inj(G).
(b) If H < G, then Fitset(H) will denote the intersection of all Fitting sets of
G that contain H. Clearly Fitset(H) is again a Fitting set of G, and so we call it
the Fitting set generated by H.
(3.2) Proposition, (a) Let H < G. Then H is an injector of G if and only if H is a
Fitset(H)-injector of G.
(b) If H is an injector of G and N < G, then HN/N is an injector of G/N, and
HN is an injector of G.
3. Normally embedded subgroups are injectors 549
Proof, (a) If H e InjFj|S€1(//)(G), clearly H e Inj(G). Now suppose that .f is a Fitting
set of G and that H e Inj,(G). Then H e Fitset(H) £ JE If К sn G, then Hr К is
-maximal in K. Consequently H n К is also Fitset(H)-maximal in K, and it follows
that H is a Fitset(H)-injector of G.
The assertions of Part(b) follow at once from (2.15) and (2.17). □
Remark. Let H be a fixed injector of a group G, and let
& = if(H) = {&: if is a Fitting set and H e Inj,(G)}.
This set is clearly of interest in relation to H and may repay further study. We
know that in general if may contain many different Fitting sets (see Example
2.16) and indeed, when if is partially ordered by inclusion, it may even have more
than one maximal element, a fact which is indicated in Exercise 9 at the end of Section
2. On the other hand. Part (a) of the preceding result obviously implies that if has
a unique minimal element, namely Fitset(H). Next we show that Fitset(H) can be
characterized as the set of subnormal subgroups of conjugates of H ; it will be useful
to have a notation for this set.
Notation. If H is an arbitrary subgroup of a group G, we put
s„HG = {S < G: S sn H9 for some some g e G}.
(3.3) Theorem. Any two of the following statements about a subgroup H of G are
equivalent:
(a) s„HG is a Fitting set of G;
(b) s„HG = Fitset(H);
(с) H is an injector of G.
Proof. It is clear from the definitions that
H e s„HG £ Fitset(H),
and it follows at once that (a) => (b).
Next we show that (b)=>(c): Let SF = Fitset(H), and let PehijHG). Because
V e & = s„HG, we have F sn He for some geG, and since H9 e if, it follows from
the ,^-maximality of V that V = H9. Therefore H = V9 ' e Inj ,(Gj, and Assertion (c)
holds. „
Finally we prove that (c) => (a): Suppose that H e Inj(G), and let JF = . We
shall prove by induction on | G| the statement that sets of the form s. PG for V e Inj(G)
are “closed under normal products’’. It will then follow that is a Fitting set of G,
since it obviously fulfils the requirements FS1 and FS3 of Definition 2.1.
Let K < G. By (2.15) we have HK/K e Inj(G/K), and therefore by induction the
set
Jf* = {S/К < G/K: S/K sn (HK/K)9 for some g e G/K]
550
VIII. Injectors and Fitting sets
is closed under normal products. Let Nt,N2eJf with N,, N2 < N1N2; we must show
that .F contains N = N, N2. It is easily verified that NtK/K e JF* for i = 1, 2. Hence
NK/K = (N, K/K)(N2 K/K) e Jf*, and in consequence we have NK/K sn HXK/K for
some x e G. By (2.14)(c) there are two cases to consider.
Cas e 1: К < H. Let i = 1,2, and recall that N,- sn H*' for some e G. Since К < H"1
in this case, we have
Ni sn KjK < NK sn HXK = Hx.
Therefore by A, 14.4 we have N = NtN2 sn Hx, and hence NeF
Cas e 2: H n К = 1. Let .F = Fitset(H), and note that F F. By (3.2)(a) we know
that H is an .F-injector of G, and so in this case the subgroup 1 is an .F-injector of
K; in particular, 1 is the only .F-subgroup of K. Since N, e £ ,F for i = 1, 2, we
have NeF, and thus N n Ke .F It follows that N n К = 1, and therefore that
N 6 lnj#(NK) by (2.11). For the same reasons the subgroup Hx is an .F-injector of
HXK, and so it meets the subnormal subgroup NK of HXK in an .F-injector of NK.
Hence N = (NK n Hx)y for some у e NK, and consequently N = NK n Hxy sn Hxy.
Therefore we again have N e УС. □
If one is merely interested in injectors and their properties, the preceding theorem
suggests that it is enough to consider just the Fitting sets of the form snHG and that
the study of general Fitting sets is superfluous. However, this is wrong for several
reasons. First, one is interested in injectors for Fitting classes, and traces of Fitting
classes need not have the stated form. Second, there are important properties which
an injector H may have and which may only be revealed by some defining Fitting
set other than s„HG, for example the property of normal embedding (see Theorem 4.6
below).
We know no general characterization of injectors without explicit use of the
concept of a Fitting set. The following theorem gives a sufficient condition for a
subgroup H of a group G to be an injector of G in a very special situation.
(3.4) Theorem. Let S and T be normal subgroups of a group G such that T <S and
S/T < Soc(G/T). Let T < H $ G and HS = G. Then H is an injector of G.
Proof. We proceed by induction on |G|, appealing frequently to the equivalence of
Statements (a) and (c) of (3.3). By that result it will be sufficient to show that the set
s„HG is N0-closed. Let g, 6 G, and let N, sn He‘ with N, < Nt N2 for i = 1,2. Then we
must show that the subgroup N = N2N2 is contained subnormally in some conjugate
of H. Since HS = G, we may suppose without loss of generality that gt. g2 e S. Let
= C°reG(H). It is straightforward to verify that the subgroup H/K satisfies the
hypotheses of the theorem in G/K, so that if К / 1, we can conclude by induction
that H/K e Inj(G/K). In this case we have H e InjfGjby (2.17), and then N is certainly
contained in some conjugate of H by the implication: (c) =» (a) of (3.3).
Therefore we can suppose that CoreG(ff) = 1 and, in particular, that T = 1 and
HoS= 1. Now let M be an arbitrary maximal normal subgroup of G containing S.
3, Normally embedded subgroups are injectors
551
Since S < Soc(G), it follows from A, 4.13(c) that S < Soc(M); furthermore we have
(H* n M)S - HXS r,M - M for all x 6 G. Therefore by induction we have
(3-«) Hr>Me Inj(M).
Because N.S sn He,S — G for i = 1,2, it follows from A, 14.4 that NS = (NtS)(N2S)
is subnormal in G. If NS / G, we can choose the maximal normal subgroup M
so that it contains NS. In this case we have N( sn № n M and 9, e S < M for
i = 1, 2; then from (3.a) and the implication: (c)=>(a) of (3.3) we conclude that
N sn (H n M)m < for some m e M, which yields the desired result. Hence we can
suppose that NS = G.
Next we show that we can suppose that H is self-normalizing. Since H complements
the abelian normal subgroup S of G, we have NG(H) = H x CS(H). Since by hypo-
thesis S, viewed as an H-module, is semisimple, we can find an H-invariant comple-
ment S* to CS(H) in S; furthermore we can clearly suppose without loss of generality
that gt e S* for i = 1, 2. If CS(H) / 1, then HS* is a proper subgroup of G for which
the hypotheses of the theorem clearly hold. Therefore by induction H e Inj(HS*), and
by (3.3) we have N sn Hx for some x e HS*, as desired. Thus we can suppose that
CS(H) = 1 and hence that NC(H) = H.
Let Я = N}r> N2, and let {1, 2} = {i, j}. If N, = then Nj < NC(N;) = Nc(Hf‘ =
H“, and therefore N = H9‘, which gives the desired conclusion. Hence we may
suppose that N,S is a proper subnormal subgroup of HB,S = G. Choose M so that
NjS < M <3 - G. Let L, = Mr. Nj. Since NS = NjNjS = G by supposition, it follows
that Lj <i-Nj sn H9‘. Furthermore we have N;, Lj <! NfL,, and so by (3.a) and (3.3) we
conclude that
NtLj sn Hm n M <
for some m e M. Thus without loss of generality we can replace N, by NjLj in the
argument. The same reasoning with i and j interchanged therefore allows us to
suppose that Я is a maximal normal subgroup of both N, and N2. Thus we have
HS < N, N2 S = G and Я = HSi r>RS< Hs‘ for i = 1,2.
At this stage it is convenient to abandon the symmetry of our notation, henceforth
supposing that g2 = 1 and writing g for g2. Let So = CS(R). Since Я < H n H", by
A, 7.9 we have g e So. Furthermore, since R<H, the subgroup So is H-invariant and
H-semisimplc, and therefore So < Soc(SGH). If So < S, the induction hypothesis
applies to the group S0H and the desired conclusion follows at once by (3.3). On the
other hand, if So = S, we have R < CH(S) < CoreG(H) = 1, and in this case N2 and
N2 both have prime order. It then follows that N = Nj x N2 is an abelian supplement
to S in G. Consequently the subgroup H (= G/S s NS/S) is a self-normalizing
nilpotent subgroup of G and is therefore an 'Ji-projector of G by III. 4.6. Since S is
abelian, it follows from IV, 5.18 that N is also an 91-projector of G, and therefore N
is conjugate to H in G. □
If H is a maximal subgroup of G and if we put T = CorcG(H) and S/T = Soc(G/T),
it is clear that the hypotheses of the preceding theorem are satisfied. Consequently
we have the following.
552
VIII. Injectors and Fitting sets
(3.5) Corollary. Maximal subgroups of a group are injectors.
Our next goal is to show that normally embedded subgroups are injectors. As
above, closure operations will be applied to sets of subgroups as well as to classes of
groups. Thus
sHG = {S < G: S < H” for some g e G[.
(3.6) Theorem (Anderson [2]). If one of the following statements about the subgroup
H of a group G is true, then they all are.
(a) sHG is a Fitting set of G;
(b) For each Sylow subgroup P of H, sPG is a Fitting set of G ;
(c) Each Sylow subgroup of H is an injector of G;
(d) H is normally embedded in G.
Proof. If Pe Sylp(H), the set sP° is the intersection of sHG with the Fitting set
consisting of all p-subgroups of G. Therefore (a) implies (b). Since sPG = s„PG, it
follows from (3.3) that (b) implies (c). If Statement (c) holds, it follows from (2.14) (a)
and (c) that a Sylow subgroup of H is a pronormal CAP subgroup of G. Therefore
by the implication: (g) => (a) of I, 7.12 Statement (d) is true.
To complete the proof we now show that (d) implies (a). Suppose then that H ne G
and put S' = sHG. Of the three defining conditions of a Fitting set we clearly have
only to verify FS2. Let R, S e S', R, S < RS, let Г be a Hall system of G reducing
into RS, and let He be the conjugate of H into which £ reduces. Let p e P, and let Gp
be the Sylow p-subgroup of G in S. Next, choose a Sylow p-subgroup P of H such
that P° = He n Gp, and put Sp = sPG. Since P e Sylp«PG>) by supposition, JF con-
sists of all p-subgroups of <PG> and is therefore a Fitting set of G. Clearly P9 is
.^-maximal and subnormal in Gp; it is therefore the J^-radical of Gp (and also the
unique .^-injector of Gp). Now by definition of J» and Sylow’s Theorem, the group
R n Gp is a p-subgroup of a conjugate of H and so it belongs to S-p; it is therefore
contained in the ^-radical of Gp and hence also in H". Similarly S n Gp < He. By
choice of £ we have RS n Gp e Sylp(RS), and from A, 6.4(b) it follows that RS r,Gp =
(R n Gp)(S n Gp). Therefore He contains a Sylow p-subgroup of RS for each prime p,
and consequently RS e S'. □
The following two consequences of this theorem are also due to Anderson [2].
(3.7) Corollary. Let H be a normally embedded subgroup of G, and let S' denote the
Fitting set sHG. Let L be a subgroup of G, let E be a Hall system of G reducing into
L^and let He be the conjugate of H into which E reduces. Then He oLisan S'-injector
Proof. It follows at once from 1, 7.4(b) that He n L is .^-maximal in L. Since by 1,
4.21 the Hall system £ reduces into any subnormal subgroup К of L, the same
reasoning shows that He л К is .‘^-maximal in K. Therefore H" L has the defining
property of an .^-injector of L. n
3. Normally embedded subgroups are injectors 553 -
If we now take L = G in this Corollary, we attain the promised objective. j
(3.8) Theorem. Each normally embedded subgroup of a finite soluble group is an |
injector for some subgroup-closed Fitting set. I
By I, 7.8 the set of normally embedded subgroups into which a given Hall system f
reduces forms a lattice of permutable subgroups. Unfortunately this does not extend ,
to the set of injectors: In Exercise 5 below we suggest an example of a group with
two injectors, which have a common Hall system reducing into them, and whose
intersection is not an injector. Furthermore, as we pointed out in (2.20), the join of
two permutable injectors need not be an injector. However, a sufficient condition for
a permuting product of two injectors to be an injector is given in Exercise 1 below.
Open question. In a given group we have the proper inclusions:
{Pronormal subgroups} z> {Injectors} о {Normally embedded subgroups}.
Can the set of injectors be described without recourse to the concept of a Fitting set? ?
One possible characterization that springs to mind is the following: By (2.6) and j
(2.14)(a) and (c), an injector H of G has the property that H n К is pronormal in G j
and is a CAP subgroup of К for every normal subgroup К of G. However, the example
described in Exercise 2 below shows that injectors are not characterized by this
property.
Exercises
1. (Anderson [1]) Let Hand К be injectorsofa group G such that (i)HK = KH and
(ii) (|HI, IK|) = 1. Show that HK is an injector of G.
2. Let S = SL(2, 3), and let V be the natural module for S. Let G be the semidirect
product [F]S, and let C be a Carter subgroup of G. Prove that for all К < G
(i) С г, К pr G,
(ii) С n К is a CAP subgroup of K, and
(iii) C is not an injector of G.
3. (Chambers [2]) Prove that the injectors of a group which has p-length 1 for all
primes p are precisely the normally embedded subgroups.
4. (Anderson [2]) Prove that an injector which is nilpotent is a normally embedded
subgroup.
5 Let E be a Hall system of G = Sym(4), and let X (sSym(3)) and Y e Syl2(G) be
subgroups of G into which E reduces. Show that X and Y are injectors of G, but
X n Y is not an injector of G. . .
6. Let g be a Fitting class, let G be a soluble group, and let V be an injector for a
Fitting set of G. Prove that FB is an injector of G.
7. If H is a normally embedded subgroup of a group G, then s„H is not necessarily
s-closed.
1
554
VIII. Injectors and Fitting sets
4. Fischer sets and Fischer subgroups
All groups considered in this section are finite and soluble.
(4.1) Definition. Let be a set of subgroups of a group G. A Fischer 3-subgroup
of G is a subgroup E of G such that
(a) E e 3, and
(b) if L is an ^-subgroup of G normalized by E, then L < E.
(4.2) Remark. If 3 is a Fitting set of G, an .^-injector of G is also a Fischer
:F-subgroup of G.
Proof. Let E e Inj,^(G). Then certainly E e 3. If L e 3 and E < NC(L), then L <
(EL)?. Since E e Inj ?(EL), it follows that L < E. □
(4.3) Definition. A Fischer set of G is a Fitting set 3 of G which has the following
property:
FS4: If К < L e 3 and if H/K is a nilpotent subgroup of L/K, then HcS.
Clearly each subgroup-closed Fitting set is a Fischer set. On the other hand, the
Fitting set of Sym(4) described in (2.2)(d) is evidently not a Fischer set. Fischer
sets may be obtained by taking traces of so-called Fischer classes. These are studied
in Section 3 of Chapter IX, where a variety of examples is given.
Our aim is to show that the Fischer 3 -subgroups of a group coincide with its
3-injectors when 3 is a Fischer set. This was first proved for Fischer classes by
Fischer himself [1]. Later, Hartley [1] gave a simpler proof, which was then extended
to cover Fitting sets and shortened still further by Anderson [1], whose approach we
follow here.
(4.4) Lemma. Let 3 be a Fischer set of a group G.
(a) If H <G, then 3H is a Fischer set of H;
(b) If N <G, the quotient Fitting set 3G/N is a Fischer set of G/N.
Proof. Part (a) is clear. To justify the second assertion, let K/N <! SN/N e 3G/N,
where S e Inj^(SN), and let H/K be a nilpotent subgroup of SN/N. Then H/K =
(H rsS)K/K S (H rs S)/(K n S), and therefore (H rs S)/(K n S) is a nilpotent subgroup
of S/(K rs S). Since S belongs to the Fischer set 3, it follows that H rsS e 3. Since
(H rsS)rsN = S rsN, which is clearly an 3 -injector of N, we conclude from (2.11)
that HnS is an ^-injector of (H rs S)N = H. Consequently H/N e 3Girt, which
proves that 3Gltl is a Fischer set. □
(4.5) Theorem. If 3 is a Fischer set of G, then the 3-injectors of G are normally
embedded. In particular, ^-injectors of G for a Fischer class g are normally embedded
in G. [Fischer classes are defined in IX, 3.3.]
Proof. Suppose that the theorem is false, and let the pair (G, 3) be a counterexample
with |G| as small as possible. Then, if V e Inj ^(G), there exists a prime p and a
P e Sylp(F) such that P ф Syl„(K), where К = <PG>. If 1 N <3 G, it follows from
4. Fischer sets and Fischer subgroups
555
(4.4)(b ) and the choice of G that VN/N ne G/N and hence that PN/N 6 Syl (KN/N).
This leads easily to a contradiction if we can take N = 0p(G) or О (G) • these
two subgroups are therefore trivial, and, in particular, we have F(G) = О (G)
and Pr>O„(G) = 1. Since .Fisa Fischer set, we have Pe.F, and P is therefore
an .'^-injector of PO„(G) by (2.11). Since PsnPO.(G), it follows that P =
(POp(G)), <) POp(G). Thus [P, 0„(G)] < P n 0„(G) = 1. But Cc(Op(G)) < OP(G) by A,
10.6(c), and therefore P = 1. Hence К = 1, and we have a contradiction. q
It follows as a special case of this theorem that injectors for subgroup-closed Fitting
sets are normally embedded. Combining this with (3.8) we have
(4.6) Theorem. A subgroup of a group G is normally embedded in G if and only if it is
an injector for some subgroup-closed Fitting set of G.
We now consider the following property of a Fitting set S of a group G:
(4.a) For all H < G the S -injectors of H are normally embedded in H.
Note that, in view of(4.4)(a), Theorem 4.5 shows that Fischer sets have this property.
(4.7) Theorem. Let S be a Fitting set of G satisfying (4.a). Then V is an S-injector
of G if and only if V is a Fischer S-subgroup.
Proof. The necessity of the condi tion was noted in (4.2). We shall prove the sufficiency
arguing by induction on |G|. Since the theorem obviously holds when G = 1, we
therefore assume that
(i) |G| > 1, and
(ii) for all groups H with \H| < |G| and for all Fitting sets Sf of H which satisfy
(4.a), a Fischer jT-subgroup of H is an Ж-injector of H.
It is clear that, if H < G, the restriction SH satisfies (4.a) for H. Furthermore, the
same is true for quotients. For let К <3 G, and let H/K <G/K. If W e lnjp-(H),
by hypothesis H'neH, and therefore WK/K neH/K by I, 7.3(b). Since WK/K
is an .Fr,;K-injector of H/K by (2.15)(b), it follows that (4.a) is satisfied by the
Fitting set SG/K of G/K. We now proceed in steps. Let V be a Fischer ^-subgroup
of G.
(1) If N*<G, we mav suppose that VN/N is not a Fischer SG/^-subgroup of G/N.
For otherwise the induction hypothesis leads to the conclusion that VN/N is an
.^.„.-injector of G/N and that LN therefore has the form W N for some W e Inj^fG)
by (2.15)(b). If N < Gf, then V = VN = WN = IV On the other hand, if N £ G,,
then Njr = 1. Since V n N < V e S, we then have V n N = 1, and it follows from
(2.11) that Vis an .F-injector of VN = WN. Because We Inj HITN), Lis conjugate
to W and hence belongs to Inj >-(G).
(2) We may suppose that G, = 1. If not, let N be a minimal normal subgroup of
G contained in G f. Then SCIK = {S/N: N < S < G, S 6 S}, andlit follows easily that
V/N is a Fischer .^-subgroup of G/N, contrary to our supposition in Step 1.
556
VIII. Injectors and Fitting sets
(3) If N <G, we may suppose that G — VSN, where 1 S c Injjr(SN) and SN <3
G By Step 1 the quotient G/N has an .'FG;N-subgroup, S* say, which is normalized
by, and yet not contained in, VN/N. Let S* = SN/N with Selnj^(SN), and let
£ = VSN. If L < G, by induction V is an S' -injector of L, and so VN/N is an
.^„-injector of L/N. Then (4.2) implies that VN/N is a Fischer .^„-subgroup of
L/N and consequently contains S*, contrary to supposition. Therefore L = G, and
then clearly SN < G.
(4) We may suppose that SN/N is a chief factor of G. By Step 3 there is a chief factor
T/N of G with T < SN. Let So = S r> T. Then T = SBN and So e Inj AT). Suppose
that VT < G. Then by induction Ve Inj ^LT), and hence Vn Те Inj^(T). Replac-
ing So by a conjugate if necessary, we may therefore suppose that V n T = So and
hence that V < NC(SO). A similar argument shows that NC(SO) contains an S-injector
W of G. Since NG(S0) < G by Step 2, the induction hypothesis implies that V is
conjugate to W. Consequently, we can suppose that VT = G and may therefore
replace S by So in Step 3.
(5) We may suppose that Coree(VN) = N, and, in particular, that G/N is primitive
with socle SN/N. Let R = CoreG(FN), and let Vo = R n V V. Then R = V0N, and
Vo n N < Vo e S; by Step 2 we therefore have Vo n N = 1 and hence Vo e Inj _F(R)
by (2.11). Since R < G, the argument used in Step 4 shows that if Vo # 1, then NC(FO)
is a proper subgroup of G containing V as well as some .^-injector W of G, and again
that by induction V is conjugate to W. Thus we can suppose that Vo = 1, that R = N,
and hence that VN/N is a maximal subgroup of G/N with trivial core.
(6) We may suppose that G is primitive. Let C = Ce{N) < G. If N < C, then SN <C
by Step 5. But this implies that SN is nilpotent and that S is a non-trivial subnormal
.^-subgroup of G, against the supposition that Gy = 1. Therefore N = Cc{N) and G
is primitive by A, 15.8(b).
(7) If p is the prime dividing |N|, we may suppose that G/N is a p'-group. It follows
from Step 6 that SN/N is a q-group for some prime q # p. Let P e Sylp(V). Since
VN < G by Step 5, we have V e Inj^(FN), and then by hypothesis Vne VN; con-
sequently P e Sylp(K), where К = (J^y. Evidently К r> N = 0„(K) n N < P r> N <
Vn N e S, and therefore К n N < Nf = 1. But then [K, N] < К n N = 1, and it
followsthatK < Cg(N) = N by Step 6. Hence К = 1,P = l,and thereforep||G : N|.
(8) The completion of the induction step: Let W e Inj > (G). Then W n N = 1 by Step
2, and so Wis a p'-group by Step 7. Therefore each Sylow p-complement of G contains
an &-injector of G. Similarly VnN = 1, and V is contained in some Sylow p-
complement Q of G. Since V e Inj AQ) by induction, V is then conjugate to an
^-injector of G contained in Q. □
Hence from Theorems 4.5 and 4.7 we derive the promised theorem of Fischer’s.
(4. 8) Corollary (Fischer [1]). If S’ is a Fischer set, the Fischer S-subgroups coincide
with the ^-injectors and thus form a single conjugacy class.
Dark [2] was the first to discover an example of a Fitting class Д and a group G
having a Fischer Д-subgroup V which is not an 8-injector of G; it is described in IX,
5.19. However, it turns out that the subgroup V in Dark’s example is an injector of
4. Fischer sets and Fischer subgroups 557
G for the Fitting set Fitset(F), despite failing to be an injector for the trace of
Therefore this example leaves still unresolved the question as to whether the set of
Fischer subgroups of a group always coincides with its set of injectors. To conclude
this section we construct an example which shows that this is not the case.
(4.9 ) Example. Let S be a non-abelian group of order 21; thus S = BA, where В < S
with |B| = 7, and A < S with | A| = 3.
First we need some facts about the representation theory of S over F2. Let U be a
trivial simple F24-module and let M = Us; then by B, 6.21(c), the special case of
Mackey’s theorem, we have MB =; (UBrJ« s (ft),)® S F2B, the regular module.
Hence it follows easily from B, 9.8(d) and Wedderburn’s Theorem B, 4.4 that
(4.Д) M« = (F2)B®l'®K’,
where V and F* are non-isomorphic simple F2B-modules of dimension 3. Let A =
<a>. Since A normalizes B, the module F2B(s MB) is self-conjugate under A, and
since the orbits of modules under Л-conjugacy have length 1 or 3, it follows that V
and F* are also self-conjugate under A. Therefore V and V* are S-invariant B-
modules, and it follows easily from B, 9.13 that V and V* can be extended to faithful,
simple S-modules over F2. (Such modules arise in another context: the socle of the
extended affine group Г(23), described in B, 12.9 and B, 12.10, is a 3-dimensional
faithful, simple module over F2 for a complement, and the complements are isomor-
phic with S.)
Let К = Fg, let 1 / 2 e K, and let C, be the КВ-module which affords the linear
representation b -> X''of В = <b> (i = 1,..., 6). Because a may be chosen so that
abcT1 = b2, it follows that U° S C'2 and U3 = similarly, the modules U}, U5, and
U6 form an Л-conjugacy class. Since F2B ® К = KB = Кв ф U, © • • • © U6, we may
therefore suppose without loss of generality that
F® № U, © U2 © C4, and
V* ® К S u3 © u5 © u6,
and it follows from B, 7.4 that, as KS-modules, F ® К S (G,)s and F* ® № (U3)s.
It is then clear that V and V* are absolutely irreducible F2 S-modules, whence
Endr S(F) s End( sfF*) S F2, and we can conclude from the degree formula in B,
4.4(c)2that V and V* are the only faithful simple S-modules over F2.
Since C, ® Uj = Ui+j when the suffices are written “modulo 7’’ and Uo = KB, a
simple calculation shows that
((F® V*)®f2 K)B S (ф ц)© 3KB,
and it follows that, as F2^*mo^u^es»
V ® J/* = V © И* © Vo,
558
VIII. Injectors and Fitting sets
where Fo is a 3-dimensional submodule on which В acts trivially. A similar type of
argument shows that К® К s К® 2P, and. in particular, Crer(B) = 0. For similar
reasons Cl-.el,(B) = 0.
Let .Sj and S2 be copies of S, let 0: S, -> S2 be an isomorphism, let Л, = <n, > e
SyljfSj), and let a2 = 0(at). Next let a denote the element (a,, a2) of the direct product
Sj x Sj, and set A = <o>. If B, e Syl7(Sf) for i = 1, 2, note that each subgroup of
В x B2 is Л-invariant. Let Ц be the irreducible F2Srmodule corresponding to the
F2 S-module V described above, and consider V, ® V2 as an F2(Sj x S2)-module in
the usual way. Since (F, ® F2)B,x, = 3(Fj)B1, it follows from B, 9.8(a) and (d) that
(4.y) {Vi®V2)e^B2= N2®N2®N3,
where each N( is an irreducible Bt x B2-module of dimension 3. Let В =
Ker (В, x B2 on N^. Since В # В, x I or I x B2, it follows from the structure of
Sj x S2 that В has 3 distinct conjugates in Sj x S2. Therefore (Ц ® F2)Bi xBz contains
3 submodules conjugate to Nt, and since these are clearly irreducible and non-
isomorphic in pairs (having different kernels), they must be Nlt N2, and N3. Hence
Nt = CFiK,,2(B), and Nt is Л-invariant since A normalizes B; similarly N2 and N3 are
also Л-invariant.
Let H = H(Vt, K2), the Hartley group corresponding to F, and V2 (see B, 12.11).
Then H is a 2-group of nilpotency class two and admits S2 x S2 as a group of
operators in such a way that there exist operator-isomorphisms
ф: H/H' -» F, © F2, and
ф : H' -> Vt ® V2.
Since the submodules {Nf} in Equation 4.y are (B, x B2)A-invariant, H' contains a
(B, x B2)Л-invariant subgroup К such that ф(К) = Nt © N2. Let P = H/K. Then P
admits (B, x B2)A as a group of operators, ф induces a (B, x B2)A-isomorphism ф
from P/P’ to Vt © V2, and we have P' = H'/K N3. For i = 1, 2 let R.P' = фГ1 (Ц),
and observe that, from the construction of the Hartley group, Rt and B2 are elemen-
tary abelian 2-groups, and that Rt n R2 = P’. Because F,, V2, and N3 are irreducible
B] x B2-modules and obviously F, £ F2, it follows that 1, P', Rt, R2, and P are the
only B, x B2-invariant subgroups of P. From this it is easy to deduce that P’ = Z(P):
for we certainly have P‘ < Z(P) since P has class two, and each of the possibilities
Z(P) = Rlt R2, and P leads quickly to the conclusion that P is abelian.
Denote P' by N, and let i 6 {1, 2}. By Maschke’s theorem there exists а (В, x В2)Л-
invariant complement И' to N in R„ and so И' s Ц as (B, x B2^-module (Ц is
naturally regarded as an S, x S2-module by inflation). Let {i, j} = {1,2}, and let
w 1 be a fixed element in H<. We want to prove that the following property holds
in P:
(4.6) For each ne N, there exists t e Wj such that n = [w, t].
Denote В, x 1 and 1 x B2 simply by B, and B2. Since P has class two, and since Bj
4. Fischer seis and Fischer subgroups
559
centralizes each element t e Wb it is clear that the map
к: t -> [w, t]
from Wj to N is an J1BJ-homomorphism. Because w ф Z(P), we have [w, W] 1 and
therefore К er (к) * W}; hence Кег(к) = 1 since is irreducible under the action of В?
It then follows that к is surjective and therefore (4.<5) holds. J
Henceforth we restrict attention to the subgroup S = 84 of the group (8, x B2)A
of operators on P. Recall that 8 = Кег(В, x B2 on NJ, whence CF1SF;(B) / 0. From
the earlier analysis of the action of В on V ® V, etcetera, it follows that Vt and V2 are
non-isomorphic as S-modules. Therefore without loss of generality we suppose that
P/N = V © V* and N = K and we change to a more suggestive notation by writing
R = N x W in place ofR, = N x Hj, and R* = N x IP* in place of R2 = N x IV2;
thus W = V and W* = V* as F2S-modules.
Now form the semidirect product
L = [P]S,
and let J denote the subgroup R* S of L. Then W" <3 J and J/W* s NS. As we
showed earlier in Example 2.19, the group NS possesses a 7-dimensional faithful
irreducible module, Y say, over F3, and Y may therefore be regarded as an F3 J-module
with Ker(J on Y) = W*. Let Z = Cr(S), and recall from (2.19) that |Z| = 3 and
Z = Cr(B).
Let X = Yl, which may be written as X = £,.ецг Y ® w since W is clearly a
transversal to J in L, and then form the semi-direct product
G = [XJL.
Since | IV| = 8, the order of X is | Y|8 = 356, and therefore G has order 29 • 357 7. Let
Z = <z>, and set
E = W*B(az);
since z commutes with W*BA, it is clear that E is a subgroup of G isomorphic with
W* BA. The following fact will play an important part in subsequent deliberations
(4.e) NC(E) = NC(W*B) = ExZ.
To prove this we first remark that
(4.0 NC(£) S WC(^*B);
this is because W* В = O2.7(E) char E. Moreover, we have
(4,„) NC(1V*B) = Cx(W*B)Nl(W*B).
560
VIII. Injectors and Fitting sets
Since В acts fixed-point-freely on each 2-chief factor of PB, we have C,,(B) = 1. Hence
В is a Carter subgroup of PB, is therefore abnormal in PB, and in consequence we
have Nfb(W*B) = IV*B; it then follows easily that NL(W* B) = W*BA. Next we
show that Cj (W*) = Y ® 1 Let 1 w e W. and let 0 ф у e Y. Since YNS is irreducible,
we have Cr(N] = 0. Therefore N contains an element n such that yn y. By (4.d)
there exists an element t of W* such that [w. t] = n, and then wt = twn = tnw since
n e Z(P). Using the fact that W* acts trivially on Y, we have
(у ® w)t = у ® wt = у ® tnw = ytn ® w = yn ® w ф у ® w,
and we conclude that СГ0Н,(1У*) = 0 when w # 1. Since each Y ® w is И'*-шvariant,
it follows that
(4.0) СЛ(Ж*)=Г®1,
as claimed. If we identify Y with Y ® 1, we obtain
СА(И'*В) = Cr(B) = Z,
and substitution in (4jj) then yields
NC(W*B) = ZW*BA.
But evidently ZW* BA = E x Z < NC(E), and so, in view of (4.4 ), Equation 4.e is now
justified.
We are now in a position to show that the subgroup E is not an injector of G. By
(3.3), (c)=>(a), it will be sufficient to prove that s„EG is not a Fitting set of G. First
observe that W* is maximal among the 2-subgroups in s„Ec. Let t e W* and n e N
with n # 1. Then by (4.0) there is an element w e W such that [w, t] = n, and therefore
W* # (IT'*)"’. However, we have (Ил*)"’ < R* < P, and since R* is abelian, W* and
(Ил*)"’ therefore normalize each other. Thus the normal product И'*(И/*)'\ as a
2-group of order greater than | W* |, cannot belong to s„Ec, and therefore s„Ec is not
a Fitting set.
In order to carry out our next task, which will be to identify certain Fitting sets of
G, we shall need two elementary results about Fitting sets in general; we temporarily
interrupt the discussion of this example in order to state and prove them.
(4.10) Lemma. Let nbe a set of primes, let X be a normal n'-subgroup of a group G,
and let L be a subgroup complementing X in G. Let 6 be a Fitting set of L consisting
entirely cf n-groups. Then the set <?c consisting of G-conjugates of the subgroups in 6
is a Fitting set of G.
Proof. Requirements FS1 and FS3 of Definition 2.1 clearly hold for <?c. To verify
FS2 let R, S < RS with R, S e <?c. Using the fact that G = LX and that <?'- = <?, we
may clearly suppose that R = Mx and S = N” for suitable M, N e <? and x, у e X.
Then we have XR = XM and XS = XN. Since XS normalizes XR, the subgroup N
4. Fischer sets and Fischer subgroups 56l
L n°rmai1,ZiS XR Г'L = XM r'L = M- Similarly M normalizes (V, and so
MN belongs to the Fitting set 6. Evidently the hypotheses imply that MN and RS
are Hall л-subgroups of XMN = XRS, and therefore RS is conjugate to MN bv I
3.3(c); in particular RS e SG. (_j
(4.11) Lemma. Let К be a normal subgroup of a group L. and let У be a subgroup-
closed Fitting set of L/K. Then the set
fi = {S: S < T for some T/K e У\
is a Fitting set of L.
Proof Again, only Requirement FS2 needs checking. If R, S < RS with R, S e fi,
then the subgroup-closure of У implies that RK/K and SK/K belong to У. Hence
RSK/K = (RK/K)(SK/K) e and consequently RS e fi. □
We now return to the example under discussion and recall that R* is a normal
subgroup of the group L = PBA. By taking R* for К in (4.11) and the set of
7-subgroups of L/R* for the set У of that lemma, we deduce that the set fi of
L-conjugates of subgroups of R*B is a Fitting set of L. Then, applying (4.10) with
n' = {3} to our group G = XL, we conclude that the set
= <?c = {S9: S < R*B, g e G}
is a Fitting set of G.
Now let У = 'S и {Ec}. We assert that У is also a Fitting set of G. Since W*B is
the unique maximal normal subgroup of E, any proper subnormal subgroup of E is
contained in IV* B, hence in R*B, and therefore belongs to У. Thus it is clear that
У is closed under taking subnormal subgroups. Next let R, S < RS with R, S e У.
If R and S both belong to '4, we have RS since *S is a Fitting set. Therefore, in
showing that RS e У, we can suppose without loss of generality that R = E. Then
we have S < AC(E) = E x Z by (4.c). If S e we have S e S.2.71, and therefore S is
contained in the unique Hall {2, 7}-subgroup H *B of E x Z; in this case RS =
EW* В = Ее У. The only other possibility to consider is that S = E9 for some g e G.
But then (IF* B)9 is a {2, 7}-subgroupofE x Z, and therefore (W*Bf = IF* B. In this
case we have g e NC(W* B), which equals NC(E) by (4.c), and again RS = E e The
claim that У is a Fitting set of G is thus clearly bom out.
Our final goal is to show that E is a Fischer .F-subgroup of G. To this end let S
be an .F-subgroup of G normalized by E; we must prove that S < E. First suppose
that S = E" for some g e G. In this case E is normalized by E9 1 and, from the analysis
of the preceding paragraph, ii follows that E = E9 and hence that S = E. Con-
sequently, we may suppose that S e fS, say S < (R*B)9. If |S| = 7 and S f E, then
721 |SE|, which is impossible by Lagrange’s theorem because 72 f| G|. Therefore 21 |S|.
Let T = O2(S). Since R*B e S2<S7, we have T e Syl2(S), whence 7/ I. We suppose
that T f E and derive a contradiction. Since T char S, clearly E normalizes T, and
we have IF* < TIF* = 02(TE) e Syl2(7E). Because T < (R*)9 < XR*, it follows that
562
VIII. Injectors and Fitting sets
XIV* < ХТИ'* < AR*,
where A'TB'* is normalized by E. This forces the conclusion XTW* = XR* since E
acts irreducibly on the factor XR*/XW* S R*/W* S N. Now set N* = XNr> TW*.
and note that N* is normal in ТЕ. From the Dedekind law we conclude that
XN* = XN and hence that АИ'*( = R*)andA'*ft'* are (abelian) Sylow 2-subgroups
of A R*. Therefore N and N* are Sylow 2-subgroups of СДЛ,(И'*)_ which equals YN
because of Equation 4.0. By Sylow’s theorem N* = Ny ‘ for some v e I, and as В
normalizes A*. it follows that both Br and В normalize N. From the definition of У
it is easy to see that C,-(A) = 1, and therefore Br = NrBy(N) = NrB(N) = B. Hence
v e Nr(B) = Cr(B) = Z. But Z centralizes E. and so N is normalized by Ey = E; N is
therefore normalized by <E, IV* BA) = E x Z. But this leads to the following con-
tradiction:
1 * Z < NV(N) = Ct(N) = 1,
which shows that after all T must be contained in E. Thus 1 / T < E, and since
W*(= 02(E)) is a minimal normal subgroup of E. it follows that T = W*. If S/W*
1, then |S/IV*| = 7, and again we have 721|SE| unless S < E. Thus we have proved
that in all possible cases E contains S. Hence E is a Fischer .F-subgroup of G, but is
not an injector for any Fitting set of G.
Open Questions. 1. Are Fischer subgroups always pronormal subgroups?
2. If J is a Fitting class and & = Ttj(G), is a Fischer .^-subgroup of G always an
injector for some Fitting set of G?
Chapter IX
Fitting classes—examples and
properties related to injectors
1. Fundamental facts
Throughout the chapter we shall always implicitly assume that any universe under
consideration is a Fitting class, and unless otherwise stated, the universe in this
section is 6. We recall from II, 2.8(a) that a Fitting class is a non-empty class of finite
groups closed under the operations s. and N„; according to II, 2.11 we therefore have
the following description:
A class g (#0) is a Fitting class if and only if the following two conditions are
satisfied:
(i) If G e g and N < G, then NejJ;
(ii) If M, N < G = MN with M and N in g, then G e g.
For n P the classes 91„, S„, and (£„ are simple examples of Fitting classes (see A,
8.2(a), 8.8(c) for 9l„ and A. 8.6(a) for (£„). According to IV, 3.14 we obtain further
examples among local formations g = LF(f) by requiring/(p) to be a Fitting class
(as well as a formation) for each p e Supp(/).
If 3 is a Fitting class and G a group, then by VIII, 2.2(a) the trace of 5 in G, that
is the set
TrB(G)={H<G:Heg},
is a Fitting set of G. It obviously has the additional property of being characteristic,
in other words, invariant as a set under the action of Aut(G). Many of the basic facts
about Fitting classes are therefore direct consequences of results about Fitting sets
already proved in Chapter VIII. We recall from II, 2.9 that each finite group has a
unique maximal normal ^-subgroup, the so-called ^-radical of G, denoted by GB.
Thus, in this notation, we have G.j, = F(G), Ge< = 0„(G), and if .^ = TrB(G), then
GB = G,. The ^-radical of a group can also be defined as the join of all its
subnormal g-subgroups, a fact which follows easily from the first part of the following
lemma.
(1.1) Lemma. Let g be a Fitting class and G a finite group.
(a) If N sn G, then NB = N n GB.
(b) If N„ .N, S G = AjNj.-.N,, and if К denotes the normal subgroup
П1=1 Wb °f 6, ‘^en - 2(G/K).
564
IX. Fitting classes—examples and properties related to injectors
Proof. Part (a) follows from VIII, 2.4(d). Using this and A, 7.4(f), we simply observe
for Part (b) that [GB, G] = [GB, NtN2... ty] = П<=< AJ < J],=i (GB n Nf =
П'=1(А.)Й = К- °
As we mentioned in the Historical Introduction to Chapter VIII, Fischer’s dualiza-
tion of the theory of projectors (then called covering subgroups), which we have
already carried through for Fitting sets, was originally formulated for Fitting classes,
and after some modification led to the following definition for the dual construct.
(1.2) Definition (Fischer, Gaschiitz, and Hartley [ 1 ]). Let ft be a class of finite groups.
An ^-injector of a finite group G is a subgroup V of G with the property that V n К
is an g-maximal subgroup of К for all subnormal subgroups К of G. We denote the
(possibly empty) set of ft-injectors of G by Inj B(G).
The following remarks follow at once from this definition.
(1.3) Remarks. Let C be a finite group and ft a class of finite groups.
(a) If Ve Injs(G) and К sn G, then Vn К e Inj-(K).
(b) If V e Inj S(G) and a: G -» Ga an isomorphism, then Va e Inj B(Ga); in particular,
Inj j(G) is a union of G-conjugacy classes.
(c) Let V be an ^-maximal subgroup of G, and assume that V n M e Inj Э(М) for all
M <• G. Then V e InjB(G).
If by ‘injective’ classes we mean those classes for which injectors always exist in
some given universe, the next result shows that for the universe S the injective classes
coincide with the Fitting classes. From this point of view Fitting classes are therefore
dual to Schunck classes (cf. Ill, 3.10).
(1.4) Theorem (Fischer, Gaschiitz, and Hartley [1]). A class ft of finite soluble groups
is a Fitting class if and only if every finite soluble group has an ^-injector.
Proof. If ft is a Fitting class and if G e S, then InjB(G) ф 0 by VIII, 2.9.
Conversely, suppose that InjB(G) 0 for all GeS, and let N sn G e ft. By the
definition of an g-injector we have {G} = Inj 5(G), therefore N = N n G e Inj S(A) by
(1.3)(a), and consequently N eg. Hence ft = s„g. Next let G = NtN2 with Nt and N2
normal ft-subgroups of G, and let V e Injs(G). Since InjB(A,) = {AJ, by (1.3)(a) we
have V n N, = Nt, and therefore A, < V for i = 1, 2. It follows that G = V e ft and
hence that ft = Nog by II, 2.11 (b). □
Next we gather together in a convenient ‘portmanteau’ theorem some of the more
important properties of g-injectors.
(1.5) Theorem. Let ft be a Fitting class and G a finite soluble group.
(a) The %-injectors of G form a characteristic conjugacy class of pronormal CAP-
subgroups of G.
(b) Let 1 = Go < Gj < < G„ = G be a series with all its factors Gi/G,_i nilpotent.
1. Fundamental facts
565
Then a subgroup V of G is an %-injector of G if and only ifVryG, is ^-maximal in G,
for i - 1,..., n. In particular, one can weaken the definition of an ^-injector for the
universe S by replacing 'subnormal' by 'normal' in (1.2).
(c) If V is an %-injector of Gand if V <H < G, then V is an ^-injector cf H.
(d) If V is an ^-injector of G and if К < G, then G = KNc(Vn K).
Proof For Part (a) see (1.3)(b) and VIII, 2.9 and VIII, 2.14(a), (c). Part (b) is a
consequence of VIII, 2.12, Part (c) follows from VIII, 2.13, and Part (d) from VIII
2.14(b). a’
The following result is essentially a restatement for Fitting classes of VIII, 2 10 and
2.11.
(1.6) Lemma. Let g be a Fitting class, and let Gbe a finite soluble group. Let N <G,
and let Lbea subgroup of G such that L r . N is an ^-injector cf N. Assume that either
(a) G/N e 91, and L is ^-maximal in G, or
(b) Le 8 and LN = G.
Then L is an ^-injector of G.
A more searching study of the properties of injectors will be undertaken later in
this chapter, particularly in Sections 3 and 4. We now turn our attention to the
behaviour of Fitting classes as classes of groups. On the question of characteristic,
we recall that the inclusion: Char(§) e o(§) can be proper when § is a Schunck class
(see Chapter III, Section 2, Exercise 8), whereas by IV, 4.3 we have Char(g) = o(g)
when g is a saturated formation. The next result shows that Fitting classes of soluble
groups behave like saturated formations in this respect.
(1.7) Lemma. Let G be a finite group which possesses a composition factor of prime
order p. Then s„nos„(G) contains a cyclic group of order p. Furthermore,
Char(g) = <r(g) when g is a Fitting class of finite soluble groups.
Proof. Let H/K be a composition factor of G of order p, and let g be an element of
H\K of p-power order. Put D = H x Z, where Z = <z> is a cyclic group of order p,
and let H* = Kfgzf Evidently we have H* < D = HH*. Each element of H can be
uniquely expressed in the form kg' with ke К and 0 < i < p — 1, and it is straight-
forward to verify that the map kg' ->• k(gz)‘ is an isomorphism from H onto H*. Hence
H and H* are in s„(G), it follows that D e nos„(G), and therefore Z e s,(C) s s„n„s„(G),
as required.
Let g be a Fitting class contained in <S. If peo(g), then p||G| for some Geg.
Because the group G is soluble, it has a composition factor of order p, and by the
above result we have Zp e s„Nos„g = g. Therefore Char(g) = <r(g). □
Remark. The final statement of Lemma 1.7 is false without the hypothesis that g £ S
(see Exercise 4 below).
(1-8) Lemma. If p 6 P, then Sp s snN0(Z,,).
566
IX. Fitting classes—examples and properties related to injectors
Proof. Let W„ = (. • • (Z„ Qj Zp) Ii...) Ii Z„ (n terms), the n-fold regular wreath power
of Z . It is obvious that W„ is generated by subgroups of order p, which are necessarily
subnormal because W„ e <5„ £ 91; therefore W„ e N0(Zp). But by A, 18.10 an arbitrary
p-group is isomorphic with a (necessarily subnormal) subgroup of W„ for sufficiently
large values of n, and so the lemma is clear. □
The following theorem follows at once from the two preceding lemmas.
(1.9) Theorem. Let ^bea Fitting class of finite soluble groups. Then g has character-
istic n if and only if 9i„ £ g —
If & is a family of Fitting classes, then, as usual for classes defined by closure
operations, the intersection Q {g: ge is again a Fitting class; and if SF is a
directed set with respect to the partial order of inclusion, then the union U {8; 8 e 5s-}
is also a Fitting class. As might be expected, the class product of two Fitting classes
need not in general be a Fitting class (see Step 7 of Example 2.14(b) below), but
fortunately this shortcoming can be rectified. A special product for Fitting classes
can be defined which is dual to the formation product of IV, 1.7 and which preserves
the Fitting class property.
(1.10) Definition (Gaschiitz [10], [15]). Let 8 be a Fitting class and ® a class of finite
groups. We define
g О ® = (G: G/Gg 6 ®)
and call 8 О 6 the Fitting product of 8 with ®.
(1.11) Remarks. It is clear that 8 О ® £ g^’. and that if ® = q®, then 8 О ® = g®.
It is also obvious that 8 — 8 О ®, although Step 8 of Example 2.14(b) below
shows that in general 6^ go ffi. Furthermore, if ® £ then g О ® £ g О §,
although when ® and § are also Fitting classes, one cannot always conclude that
® О g £ § c g as is shown also by Step 1 of Example 2.5(a). However, if g = Qg,
then ® О g = ®g <= §g = § о g.
Comparison of the next result with IV, 1.8 reveals a close duality between Fitting
and formation products
(1.12) Theorem. Let g and ® be Fitting classes and § a class of groups. Then
(a) g О 6 is a Fitting class;
(b) For any Ge® the ^-radical of G/G% is G^/G^
(c) (g О ®) о § = g о (® о §).
Proof, (a) Let N < G e g о ®, and put К = Gg. By (1. l)(a) we have Nr,K = Ns,
and so A'/Ag = NK/K < G/K e ®. Hence N e g О (s„S) = g о ®, and therefore
»® IS s.'closed. Next suppose that G = NtN2 with N, < G and N, e g О ® for
1 ~ Fut = and let i e {1,2}. Again using (l.l)(a), we obtain NtK/K =
1. Fundamental facts
567
W’f®' a"d “nSequentl>' G/^ = (MK/K)(N2K/K)6Nos = (S. Therefore
G e g О ®, and we have shown that g о ft is also N0-closed.
(b) Since g £ g о G, we have GB < GB. .e, and hence GB = GBn GBnl6 = (G,nlB)»
by (1.1)(а)д Therefore Gs,ffi/GB e <5, and so GBce/GB < (G/GBL Let R/G denote
(G/GB)e. Again by (1.1 )(a) we have RB = R n GB = GB, and therefore R e g о ft. It
follows that (G/GB)ffi is contained in, and hence equal to, Св,.й/Сй, as required.
(c) Using the definition of a Fitting product, an isomorphism theorem, and Asser-
tion (b), we obtain the following sequence of equivalent statements-
(1) G e (g О ®) О ft
(2) G/GBo(Beft
(3) (G/GB)/(GBC(B/GB)eft
(4) (G/GB)/(G/GB)eeft
(5) G/GB e ® о ft
(6) G e g О (® O §).
□
Although Step 5 of Example 2.14(b) shows that in general Fitting classes are not
R„-closed, the following result, known as the *quasi-R0 lemma’, can sometimes be used
as a substitute for R0-closure. In Theorem 1.24 of Chapter X a stronger version of
this lemma is proved for Lockett classes.
(1.13) Lemma. Let A, and N2 be normal subgroups of a group G such that Nv n N2 = 1
and G/NiN2 is nilpotent, and let g be a Fitting class containing G/Nl, Then G eg if
and only if G/N2 e g.
Proof Put N = NtN2, let D = (G/Nl) x (G/A2), and define the familiar homo-
morphism p: G -> D by p(g) = (gNt, gN2) for all geG. The hypothesis that
A2 ci N2 = 1 implies that p is a monomorphism. Since p(N) = (A/AJ x (A/A2) < D,
we have D/p(N) = (G/N) x (G/N) e 'Ji. Consequently G s p(G) sn D, and therefore
G e s„dg(G/Ai, G/A2). Hence if G/A2 e g, we have G e s,D„g = g.
On the other hand, it is obvious that D = ((G/Nf) x 1, p(G)}, and therefore
D e NpfG/Aj, G). If G e g, it follows that G/A2 s 1 x (G/A2) e s„(D) <= s,N„g = g.
□
Remark. It is clear from the proof that we only require g to be <s„ D0>-closed in
order to prove the sufficiency of the condition.
The rest of this section is devoted to a description of the injectors for a Fitting class
product go®, and naturally for this purpose we must restrict ourselves to the universe
S. The description is due to Lockett, and the essential ingredient is the following
construction.
(1.14) Definition (Lockett [2]). The operator L„( ). Let л be a set of primes, and let
ft be a Fitting class of finite soluble groups. Then define
(l.a) £n(g) = (G g <5: the ft-injectors of G have л'-index in G).
568 IX. Fitting classes- examples and properties related to injectors
Thus L,(g) consists of all finite soluble groups whose g-injectors contain a Hall
n-subgroup: in particular, we have L0(g) = © and LP(g) = g.
(1.15) Theorem (Lockett [2]). Let % be a Fitting class, and let fi = L„(g), as defined
in (La). Then
(a) fi is a Fitting class;
(b) gue,.sge. £e = cs,-;
(c) L,(fi) = fi;
(d) Any two of the following statements are equivalent:
(i) g = f-JS);
(ii) g = g6„.;
(iii) For each G e S, the ^-injectors of G have n-index in G;
(iv) MS) = ©•
Proof, (a) Let G 6 fi, and let V e Injs(G). Then there exists an S e HallJG) such
that S<F. If N < G, then VnN eInjs(iV) by (1.3)(a), and V r. N contains
S n N 6 Hall„(W). Therefore N e fi and £ = s„£. To prove that fi is N0-closed,
suppose that a group G is the product of two normal fi-subgroups and N2, and
let V e Inj jj(G). Let H e Hall„(F), and let H* be a Hall л-subgroup of G containing
H. Since V n N; is an g-injector of 6 fi, for i = 1,2 we have H N; e На11я(Л/;),
and therefore H* rt = H c\ iV,. It follows from 1,3.2(d) that (H n TV, )(H n N2) is a
Hall л-subgroup of NtN2 = G. Consequently H* = H < V, and G e fi.
The inclusions of Part (b) follow at once from the definition of fi. To prove Part
(c), suppose that, if possible, L„(fi) Ф fi, and let G be a group of minimal order in
L„(fi)\fi. Let N be a maximal normal subgroup of G, which belongs to fi by Part (a)
and the choice of G. Let |G: N| = p. Ifp e n', then G e fiS„- = fi, contrary to supposi-
tion. Hence pen. But an C-injector V of G contains N and has л'-index in G.
Therefore G = V e fi, a contradiction which proves Part (c).
Finally we prove a circle of implications for Part (d). The implication: (i) => (ii) is
clear from Part (b). Next we show that Condition (ii) implies (iii). Suppose that
g = g<S„., and if G e S, assume inductively that in soluble groups of order less than
IG| the g-injectors have л-index. Let Ve Inja(G) and К < G. Then |K: (Fn K)| is
a л-number, which coincides with | G: V\ if VK = G. Therefore suppose that V < K,
and observe that in this case G = KNG(V) by (1.5)(d). Consequently |G: Nc(F)| =
IК: NK(F)| is a л-number, and therefore V is normalized by a Hall л'-subgroup, H
say, of G. But then VH e g<5„. = g, and it follows that V = VH by the g-maximality
of V. Hence Condition (iii) holds. Since Condition (iii) obviously implies (iv), we can
now complete the circle by showing that: (iv) => (i). Suppose that L„.(g) = S, and let
G E l-«(g). Then an g-injector V of G has n'-index and at the same time л-index in
G. Hence G = V e g, and it follows that g = L„(g). □
Next we study the injectors for the Fitting classes L„(g), and then apply our
observations to the promised description of the injectors for a Fitting product g О (5.
Let g be a Fitting class and n a set of primes. If W is an L„(g)-injector of a finite
soluble group G, it follows from (1.15)(c) and (d) that W has л-index in G. Thus W
contains a Hall n'-subgroup, G„. say, of G. Furthermore, by definition of L„(g),
1. Fundamental facts
569
an g-injector V of W contains a Hall л-subgroup of W, and therefore by order
considerations we have
(1Д) FG„. = G„.|<
Consequently, in order to determine an I-„(g)-injector, one must identify the sub-
group И It would be natural to hope that V is in fact an g-injector of G. Although
this is indeed often the case as we shall see in Section 3 below, unfortunately it is not
always true, for in (5.19) we describe an example of a Fitting class g and a group G
such that an g-injector of G permutes with no Hall л'-subgroup of G. The next result
shows that this permutability property holds the key.
(1.16) Theorem (Lockett [2]). Let % be a Fitting class, let л £ P, and let Gets. Let
V and G„. denote respectively an ^-injector and a Hall ^-subgroup of G, and put
W = <K G„.>. Then W is an L^finjector of G if and only if VG„. = G„. V.
Proof. Write C = L„(g). By (1.5)(c) we have V e Inj s(IV), and so if IV e Injt,(G), then
Equation (1./?) of the preceding discussion yields the desired conclusion.
Conversely, suppose that V permutes with G„.. We proceed by induction on |G| to
show that IV e Inji,(G). The result is obviously true if |G| = 1; therefore let A be a
maximal normal subgroup of G. If Vn e HallJF), then we obviously have W = l„G„.,
and since n N and G„. n N are Hall subgroups of IV n N < IV, it follows that
(Vci N)(G„. ci A') = IFci N. Consequently the g-injector Fn N of N permutes with
G„. c N, and by induction Wn N is an C-injector of N. Therefore by (1.6) it will be
enough to show that Ж is C-maximal in G. Let W < U <G with U e £. Since
V e Injs(G) and since U e C, we conclude that V contains a Hall л-subgroup of U,
and hence U = VGr. = W. □
The conclusions of the preceding discussion can be more elegantly formulated by
using the concept of strong containment, which we investigated for Schunck classes
in Section 1 of Chapter VI, for saturated formations in Section 5 of Chapter VII, and
which we now define for soluble Fitting classes in the obvious way.
(1.17) Definition. Let g and S be Fitting classes of finite soluble groups. We say that
g is strongly contained in (5 (and write g « S) if a (5-injector of each group G in <S
contains an g-injector of G. The relation ‘«’ of strong containment is clearly a partial
order on the family of Fitting classes of finite soluble groups. Of course, we have
already used the notation *«’ for strong containment between Schunck classes, but
as usual we shall rely on the context to make the meaning clear, in particular « will
have the meaning of this definition throughout Chapters IX, X, and XI.
Theorem 1.16 and the remarks preceding it have the following formulation in this
terminology.
(1.18) Theorem. Let g be a Fitting class in S, and let л <= P. Then
(a) 6,. « L,(g), and
570
IX. Fitting classes—examples and properties related to injectors
(b) 5 « L„(g) if and only if for each soluble group G an ^-injector of G permutes
with some Hall n'-subgroup of G.
Our next important objective is Lockett’s description of the g о ©-injectors of a
group (Theorem 1.22), but first we give a description of the Lp(g)-injectors of a group
which has p-normally embedded g-injectors.
(1.19) Proposition. Let g be a Fitting class, let V be an ^-injector of a finite soluble
group G, and assume that V is p-normally embedded in G. Let PeSylp(F), put
К = <PC>, and let L/K = OP(G/K). Then
(a) L is the Lp(8)-radical of G,
(b) the subgroups {LQ: Q e Hallp.(G)} are the Lp(8)-injectors of G,
(c) g«Lp(g),and
(d) the LplSfinjectors of G are p-normally embedded in G.
Proof (a) Let R denote the Lp(g)-radical of G. Since V p-ne G, we have P e Sylp(K),
and therefore |L:(Ln F)|, which divides |L: K||K : (K n F)|, is a p'-number. Since
LriVe Injs(L), it follows that L e Lp(g) and hence that L < R. On the other hand,
the g-injector I'riR of R contains a Sylow p-subgroup of R e Lp(g). Since
P e Sylp(F n R), it follows that P e Sylp(R). In particular, |R : K| is a p'-number, and
consequently R < L. Therefore R = L, as desired.
(b) By the implication: (a) =>(b) of 1, 7.12 there exists a Q e Hallp.(G) and a
P 6 Sylp(G) such that PQ = QP and P < P. Let Qo e Hallp.(F). Then Qo < Q9 for
some geG, and since G = QP, we may suppose that g e P. It follows that VQ9 =
P(?oCe = = (PQ)9 < G, and therefore by (1.16) the subgroup VQ9 is an
Lp(g)-injector of G. Since L < VQ9, it follows that VQ9 = LVQ9 = LPQ9 = LQ9, and
Assertion (b) is now clear by the conjugacy of injectors.
(c) Since VL/L is a p'-group, Assertion (c) follows from Hall’s theorem and Asser-
tion (b).
(d) Since \LQ: K| is a p'-number for Q e Hallp.(G), it follows from Part (b) that P
is a Sylow p-subgroup of each Lp(g)-injector of G. □
(1.20) Lemma. Let g be a Fitting class, let T/R be a normal factor of a group G such
that Cg(T/R) < T. and let X be a subgroup of G which satisfies the following two
conditions:
(i) (X n T)R = T, and
(ii) (X n Т)й < R.
Then Хй = (X л Т)й. In particular, if G/G^ is soluble, these hypotheses are satisfied
with R = Gs, T/R = F(G/R), and X >T. and in this case we have X% = Тй.
Proof. Put К = Xs. Since X r> T < X, we have К n T = (X Т)й, and therefore
К n T < R by Hypothesis (ii). Using Hypothesis (i) and the fact that К and X n T
normalize each other, we obtain [К, T] = [К, (X r> T)][K, R] < (K n T)R = R.
Consequently К < CO(T/R) < T, and it follows that К = К Г.Т = (X n Т)й. □
(1.21) Lemma. Let g and © be Fitting classes, and let я = Char(©). If G e LJg),
then InjB(G) = Inj8c<s(G).
1. Fundamental facts
571
Proof. Let U be an g-injector of G e L„(g). First we assert that 17 is g 0 ©-maximal
m G. Suppose that U < H < G with H e g о ©. Then Н/Нл e © n G <= G„. Since
U e InjR(H), we have H8 < U, and consequently |Я: G| is a л-number. On the other
hand, \H: l/| divides |G: G|, which is a л'-number because G e L (g) Hence H = U
and the assertion is proved. If К sn G, then К e L„(g) and U n К e InjR(K) and
the preceding argument shows that Ur К is g 0 ©-maximal in K. Therefore
U e Inj and the conclusion of the lemma now follows from the conjugacy of
injectors. Q
(1.22) Theorem (Lockett [2]). Let g and 6 be Fitting classes, let л = Char(©),
and set C = L„(g). Let G be a soluble group, let V be an fi-injector of Gc, let
S e Hall„(Nc(F)), and finally let U/V be a (b-injector of SV/V. Then
(a) U is an g О (b-injector of G,
(b) UG^/G^ is a (b-injector of G/Ge, and
(c) S V is an g о Qn-injector of G.
Proof. First observe that by (1.21) we have
(1) V is an g o ©-injector of GE.
Next put К = GE, and let T/K = F(G/R). By (L15)(b) we have O„.(G/R) = 1, and
therefore, since Char(©) = л, we have
(2) T/R 6 91„ S ©.
Now let N = NT(F) and He Hall„(N). By the Frattini argument NR = T, and
so HR = T. The index |(Hr>R)F: F|, which equals |(H r> R): (H n F)|, is clearly
a л-number and at the same time divides |R: V\, which is a л’-number because
R e C = L„(g). Therefore H r R = H n V, and it follows that HV/V s H/(H г V} =
H/(H r R) s HR/R = T/R. Hence, from Step 2 and the fact that |T:H| =
|R: F||F:(Hn F)|, we have
(3) HF/Fg91„s6 and HeHall„(T).
Put К = (HK)S. Since V < HV and V e g, we have V < К r R e s„g = g, and
because V is g-maximal in R, thence К n R = V e Inj8(R). It therefore follows from
(L6)(b) that К g InjR(KR). Since |KR: K| = |R: L|, a л'-number, we conclude that
KR 6 C, and consequently KR < Ts = R because KR sn T. Thus К — К r R — V,
and we have shown that
(4) F = (WnB-
From Steps 3 and 4 we can deduce that HV e g 0 ©, and therefore by Step 1 and
(1.6)(b) we have
(5)
HV is an g 0 ©-injector of T.
572 IX. Fitting classes—examples and properties related to injectors
It follows from Step 5 that G has an 5 0 ©-injector, U* say, such that U* n T =
//Г. Applying (1.20) with V* in the role of X, we obtain (using (4))
(6) V = (U*)S.
In particular, V*/V e ® £ S„, and therefore l/*/F is contained in some Hall
я-subgroup of NO(V)/V, which we can suppose to be the group SV/V mentioned
in the statement of the theorem; furthermore we can suppose that H < S by the
conjugacy of Hall subgroups. Next we aim to show that
(7) U*/V is a ©-injector of S И/И
Let 1 < HV/V = Gj < < G„ = SV/V be a normal series of SV/V with nilpotent
factors. (Note that G, 6 91 by Step 3.) Let ie{2, ...,n} and let W/V = Gf. Since
(l7*/F)nGj = Gj, which belongs to ® by Step 3, in order to verify Step 7 it will
be enough to show that (17* r> W)/V is ©-maximal in W/V by (1.5)(b). Let X/V
be a ©-subgroup of W/V containing (17* n W)/V. Then VH < X r , T < VS r , T =
V(Sr\T) = VH because HeHall„(T). Therefore (X n T)s = (VH)S = V, and it
follows from (L20) that = V. Consequently X 6 ft 0 ®. But U* is an JO ©-
injector of SV, and therefore U* n W is J 0 ©-maximal in W. Consequently
X = 17* n W, and Step 7 is complete. Statement (a) of the theorem now follows from
the fact that U is conjugate to U* by the conjugacy of ©-injectors.
(b) Since SryR = Hr\R = HcyV, we have SVr, R = (Sr\R)V = V, and conse-
quently for s g S the map sV -> sK is an isomorphism from SV/V onto SR/R. There-
fore UR/R is a ©-injector of SR/R. But since Char(®) = я and SR/R e Hall„(G/K),
each ®-injector of G/R is contained in some conjugate of SR/R, and Statement (b)
now follows from the conjugacy of injectors.
(c) This is a special case of (a) with © = G„. □
If we assume in the hypotheses of the preceding theorem that ft = ftG„. (this always
holds, for example, if Char(®) = P, because then G„. = (1)), then by (1.15) (d) we have
/-„(ft) = 5. an£l the subgroup V coincides with Gc. Hence we obtain the following
corollary.
(1.23) Corollary. If the Fitting class J in Theorem 1.22 satisfies the condition
5 = ftG„. (for example, if 91 ®), then a subgroup U is an ft i> (9-injector of G if
and only if G/GB is a (9-injector of G/G^.
Further properties of the operator Ln( ) can be found in Sections 3 and 4 of this
chapter and in Section 1 of Chapter X. We bring this section to a close with a few
words about another operator which produces new Fitting classes from known ones.
It was introduced by Hauck [3] and has been studied in detail by Brison [2]. It may
be variously regarded as the dual of a construction of Blessenohl [2], as a special
case of the operator L„( ), or as a special case of Construction В described in Section 2
below.
1. Fundamental facts
573
(1 24) Definition. Let я be a set of primes, and letg be a Fitting class of finite groups
Then define r '
K„(S) = (G g G: if H e HallJG), then H e g).
The following observations follow easily from the relevant definitions.
(1.25) Remarks. Let g be a Fitting class of finite groups.
(a) K„(5) = n G„), and so K„(g) is a Fitting class by (1.15)(a);
(b) K„(g) = G„. о K„(g) о G„..
In contrast to the situation for the operator L„( ), it is not always true that
8 S K„(g). In fact, it is obvious that this inclusion holds if and only if g is closed
under taking Hall я-subgroups, a situation which has been sufficiently well studied
to justify the following definition.
(1.26) Definition. If я is a set of primes, a class g of finite soluble groups is said
to be Hall л-closed provided that whenever H e Hall„(G) and G e g, then H e g.
Furthermore, g is said to be Hall-closed if it is Hall я-closed for all я g P. It is not
difficult to find examples of Fitting classes which are not Hall closed.
Bryce and Cossey [5] and Hauck [3] have proved that certain normal Fitting
classes are Hall-closed (see X, 6.7 for example). Brison has studied the Hall-closure
of Fitting classes extensively in [5] and [6] (see Chapter X, Section 6, Exercise 2).
We end the section with a striking description of the K„(g)-radical of a group.
(1.27) Proposition (Brison [2]). Let л be a set of primes, letft be a Fitting class, and
let S3 = K„(g). If G is a finite soluble group, and if H is a Hall л-subgroup of G,
then
(a) H n GB = Hg, and
(b) GB/<(HR)C> = O,.(G/<(Hr)c>).
Proof, (a) Write К = Gs. Since К < G, we have H n К e Hall„(K), and therefore
HoK <HS. Next put F/K = F(G/K), and note that F/K e G„ by (1.25)(b). Thus
F/K < HK/K e Hall„(G/K), and in particular F normalizes НЯК. Since H n К =
HsnK, evidently e На11„(НлК), and in consequence //rKgR. Thus
F n НЪК g s„R = R. On the other hand, since F/K e 91, we have F r> HXK sn G, and
therefore F n HBK < GR = K. From these remarks we then deduce that
[F.HBK]<FnHsK<K.
Hence НЪК < C^F/K) < F by A, 10.6(a), and from this we conclude that Ня <
H n F n H~K < H n K. Thus Assertion (a) is true.
(b) By Part (a) the index |K : HB| is a я'-number, and if J denotes the subgroup
<(HB)G>, it is clear that J < К and that K/J < O„m On the other hand putting
L/J = O,.(G/J), we have L e RG„- because J e s„(K) S R. Hence L e R by (1.25)0,
and therefore L < K.
574 IX. Fitting classes -examples and properties related to injectors
The K„(ft)-radical of an arbitrary finite group is now obtained by applying the
description of (1.27)(b) to the soluble radical of G.
Exercises
1. Find Fitting classes ft and (5 such that fts® and L„(ft) £ L„(G) for some set я
of primes. Compare with the corresponding behaviour of the operation K„( ).
2. Show that, whereas Kn(ft) n K„(®) = K„(ft r> ®), the corresponding equation for
) does not hold for all Fitting classes ft and ®.
3. (Brison [2]). Let ft and ® be Fitting classes. Prove that for all sets я of primes the
equation K„(ft О ®) = К„(ft) О Кг((5) holds, whereas for suitable choices of я the
corresponding equation for L„( ) does not hold, even for ft = ® = 91.
4. Let E be a finite simple non-abelian group, and let ft = (s„, N0>(E). Then
Char(ft) = 0 <= <r(ft).
5. Let ft be an s-closed Fitting class, and let H < G. A necessary condition for H < GR
is that: <Я, №> e ft for all g e G. Show that this condition is not sufficient
(although by A, 14.11 it is sufficient when ft = 91„).
6. (Lockett [1]). Let ft (£S) be a Fitting class with the property that whenever
a soluble group G is the product of normal subgroups and N2, then F =
(F n Ai)(F n N2) for each ft-injector F of C. Show that ft = S„ for some set я of
primes. (Compare this result with Theorem IV, 5.3.)
2. Constructions and examples
This is a reference section and is not intended to be read as part of the continuing
narrative. It is a repository of general methods for constructing Fitting classes, and
from it we draw examples which will be used throughout the remaining chapters to
illustrate the existing theory of Fitting classes and to test the limits of its validity. We
hope that it may also provide a useful source of counterexamples for those engaged
in research in this still developing field. It contains a fairly representative selection of
known Fitting classes, with two important exceptions: the Dark construction, to
which we devote two sections at the end of this chapter; and the classes defined by
the Fitting pairs of Berger, Laue, Lausch, and Pain, which are dealt with in Chapter X,
Section 5. Except where otherwise stated, the universe throughout this section is
assumed to be G.
The first construction that we describe provides a useful upper bound for the join
of two Fitting classes.
Construction A (Hauck—see Cusack [1]). Let ft and G be Fitting classes, and let
<r = {p g P: Zp is a composition factor of G/Ga for some G e ft}, and
т = {p g P: Zp is a composition factor of G/Gg for some G e ®}.
2. Constructions and examples
575
Further, let я be a set of primes containing an t, and then define
P 4(S, ®) = (G: G/(GBG(f)) g 91,).
(2.1) Theorem (Hauck). The class N&, &)defined in(2.a} is a Fitting class containing
Д and <5.
Proof. Let К < G e N„(g, ®). Then K/(K n GBGa) = KGBG(6/GBG(6 e s.91, = 91„.
Furthermore we have КЯКЙ = (К n GB)(K n G(6) < К n G,Gffi, and
[K,(KfiGjGe)]S[K, GB][K, Gltl]<(Kr Gj)(KnG„J. Thus (КпСяСй)/КвКв
is a central factor of K. Next observe that K(6 = (KnG(tl)(r, <(KnGBG(6)(6 < Km.
By definition of a, we have GBG,fl G ® 0 T,, where 3E„ is the Fitting formation
of all groups whose soluble composition factors are о-groups. Therefore К n G BGlfl g
s„(® 0 I„) = ® o Hence (К nGBGlf,)/K(r, = (К nGBGW)/(K nGBGa)(6 gI,. A
symmetrical argument for r then allows us to conclude that (K n GBG(f,)/(KBK(6) g
I„nl, S X„. Thus we have shown that К/КВКЙ is a nilpotent я-group, and
N„(g, ®) is therefore s„-closed.
To prove No-closure, let G = MN with M,N <G and M,N e N„ffi, ®). Let M* =
MBMV, and N* = NBN(r,. By supposition M/M* e 91„ and therefore MN*/M*N*
M/(M ri M*N*) e q(M/M*) S 91,. Similarly we have NM*/M*N* e 91„ and conse-
quently G/M*N* G n„91„ = 91„. Since GB > M^N^ and G((i > M^N®, we conclude
that G/GBGe g q(G/M*/V*) £ 91,. Hence G g A',(g, ®), and therefore N,(g, ®) is
N0-closed. □
Remarks, (a) It is clear from the proof of (2.1) that if an arbitrary <Q, E,>-closed
Fitting class is substituted for 91„ in (2.a), then the resulting class is again a Fitting
class.
(b) In Exercise 2 below we suggest an example to show that if g and ® are Fitting
classes, then the class
I = (G:G = GBGa)
is not in general a Fitting class. However, if g v ® denotes the smallest Fitting class
containing 5 and ®, it is clear that
I c g v ® S Nan(%, ®).
In particular, if о т = 0, then these three classes coincide.
The next construction is dual to the construction for formations described in
IV. 1.2.
Construction В (Hauck [3]). Let g be a Fitting class and © an arbitrary class of finite
groups. Then define a new class g | ® as follows:
(2.0) gf® = (G6G:InjB(G)S®).
576 IX. Fitting classes—examples and properties related to injectors
It is obvious from this definition that
(gn£)f(®nS)= gf® = gf(gnG).
Furthermore, if® is s„-closed, then clearly gf G is also s„-closed. However, g f ® is
not in general N0-closed even if G is a Fitting class; for example, the class 91 f G(
does not contain Sym(3) x Z2, which is the normal product of two copies of the
(91 f >3 , (-group Sym(3).
In order to justify this construction, we now describe two special situations in which
g f ® is nevertheless a Fitting class. The first of these shows that the Lockett operator
L„( ) arises as a special case of this construction.
(2.2) Proposition. Let n S P, and let ft be a Fitting class of finite soluble groups.
Then
(a) gGjg = L„(g),and
(b) G„fg = N,(g).
In particular, gS„ f g and G„ f g are Fitting classes.
Proof. Let ® = gS„, and let V be a ©-injector of a group GeS. Since © = ®S„,
an application of (1.15)(d) shows that V has я'-index in G. If GegG„f g, then
obviously V e Injs(G), and consequently G e L„(g). Conversely, if G 6 L„(g), then
an g-injector of G is evidently also an gS„-injector, and so G g gG„ f g. Hence
gS„ f g = LP(g). The remaining assertions follow at once from (1.15)(a) and (1.25)(a).
□
(2.3) Theorem (Hauck [3]). Let X and 9) be Fitting classes of finite soluble groups,
and let я S P. Then (XS„)f ('£)£„.) is a Fitting class.
Proof. Let ® and § denote the Fitting classes XG„ and 9)G„. respectively, and
put g = G f Since it is clear that g is s„-closed, it remains to prove that
3 = N0g. Let G be the product of two maximal normal subgroups Nt and N2
which belong to g, and let V be a G-injector of G. Let ie {1,2}. Since
Pn N, G Inj(6(N,), we have Pn N, g f>. If (|G: NJ, |G: NJ) = 1, then by A, 1.6(c) we
have V = (Pn NJ(F n N2) g n0§ = §, and consequently Geg. Therefore we can
suppose that IG: NJ = |G: N2| = peP, and further that G = I'N, = VN2, for if
F < N( for some i, then I'g§ and again Geg. Since F n N, <• Vh we can further
suppose that Fn N, = Pn N2, for otherwise P = (Pn NJ(Pn N2) g f>. Since the
subgroup PnN], as an XS„-injector of Nl; contains a Hall я-subgroup of N,
by (1.15)(d), and since V r\ N2 = V N2 < Nt r> N2, it follows that the prime p =
IN,: N, n NJ belongs to я'. But Pn N, is a normal ^-subgroup of P of index p, and
therefore V g = £>; consequently, in any case we have Geg. Hence by II, 2.11 (b)
we conclude that g = Nog. □
For the benefit of the next two constructions we recall from Definition IV, 4.9(b)
that a Baer function associates with each simple group J a class f(J) of groups such
that when J г Zp for some p e P, then /(J) is either empty or a formation. A chief
2. Constructions and examples
577
factor H/K of a finite group G is /-central if Autc(H/K) e f(J). where J is the
composition type of H/K. The expression “/-hypercemral m G ’ will thTn have its
obvious meaning when applied to a normal section of G.
Construction C (Gaschiitz). Let I and 9) be Fitting classes with I с and let
R =. « denote the function which assigns to each finite group G the characteristic
section R(G) = Gvl/Gj of G. We then define an associated class HR( f, R) of groups
as follows:
Pt) HR if, R) = (G e G: R(G) is /-hypercentral in G).
Remarks on Notation, (a) The letters HR stand for “Hypercentral Radical (section)”.
(b) For a universe S other than G, we shall use HR(/ R) also to denote what is,
strictly speaking, the class HR{ f. R)nfB.
(c) If / is a formation function in the sense of Definition IV, 3.1(a), we shall regard
/ as a Baer function by setting /(Zp) = /(p) for all p e P and /(J) = ГЪ, ,,/(p) when
(2.4) Theorem. Let I and /) be Fitting classes such that 3i S 'J, and let p =
{p6 P:pIlG/G’GjI for some Ge'!)}. Let f be a Baer function which fulfils the
following two requirements:
(a) For all J 6 3. the class f(J) is either empty or a Fitting formation;
(b) f(J) + 0 whenever |J| = p e p.
Then the class HR(f, R) defined in (2.y) is a Fitting class.
Proof. Put g = HR(f R). We prove in turn the two closure properties.
^-closure: Let N<Ge“fi. By (l.l)(a) we have R(N) = (N n GV)/(N n GJ =
(N ci G^JGj/Gj, and this last group is a normal subgroup of G/Gj contained in R(G).
It follows from the definition of g that G acts /-hypercentrally on the normal section
R(N) of G, and by refining a G-chief series through N n G?l and N c Gx to an N-chief
series, it follows easily that N induces on N-chief factors within the normal section
R(N) groups of automorphisms, each of which belongs to QS./(J) = f(J) for the
appropriate J 6 3- Consequently N e J, and g is s„-closed.
N0-closure: Let N, and N2 be distinct maximal normal g-subgroups of a group G.
By II, 2.11(b) it will suffice to show that Geg, in other words, that R(G) is/-
hypercentral in G. Set R = (IVJ^fWj)^ and S = (A'JjfA'iIr-
First we show that GV/G3R is/-hypercentral in G. By (l.l)(b) we have Gv/R <
Z(G/R). Therefore all chief factors of G between Gw and GXR are central, and since
the prime divisors of GV/GXR clearly all belong to p, it follows that GV/GXR is
/-hypercentral in G. , . , , „ _, „ ~
It remains to show that GXR/Gj is /-hypercentral in G, and because GjR/G.
R/(G* c R) and GrcR > S, it will be enough to show that R/S is /-hypercentral in
G. Let L = (N2)v ri S(N^}. Then R/S(N^ S (N2}v/L and L > (N2)x. Moreover,
578 IX. Fitting classes—examples and properties related to injectors
since [(Л'2)ц, iVJ < L, the groups G and N, induce isomorphic groups of auto-
morphisms on (N2)v/L, which is part of the section K(/V2). Therefore R/S(Nt)v is
/-hypercentral in G. Similarly the section R/S(N2)V is /-hypercentral in G. Hence, if
D = S(N, )vlnS(A'2)vl, we can conclude that R/D is also /-hypercentral in G.
Finally, we must deal with the normal section D/S, which is contained in
S(Ni)^/S s for i = 1,2. Thus, if H/K is a G-chief factor in the section D/S,
there is a G-isomorphic chief factor H,/K, in the section (1Ч)ч)/(/Ч)х. By A, 4.13(c)
this section HJK, is a direct product of minimal normal subgroups of HJK,, and
these can be regarded as Л',-chief factors of Е(Л',). Since Л', e ft, it follows that
Ni/Cf/XHj/Kj) e R0/(J) = /(/)> where J is the composition type of H/K. Setting
C = CG(H/K), we then have
C/C = (NtC/C)(N2C/C) = (N,/(N, n CYi(N2/(N2 n C))
= (N1/CNl(H,/KMN2/Cfll{H2/K2)) e n0/(J) = /(J).
Thus we have shown that K(G) is /-hypercentral in G, and hence that ft is No-
closed □
We now illustrate Construction C by studying some special cases in more detail.
(2.5) Examples, (a) Set I = (1) and 'J = for some л £ P. Then K(G) = O„(G). If
f is the Baer function defined by f(J) = (1) for all J 6 3. then we obtain
HR(f. R) = (Ge(£: On(G) < ZJG)),
which is therefore a Fitting class by (2.4). If л = 0, then HR(f, R) = &, and if л = P,
we obtain HR(f, R) = 91. Suppose that 0 л ф P, and write ft for HR(f, R). Then
ft has the following properties:
(1) Qft + ft + Sft, ft £ 91 0 ft and 6p О ft £ 6p 0 (E, 0 ft: Let p e л, q e л'. and
recall from B, 10.7 that the group E(q/p) has a faithful irreducible module N over Fe.
Let S denote the semidirect product [W]E(q/p). Since O„(S) = 1, then certainly
5 e ft, and hence E(q/p) e (Qft n sft)\ft. Obviously S g ft\(9i <> ft) and S e (Gp о ft)\
(6P <’ 6, О ft).
(2) ft + E»ft: With S as defined above in (1), let P denote the principal inde-
composable FpS-module such that P/J(P) = (Fp)s the trivial FpS-module. By B, 6.18
the module P has a composition factor not isomorphic with (Fp)s, and therefore
[P]S6Eoft\ft.
(3) ft = Roft: Let N, be a normal subgroup of a group G such that G/Nt e ft for
i = 1,2 and N, n N2 = 1. Then for i = 1,2 we have O„(G)/(O„(G) r> N, ) s OJONj/ty <
O^G/Nj) < Z^iG/Ni). Hence O„(G)/(O„(G) N() is G-hypercentral, 'and it follows
easily that O„(G) is hypercentral in G.
Finally, we characterize the g-radical of a group G, asserting that Gg is the
intersection of the centralizers in G of the G-chief factors below O„(G). Denote this
intersection by K. Since O„(K) =Kr Or[G), it is clear that O„(K) is hypercentral
in K, and therefore that К < Gg. To prove the reverse inclusion, let S/T be a G-chief
2. Constructions and examples
579
»Tr?^-r°”(G)uin a SerieS paSSing through °"(G> and throu^h C«"O,(G) =
O„(GS). If S/T is above O„(GS), we have [S, G.,] < [0„(G), Gs] = O,IC)r G, < T,
and so Gs centralizes S/T. On the other hand, if S/T is below O„(GB) then S/T is
a product of minimal normal subgroups of GB/T and is therefore centralized by G.
because GB e g. Consequently GB < K, and the assertion is justified.
(b) For our second illustration of Construction C we take R(G) = O„(G) as in the
preceding example, and define a Baer function g by setting g(J} = for all J e 3. If
G is a finite group, and if К denotes the intersection of the centralizers in G of the
G-chief factors below O„(G), it follows from IV, 6.9 (with g = 91„in the theorem) that
K/QfOJG)) e 'Jl„. It therefore follows that
HR(g, K) = (G e 6 : G/Cc(O,(G)) e (£„)
By (2.4) we know that HR(g, R) is a Fitting class, and it is clear that
£ n HR(g, R) = (G 6 to : O„(G) is centralized by some H e Hall„.(G)).
It is not hard to verify that this class HR(g, R) coincides with g6„, where g is the
Fitting class described in the preceding example, and that, as before, it is R^-closed
but not Q-, S-, or Еф-closed when 0 я P.
(c) (Blessenohl and H. Laue). Let I = (1) and 'J = 6, so that R(G) = G for all
Ge6. Let h be the Baer function defined as follows:
(2.Й)
h(J} =
(1)
P„(J)
for all J e 3 n 21. and
for all J e 3\2l.
When J is a non-abelian simple group, then Dtl(J) is a Fitting formation by II, 2.13.
Therefore by (2.4) the class HR(h. R) is a Fitting class and, by IV, 4.17, also a solubly
saturated formation; we denote this class by 23. In order to characterize 23 in terms
of the description in Blessenohl and Laue [2], we need the following facts.
(2.6) Lemma. Let h be the Baer function defined in (2.<5), and l.t N be a minimal normal
subgroup of a group G. Then any two of the following statements are equivalent.
(a) The automorphisms of N induced by elements of G are inner,
(b) N is simple and G = NCG(N);
(c) N is h-central in G.
Proof Put C = Cc(N), and let J denote the composition type of N. First we prove
the implication: (a) => (b). Suppose that Statement (a) is true. If J e 21, then Inn(M = 1,
and so G = C = NC. In this case N < Z(G). and therefore N S Zp = J. On the other
hand, if J ф 21, then Z(1V) = 1, and so Co N = 1. Hence, in this case,
/V S NC/C < G/C £ lnn(M s iv,
N, it follows that No < NC = G, and from the
= N0S J. Thus, in either case, Statement (b) is
and consequently G = NC. If No <
minimality of N we conclude that N
true.
580 IX. Fitting classes—examples and properties related to injectors
Next assume that Statement (b) holds. Then
[1 if J e 31
Autc(N)SG/Cc(N)S{N
and therefore Statement (c) holds. Finally, assume the truth of Statement (c). If
J e®, then h(J) = (l), and so Autc(N) = 1 = Inn(N). If J e 3\4I, then N s
NC/C < G/C e d0(J). But if A is a normal subgroup of a group D e d0(J), it follows
from A, 4.13(b) that D = К x K* with К, K* e d0(J). Thus G/C = NC/C x N*/C,
and therefore [N, N*] < N n N* = 1. It follows that N* = C and hence that G/C =
A'C/C = N = Inn(JV). Hence Statement (a) is a consequence of Statement (c), and the
circle of implications is complete. □
From this lemma the following characterization of the Fitting class ® = HR(h, R)
can be at once inferred:
® = (G e G: G induces inner automorphisms on each of its chief factors).
For further analysis of the class ® we need the following observations.
(2.7) Remarks.
(a) If G e ® and F(G) = 1, then G = Soc(G) 6 d„(3\®).
(b) ® is a solubly saturated formation but is not r^-closed.
Procf. (a) We proceed by induction on |G|. Let N < G, and let C = Cc(N). Since
F(G) = 1, we have Z(N) = 1, and so G = N x C by the implication: (c) => (b) of (2.6).
Since C e s„® = ®, and since F(C) = F(G) r> C = 1, it follows by induction that
C e d0(3\9I). The conclusion of Part (a) is now clear.
(b) As noted earlier, Theorem IV, 4.17 implies that ® is solubly saturated. On the
other hand, it is not difficult to see that ® # Еф®. For example, it can be shown (see
Appendix [1] that if p|| J|, a non-abelian simple group J has a faithful module M over
Fp such that there exists an extension E such that (i) l-»Af-»E->J->l and
(ii) M < Ф(Е). In this case E e еф®\®. □
In view of the fact that 91 = G n ® and that the S-radical of an arbitrary finite
group has similar properties to the Fitting subgroup of a soluble group, we can regard
the class ® as a generalization of the class of nilpotent groups. It was first considered
by Bender in [1]> where he introduces the so-called generalized Fitting subgroup F*(G)
of a finite group G, defined as follows:
F*(G)/F(G) = Soc(Cc(F(G))F(G)/F(G)).
In fact, it turns out that F*(G) is just the ®-radical of G, and, in particular, that
® = (G e G: G = F*(G)). To see this, put F* — F*(G), and observe that (i) F*/F(G)
is a direct product of non-abelian simple groups and (ii) F* = F(G)Cf«(F(G)). From
these observations it is clear that F* e ® and hence that F* < G®. On the other hand,
each soluble normal subgroup of G® is nilpotent, and so F(G®) = (G®)s. It follows
2. Constructions and examples
581
from (2.7)(a) that the quotient Ga/F(Ggl) is a «-group with trivial Fitting subgroup
and is therefore a direct product of non-abelian simple groups. Since 91 c «, we
have F(G) = F(Gi8), and therefore, in the notation of IV, 6.8(b), it follows that
F(G) < Z4I(GV). Consequently IV, 6.10 implies that GB/Cc (F(G)) e 91 and we
can therefore conclude that Ga/(F(G)CCe(F(G))) e 91 n d0(3\91) = (1). Thus Gs =
F(G)CCe(F(G)) < F*, and our assertion that F*(G) = Gs is proved.
One of the important properties of the Fitting subgroup in the universe S that is
fulfilled by the generalized Fitting subgroup in the universe G is the following:
Cc(F*(G)) < F*(G).
A proof of this fact, together with other descriptions and properties of F*(G), can be
found in Chapter X, Section 13 of Huppert and Blackburn [2]. We shall have more
to say about the class SB in Section 4, where, in particular, we shall prove that every
(not necessarily soluble) finite group possesses a unique conjugacy class of SB-injectors
(see Theorem 4.15).
In the next construction, which resembles Construction C, the radical section
G^/Gx is replaced by a smaller section associated with a certain socle. Recall that the
л-socle, Soc„(X), of an arbitrary finite group X is defined to be the product of all
minimal normal л-subgroups of X.
Construction D (Gaschiitz). Let X and 9) be Fitting classes with X s 9), let n s P,
and let S = SI 4I „ be the function which assigns to each finite group G the following
characteristic section of G:
S(G) = (G^/Gx) n SocJG/Gx).
If f is a Baer function, we then define a class HS(f S) as follows:
(2.c) HS(f S) = (G e G: S(G) is /-hypercentral in G).
(The letters HS are intended to suggest “Hypercentral Socle (section)”, and the
remarks on notation in Construction C apply equally well here.)
(2.8) Theorem. The class HS(f, S) defined by Equation (2.r) above is a Fitting class,
provided that
(a) /(J) is either empty or a Fitting formation for each J eg, and
(b) f(J) # 0 for each such J whose order belongs to the set
p = {p e P : pllG/G'GxI for some G e 9)}.
Proof Write Я = HSif, S). First we show that g = s.g. Let N < G e g, and let
M/Nx be a minimal normal п-subgroup оГВД contained in Nv/Nx and of compos.-
. .; c and = N Сч) G, for each о 6 trie
582 IX. Fitting classes—examples and properties related to injectors
Me/Nx e then by definition we have
(2.0 K/Nx < Socn(N/Nx) n Nv/Nx.
Let L/Nx be a minimal normal subgroup of G/Nx contained in K/Nx. Since
L n Gx < N n Gx = Nx, we have L/Nx LGX/GX < Gv/Gx, and therefore LGX/GX
is a G-chief factor of composition type J in the section S(G). Consequently
AutN(L/A'I) e s„/(J) = /(J). Then, if L0/Nx is a minimal normal subgroup of N/Nx
contained in L/Nx, it follows that AutN(L0/Nx) e Qf(J) = f(J], From the definition
of К we see that L0/Nx is Al-isomorphic with M"/Nx for some geG, and so
NutK(M/Nx) S AutN(M9/A'jc) e f(J). This proves that Neg and hence that g is
s„-closed.
To complete the proof we now show that g = Nog. Let Nj and N2 be maximal
normal subgroups of a group G such that G = Nl N2 and N, e g for i = 1, 2. By
II, 2.11 (b) it will suffice to prove that G e g. Let M = M/Gx denote a minimal normal
zt-subgroup of G/Gx contained in Gv/Gx and of composition type J. For i e {1,2}
we have
(2.//) (А4)$/(А4)х = (A/J^Gx/Gx G^/Gx-
Let Kj = M n^Gx/Gx), i = 1, 2. By the choice of M, either M < Kt or Kt = 1.
И Kj = 1, then M n Nj = M r> G^c\ Nt — M n < Gx, and it follows that
Nj < Cc(M). Therefore, if K, = K2 = 1, we conclude that M is centralized by
Nj N2 — G. Hence in this case M = Zp for some p e P, and, in particular, J = Zp.
Since M e 9), we have Zp = M/Gx = My/Mx, and therefore/(J) 0 by Hypothesis
(b). Consequently M is /-central in this case. Next suppose that M < K,. Since M
is completely reducible under the action of A', and since A, e g, it follows from
that Nl/Cfji(M)eRBf(J) = f(J). If K2 = 1, we conclude that G/Cc(M) =
NjCG{M)IC^M) s Nj/СцДМ) and hence that Autc(Al) e/(J). A similar argument
applies if M < K2_and Kj = 1. Finally, if M < K, n K2, we have Autc(M) =
П?=1 N,Cc(M)/Cc(M) g n„/(J) = /(J). Hence G e g. □
Remark on Notation. To draw attention to the set n of primes involved in this
construction, we may sometimes write S) instead of HS(f, S).
(2.9) Examples.
(a) Let ® be a fixed <s, Q, N0>-closed universe, let 3E — (1) and 9) = ®, and let
/(J) = (1) for all J e 3- The special case of the class S) thus defined will appear
quite often in the sequel, and therefore we denote it by the fixed symbol 3”; further-
more, if n = P, we simply write 3 instead of 3P- Thus we obtain
3” = (G e ®: SocJG) < Z(G)), and
3 = (G e 9?: Soc(G) < Z(G)).
By (2.8) the class 3” is a Fitting class, and if zt = 0, then 3" = We now point out
some useful facts about 3"-
1. Constructions and examples
583
(1) If ® = S, then
easily seen to be false
group.
3 A 1'3'’: p e a}. This is obvious for the universe G and is
m general for a universe which contains a non-abelian simple
a"^G6®’.the ,3?radical of G is Q(Soc,(G)). To see this, write
c cc(»oc„(O)j, and let N be a minimal normal л-subgroup of C < G. Then
> — N 1 x • x №' < C for suitable e G, and every minimal normal sub-
group of C contained in <A'C> is C-isomorphic with №' for some i e {1,..., r}. Let
К be a minimal normal л-subgroup of G contained in <№>. By definition the
subgroup Ccentralizes K, and so №‘ < Z(C) for some i e {1,r}. Since Z(C) < G,
it follows that N < Z(C) and hence that Cc(Soc„(G)) < G3..
To prove the reverse inclusion, consider now a minimal normal л-subgroup M of
G- If M Gj., then M n Gj. = 1, and therefore G3. < Cc(M). On the other hand, if
Л1 < Gj., then the subgroup M r, Z(G3.) is clearly normal in G and is non-trivial
because G3. e J". Thus M < Z(G3„) and again we have Gj. < CG(M). Consequently
G j. < Cc(Soc„(G)), and the assertion that these two subgroups coincide is justified.
(3) 3" = КоЗ" To prove this, let N>, N2 < G with N, n N2 = 1 and G/ty e 3" for
i = 1,2. Let N be a minimal normal л-subgroup of G. Since A\ n W2 = 1, there exists
a j e {1, 2} such that N n A} = 1. Then we have
N S NNj/Nj < Soc„(G/M) < Z(G/N.}.
G
Consequently N < Z(G), and G e 3"-
(4) If л Z 0, it is not difficult to construct examples which show that 3” is not
closed under any of the operations s, q, and еф.
(b) It is natural to ask whether the inclusion of the lower Fitting class X really
brings additional generality to Constructions C and D, or formulating the question
more precisely:
Given Fitting classes X and 'll with X S 9) and a Baer function f does there always
exist a Fitting class SB and a Baer function g such that if R = Rj% and R = R,i).so,
then HR(f, R) = HR(g, Rfl
With the help of the preceding example we can now give a negative answer to this
question. Let R = R„ 3, and let f(J) = (1) for all J e 3. In view of the characterization
of the 3-radical given in Part (2) of Example (2.9)(a), we obtain the following
description of the Fitting class HR(f R):
HR(/, R) = (G e G: Cc(Soc(G))/F(G) is hypercentral in G).
Put R = HR( f R), and suppose, by way of contradiction, that for some Baerfunction
g and some Fitting class SB, the class & has the form 3 = HR(g. R), where R = R(1
First we show that Char(SB) = P. If not, let p e P\Char(9B), and let q e P\(2, p}.By
A 20 8 and A 21 4 (with special pleading for the case p - 2) there exists an extra-
[Z(E), ()„(/))] = 1. Let S denote the semidirect product [E]f>. Then Soc(S) - Z(E),
f(J) = |
584 IX. Fitting classes—examples and properties related to injectors
and clearly S / g. Moreover, since p f Char(9B), we obviously have Sffi = 1 and
therefore 5 e HR(g, R) = g. This contradiction shows that Char(9B) = P.
If P e P and G e the class of primitive groups whose socles are p-groups, then
R[G) = 1 and Geg. Hence by supposition G e HR(g, R). Since Char(9B) = P, we
have Soc(G) < Gffi, and therefore the Fitting formation g(Zp) contains the following
class
I = (G/Soc„(G): G e ф" n 6).
But qI = <5 by A, 18.5(c); therefore S £ g(Zp), and consequently, since p was arbi-
trary, we have E £ HR(g, R) = g. Since the group S defined above belongs to G\g,
we have reached a final contradiction. Hence we conclude that g cannot be expressed
in the form HR(g, R), and therefore that new Fitting classes are obtained by allowing
the parameter X to be non-trivial in Construction C. This example can also be used
to show that the role of X is not redundant in Construction D either.
(c) For our final illustration of Construction D, we take X = (1), 9) = G, andzt = P;
thus S(G) = Soc(G) for all G e G. If /is the Baer function defined by
G forJegn'H
d0(J) for J e 3\91,
one can easily deduce from the proof of (2.6) the following description of the Fitting
class thus obtained:
HS(f, S) = (G e G: [Soc(G), Soc(G)] is a direct factor of G).
This is a special case of a construction described by Blessenohl and H. Laue [2],
which in turn is a special case of Construction D.
Before we describe the next type of Fitting class construction, we need the following
definitions, which will be extensively used in the sequel. The definition of a Fitting
pair is particularly important.
(2.10) Definitions, (a) Fix once and for all a set containing one and only one
representative of each isomorphism class of finite groups. Thus for each G e G there
exists a unique Go in such that G s Go. Then for each class I eG we define
Set(X) = I n S, and call Set(JE) the underlying set of X. Obviously the theory of classes
of groups could be equally well formulated in terms of their underlying sets, with
statements of the form “G e JE” replaced by “G is isomorphic with a group in Set(JE)”.
(b) If G and Ii are groups, a subnormal embedding of G in H is a monomorphism
a . G -> H such that Ga sn H; it is called a normal embedding if Ga < H.
(c) Let g be a Fitting class of finite groups. A pair (4, d) is called a Fitting pair for
g (or an g-Fitting pair) if A is an abelian group (possibly infinite) and
d: Set(g) - U {Hom(G, A) .Ge Set (g)}
is a map with the property that the image dc of each G in Set(g) is a homomorphism
2. Constructions and examples
585
from G to A fulfilling the following two conditions-
Construction E (Blessenohl and Gaschiitz [1]). Let g be a Fitting class, and let (A. d)
be an g-Fitting pair. Then define an associated class Я(Л, d) as follows:
Я(Л, d) = (G e Set(g): Gdc = 1).
(Recall that if SC e S, then (T) denotes the class of groups generated by
(2.11) Theorem. Let g be a Fitting class, and let (A,d} be a pair which satisfies
Property FP1 of Definition 2.10(c). Then the class Я = Я(Л, d) is a Fitting class If
G e Set(g), then
(i) Ся = Ker(dc),
(ii) [G, Aut(G)] < G«,
(iii) G/Ga is abelian,
(iv) Ga is Я-maximal in G, and
(v) Ga is the unique Я-injector of G.
Proof To show that Я is s.-closed, let A < Ge Я. Since Я £ g, we can find N
and G in Set(g) such that N = N and G s G, and the fact that N < G implies the
existence of a normal embedding a from N to G. From Property FP1 we then
deduce that
Nd,; = N(a ° dG) = (iVa)dg < Gdg = 1;
thus N = N e Я and Я is s„-closed.
To prove the N0-closure let G = N2 with A' < G and e Я for i = I, 2. Since
Ni, N2 and G are in g, we can find ‘barred’ isomorphic copies in Set(g) and, for
i = 1, 2, normal embeddings a, from to G such that G = (Ni)allN2)a2. From
Property FP1 we then deduce that
Gdg = (AIa1)(lV2a2)dg = (Ntd^HN.d^) = 1.
Hence G 5 Ge H,and it follows from II, 2.11(b) that Я = м0Я.
Next we show that Assertion (i) holds. Let G e Set(g), and put К = Ker(dc). Then
we can find a group К e Set(g) and a normal embedding a from К to G such that
Ka = K. Since Kdg = A(a ° dc) = (Ka)dc = KdG = 1, it follows that К e Я and
hence that К < G„. Now let G« S R e Set(g), and let a be a normal embedding from
R to G such that Ra = G„. Since R e Я, we have 1 = RdK = R(a ° dc) = G„dG. which
means that GB < A Hence G« = Ker(dc).
If G e Set(g) and a e Aut(G), then a is a normal embedding from G to G, and so
for each g e G we have gdG = (ga)dG by FPL Thus [g, a] - g (ga) e Ker(dG) — G«,
which implies that [G, Aut(G)] < G«. Hence Assertion (ii) is true, and Assertions (in),
(iv), and (v) follow directly from it. u
586
IX. Fitting classes—examples and properties related to injectors
(2.12) Remarks, (a) It will be noted that Property FP2 is not used in the proof of
(2.11).
(b) The use of underlying sets in place of group classes facilitates a rigorous
development of this material, and we have found it especially helpful in the treatment
of the Lausch group in Section 4 of Chapter X.
(c) Although the requirement that the group A in the Fitting pair (A, d) be abelian
is not used explicitly in the proof of (2.11), it is worth remarking that this is not an
independent axiom but rather a consequence of FP1 and FP2. To see this, let x, у e A.
By FP2 there is an element ge G e Set(g) and an element he H e Set(5) such that
x = gdG and у = hdH. Let G x H = D e Set(g), and let a and fi be isomorphisms of
G and H such that D = Ga x II fi. Then by FP1 we have
O, y] = lgadD, h[ldD] = [да, hfj]d„ = idD = 1.
(d) Requirement FP1 of the definition of an Д-Fitting pair ((2.101(c)) may be
replaced by the following condition:
FP1*: dG — a о dBforallG, H e Set(g) and for all subnormal embeddings a: G -»H.
Clearly FP1* implies FPL Conversely, suppose that FPI holds, and let a: G -» H
be a subnormal embedding. Then H has subgroups H, such that
a(G) = <H2< <Hr = H.
Since Hi sn H e ft we have Hj e Д, and so there exist groups Gj (= G), G2,. . ., G,( = H)
in Set(g) and normal embeddings a,: Gj -► Gj+1 (i = 1,..., r — 1) such that
a = a2 ° a2 ° ° ar_j.
Since dGi = о dGi t by FPI, it follows that dG = dGl = a, ° a2 ° ° a,_| ° dH =
a ° dH. Therefore FPI* is equivalent to FPL
Theorem 2.11 motivates the following definitions. Of particular importance is the
definition of a normal Fitting class. It will be studied in depth in Section 3 of Chapter
X, but will crop up in various contexts before then.
(2.13) Definitions, (a) If (4, d) is an Д-Fitting pair, we call the Fitting class R(4, d),
which is defined at the beginning of Construction E, the kernel of (4, d).
(b) Let X s G, and let 9) be a Fitting class. We shall say that 9) is normal in X
(or simply that 9) is X-normal) if
(a) (1) # 9) s x, and
(b) G4I is 9)-maximal in G for all G e X.
Obviously, if X = s„X and 9) is X-normal, then every group in X has a unique
?)-injector, namely the ?)-radical.
If Д # (1) is a Fitting class, it follows from (2.11), (iii) and (iv), that the kernel of
an Д-Fitting pair is an Д-normal Fitting class. To illustrate the significance of this
observation, we now describe a family of Д-Fitting pairs. They give rise to some
non-trivial Д-normal Fitting classes, one of which we shall then analyse more
closely.
2. Constructions and examples
587
(2.14) Examples (a) Let g be a fixed Fitting class of finite groups, and let n be a
non-empty set of primes. For each q e P, then define
e(q) = sup{n: q"|(p - 1) for some p e it}.
Thus e(q)e M и {0, co}. Next let A = /1(zt) denote the restricted direct product of the
groups q e P}, where Zr is the Priifer q-group. For each pen, the abelian
group A contains a unique subgroup of order (p - 1), and this we identify with the
multiplicative group Fp.
We now construct a map d so that (A, d) satisfies Property FP1. Let Geg. If Pi e it,
let C‘(G) = {M;,..., Л1;(0} denote the set of (abelian) p,-chief factors in a given chief
series of G. If g e G, let dy(q) denote the determinant of the linear map which g induces
on the [Fpi G-module Л/j. By means of the embedding of Fp* into A we consider d^g)
as an element of A, and then define ' J
n(i)
9dG = П П d№ 6 A
р,-ея j=l
where gdG = 1 if C‘(G) = 0 for all p; e n. Since G is finite, this product is well defined,
and by the Jordan-Holder theorem it is independent of the chosen chief series of G.
Because the determinant of a linear transformation induces a group-homomorphism,
it follows that dG is a well-defined homomorphism from G into A. It is easy to
deduce from these observations that the map d, although defined on the class g, is
invariant on each isomorphism class; or, equivalently, that if Gj, G2 e g, a: Gj -»G2
an isomorphism, then a ° dGi = dGl. In order to prove that this pair (Л, d) satisfies
Property FP1, it will therefore be sufficient to verify that if N < G e g, then
xdN = xdc for all x e N. Choose a chief series of G passing through N, and for p, e n
let C‘ = {M\,Л1'(||} denote the p,-chief factors in this chief series, numbered so
that {Afj,..., М^10} are precisely the ones below N. Let x e N. Since x induces the
identity map on chief factors above N, we have
(2.0)
m(i)
xdG = П fl
If M is a simple G-module, by B, 7.1 we have MK — Lt ® • • • ® L„ and so the
determinant of “x on M" is the product of the determinants of “x on L," as i runs
from 1 up to t. Therefore, if we refine the given G-chief series below N to a chief series
of /V, each term dy(x) in (2.0) can be replaced by the product of the determinants of
the linear maps induced by x on the A'-chief factors of M‘ in the refined series, and
it is clear that the right-hand side of Equation 2.0 then becomes the required
expression for xdK. , . . .
Thus the pair (A d) satisfies FPL and the associated class «(A, d) is a Fitting class.
We shall denote this class by T(n). Before leaving this example, we show that if
iii2 c g then the pair (Л, d) that we have just constructed also satisfies FP2 and is
therefore an g-Fitting pair. Let x e A. Then there exists a finite subset {p. p }
of it such that x = x, ...x„ with each x, e Fp}. With respect to the action of <x;>
588 IX. Fitting classes—examples and properties related to injectors
on the additive group F* by field multiplication, form the semidirect product Gf =
[Рр+]<х,> for i = 1,..., n. Since each G; is in 912 £ 8, we can find a group G e Set(8)
and an isomorphism g -»g from G onto the direct product G = G, x • • • x G„. If
x- = (1,1, x;, 1,..., 1) e G and Ug — x1...xn, then from the fact that dc satisfies
FP1 we conclude that gdG = Xj... x„ = x. This proves that (4, d) satisfies Condition
FP2 of Definition 2.10(c).
Remark. A more general version of the above class has been studied by Zappa [2].
It is constructed in the same way as T>(n), except that only those chief factors which
occur in a fixed radical section R(G) = Gz/Gy are included in the sets C'(G).
(b) We now study a special case of the preceding example in more detail. Let 8 = G
and л = {3}. Then the class Ф = T({3}) consists of all finite groups G such that
P|"=1 Det(g on Л1,) = 1 for all g e G, where the product is taken over the 3-chief factors
Mu..., M„ of a given chief series of G. We describe some of the properties of the
class T>. Since the corresponding group A is just F3X, we have
(1) |4| = 2, and therefore | G/GiJ = 1 or 2 for all GeG.
(2) If G = Sym(3), then Gr = Alt(3), and from (1) it follows that
Gr x Gj, < (G x G)j,.
Therefore the radical of a Fitting class does not in general respect direct products.
This observation is the starting point for the theory of Lockett sections, which we
discuss in Section 1 of Chapter X.
(3) If|G/O3(G)| = 2, thenG e if and only if the number of eccentric 3-chief factors
in a chief series of G is even.
(4) qT> sT>. To justify this assertion let P be an elementary abelian group
of order 9, let a be the inverting automorphism of P defined by x° = x-1 for all x g P,
and let G = By (3) we have G e Ф, and therefore Sym(3) e (sT r> qT)\T.
(5) T> # r0T>. To see this, let Q be an elementary abelian group of order 27, let (3
be the inverting automorphism, and let H = Then by (3) we have H £ T, but
if G is the group defined in (4), then clearly H e r0(G) £ R0T>.
(6) Ф # ефТ>. This is most easily seen by taking R = Z3 x Z9t letting у be the
inverting automorphism, and putting К = [R](y\ Then by (3) we have К/Ф(К) e T>
and К ф Ф.
(7) If 8 = S3, then 8T n0(8T). In proving this we shall have demonstrated that
in general the class product of two Fitting classes is not again a Fitting class.
Let L = Sym(3) x Sym(4), and let R = £t. Since Sym(3) £ T>, we have L £ T>, and
therefore |L: R| = 2 by (1). Because Sym(3) e 8T and R e D £ 8^\ we therefore
conclude that L = Sym(3)R e n0(8T). But if К < L and К g 8, we must either have
К = 1 (in which case L/K s L ф Г>), or |K| = 3 and then L/K ^Z2x Sym(4) ф D.
Hence L $ 8^-
(8) If 8 = S3, then T> £ 8 0 To prove this assertion, which justifies a remark
made in (1.11), let N be a simple F3Sym(4)-module such that Ker(Sym(4) on N) =
Alt(4), and let T = [A]Sym(4). Then clearly T e T>\(8 0 ^)-
2. Constructions and examples
589
Construction F. The method of constructing Fitting classes described here is due to
Cossey and Kanes [1]. The idea is to define a class of p-soluble groups by specifying
properties of certain modules associated with the p-chief factors. This is reminiscent
of the local definition in the theory of formations, and so it is not surprising that
the method often yields Fitting classes which are also formations (called “Fitting
formations ). The construction has evolved from early attempts to find examples of
non-saturated Fitting formations (see Hawkes [5] and Berger and Cossey [2]), and
it also exploits ideas of Gajendragadkhar [1] and Isaacs [3] from the theory of
characters in a surprising way.
We present this construction in two stages. First we postulate a class of modules
(called a Fitting family’) satisfying four axioms. Second we show how to associate
with each Fitting family a sequence of Q-closed Fitting classes, among which there
is always at least one formation. Afterwards we analyse some concrete examples
and describe a general method of generating Fitting families using the idea of
л-factorability for modules.
(2.15) Definition. Let К be a field, and assume that, for each group G in a suitable
fixed universe 33 (usually the class of soluble groups or the class of p-soluble groups),
we are given a class 991(G) of simple KG-modules. Then the class 932 = (JCeB99l(G)
is called a Fitting family if it satisfies the following closure properties:
FF1: If V e 991(G) and N < G with N < Ker(G on V), then V, defined as a G/N-
module by the equation v(gN) = vg, belongs to 9Jl(G/N). (We say 9J2 is “closed
under deflation” of modules.)
FF2: If V e 931(H) and e: G -»H is an epimorphism, then К defined as a G-module
by the equation vg = v(eg\ belongs to 991(G). (We say 931 is “closed under
inflation’’ of modules.)
FF3: If V e 931(G) and A' < G, then the composition factors of V„ are in 931(H). (By
iteration, the same condition holds for subnormal N, and so we say 931 is closed
under subnormal restrictions”.)
FF4: If Nt and N2 are distinct maximal normal subgroups of G and if V is a simple
KG-module such that the composition factors of FN1 all belong to 991(14) for
i = 1, 2, then V belongs to 931(G). (We say that 931 is “closed under normal-
product extensions”.)
Remark. We note that, if 991(G) is non-empty, then Axioms FF2 and FF3 together
ensure that 991(G) contains the trivial module Kc.
To describe our candidates for Fitting classes associated with a given Fitting family
of modules some special notation will be helpful.
(2 163 Definition. Let r be a natural number, p a prime, and К an extension field of
F If G is a p-soluble group, the symbol 11(G) will denote the class of all KG-modules
which arise as composition factors of modules of the form
(V, ® •® K)®K-
590
IX. Fitting classes—examples and properties related to injectors
where each Vi is isomorphic with some p-chief factor of G or with the trivial module
(Fp)c, and where the tensor products are over Fp. (Thus 34(G) S IrrA(G), the class of
all simple KG-modules, and in particular, Ttp(G) is the class of all modules occurring
as p-chief factors of G, denoted also by Chiefp(G).)
(2.17) Lemma. Let К be a field of characteristic p > 0, and let N be a normal subgroup
of a p-soluble group G.
(a) If W e L'K(N)y then ’L'k(G) contains a module V such that If has W as a direct
summand.
(b) If V e 24(G) and W is a composition factor of then W e TrK(N).
Proof, (a) By definition of there exist modules ,..., Wre Chiefp(N) о (Fp)N
such that IT is a composition factor of (Wl ® • • • ® Wr) ® K. Since all p-chief factors
of N appear in a refined chief series of G below N by the Jordan-Holder theorem,
and since G-chief factors are semisimple on restriction to N by B, 7.1, it follows that
we can find Ц,..., Vr e Chiefp(G) о (Fp)e such that (Ф W<* for i = 1,..., r.
Thus
(Ц ® ® К)л = (Ц)л ® • ® (K)n
= (wt ® ••® и;)® w*
for a suitable submodule W* by the distributive law for tensor products. Consequently
W is a composition factor of ((Fj ® ® I')® f'l/v- Since by the Jordan-Holder
theorem all composition factors of ((Г] ® • • • ® K) ® K)n can be obtained by restrict-
ing to N a composition series of (Ц ® • • • ® If) ® K, Assertion (a) now follows (with
a further appeal to B, 7.1).
(b) Let V be a composition factor of (F, ® ® FJ ® K, where by the Jordan-
Holder theorem we may suppose without loss of generality that the non-trivial
Vj are p-chief factors of G in a chief series passing through N. Then If is
isomorphic with a section of ((14)л ® ••• ® (1^)л) ® K. If a non-trivial Ц lies
above N, then N < Ker(G on Ц), and so (Ц)л. is a direct sum of trivial simple
/V-modules. If, on the other hand, V, is below N, then (И)л *s a direct sum of
p-chief factors of N by В, 7.1. Thus it follows from the distributivity of ®
over ф and from the Jordan-Holder theorem once more, that a composition
factor of VN is isomorphic with a composition factor of a module of the form
(Gj ® ® Ur) ® K, where each If is either trivial or isomorphic with a p-chief
factor of N. □
The above lemma provides the key to proving that the classes of groups which we
are about to create out of a given Fitting family of modules all have the closure
properties of a Fitting class.
(2.18) Theorem (Cossey and Kanes [1]). Let r > 1, let К be a field of characteristic
p > 0, and let УЯЬе a Fitting family of modules over K. Set
2. Constructions and examples 591
(2j) 3-(r> = (G: G is p-soluble and 3i(G) £ 'JJi(G)).
Then 3(r, 9Ji) is a Q-closed Fitting class and is even a formation ifr = l.
Proof. Set I = 3(r,®i). First we prove that 3 = s>,3. Let N< Ge3, and let
w E By (2.17)(a) there is a Kin 3^(G) such that TV is a direct summand of VN.
Since V e ®i(G), it follows from Axiom FF3 that W e ®l(N). Thus 3UA) s 'JJi(N)
and therefore N e 3.
Next we show that 3 is N0-dosed. Assume that G = N, N2, where Nt and N2 are
maximal normal 3-subgroups of G. By II, 2.11(b) it will suffice to prove that G e 3.
Let i e {1, 2}, let V e 3g(G), and let W be a composition factor of By (2.17)(b)
we have W e 3J(Nj) and therefore W e ®l(Af) by the assumption that N, e 3. From
Axiom FF4 it then follows that V e 9R(G) and hence that G e 3 as required.
We will now justify the claim that 3 = q3. Let N < G. By regarding the modules
in 3i(G/N) as G-modules by inflation, the definition of 3'K( ) clearly implies that
3j.(G/A) s 3^(G). Thus, if G e 3, from this viewpoint we have Z'K(G/N) s 9Ji(G).
But now viewing the elements of 3i(G/A) more naturally as G/N-modules, we can
deduce from Axiom FF1 that 3^(G/N) £ W1(G/7V). Therefore G/N e 3, and we have
shown that 3 is Q-closed.
Finally, we consider the special case where r = 1. Then G e 3 if and only if for all
V e Chiefp(G) all composition factors of К® К belong to ®1(G). Let Nt, N2 < G with
G/Nj e 3 and Nt n N2 = 1. If we regard the chief factors of G/7Vf as G-modules, by
the Jordan-Holder theorem we have
Chiefp(G) = ChiefpjG/NJuChiefpfG/Nj),
and therefore
3£(G) = 2к(С/М) u 3J(G/N2).
Because G/Nt e 3 by supposition, it follows from Axiom FF2 that, viewed as a class
of G-modules, З^б'/А,) E ®i(G) for i = 1, 2. Thus 3}(G) s 9Ji(G), and so Gel.
Hence, when r = 1, the class 3 is Ro-closed and consequently a formation. □
Next we give some explicit examples of Fitting families (in fact, one for each prime
p) and then investigate some of the Fitting classes which, according to Theorem 2.18,
they give rise to.
(2.19) Definition. Let p be a prime, and let К = Fp, the algebraic closure of Fp. For
each group G define
®1P(G) = (Ve IrrK(G): p|DimK(F)),
and set 9!Rp = U ®K(G), where the union is over the universe of p-soluble groups.
(2.20) Proposition. The class W defined in (2.19) is a Fitting family of modules over
Fp in the universe of p-soluble groups.
592 IX- Fitting classes—examples and properties related to injectors
Proof. It is at once clear that Axioms FF1 and FF2 are satisfied for 4)1'’. Moreover,
if V is a simple KG-module (К = Рр) and N < G, by B, 7.1 the simple summands of
Fn all have the same dimension, which therefore divides DimK(F); thus Axiom FF3
also holds.
To verify Axiom FF4, let V e IrrK(G), let Nt and N2 be distinct maximal normal
subgroups of G, and assume that the composition factors of Ьл. have K-dimension
prime to p for i = 1, 2. Since G is p-soluble, the groups G/Nj are either p'-groups or
else cyclic groups of order p. First suppose that G/N2 is a p'-group. By B, 7.14 the
К-dimension of V divides dlG/NJ, where d is the К-dimension of a composition
factor of KN1, and in this case DimK(F) is certainly a p'-number. Clearly the same is
true if G/N2 is a p'-group, and so we can suppose that IG/NJ = p for i = 1, 2 and
hence that G/N = Zp x Zp, where N = N, n N2. Let
if = ^©-©и;
be a decomposition of If into the sum of homogeneous components I4j, and let T
denote the stabilizer of W,. By Clifford’s Theorem B, 7.3 we can regard Wt as a simple
T-module and obtain I'sIHj )G. Since T/N is a p-group and since К contains a finite
splitting field for the subgroups of G by B, 5.21, it follows from B, 8.3 that W, is already
simple as an K-module.
Suppose that Nt f T. Then N = N2 n T, and N is the stabilizer in Nt of the simple
N-module Wt. By B, 7.4 the module U = (И/1)л' is simple, and if 1, x2, ..., xp is
a transversal to N in N,, then
V =? I4j ф ® Wtxp < V.
It follows that U is isomorphic with a composition factor of F^i, and so by assumption
its К-dimension is a p'-number. But this contradicts the fact DimK(U) = p Dim^-flF,),
and so we must have Nt < T. Similarly N2 <, T, and therefore T = G. Consequently
V = whence if is simple and DimK(V) is a p'-number. Thus V e ®lp(G), and we
have shown that Axiom FF4 is satisfied. □
(2.21) Examples. By (2.18) and (2.20) the class S(r, 4Jlp) defined in (2.i) is a Q-closed
Fitting class and, for r = 1, even a formation. For notational convenience we will
denote this class simply by T(r, p) below.
(a) (Hawkes [5]). As our first illustration, we show that the class
8 = 1(1,3)пб,е2е,
is not Еф-closed. Since G3 G2 G3 is a Fitting formation, we therefore obtain an example
of a Fitting subformation of 'Ji3 which is not saturated. (We shall see in XI, 1.8 that
Fitting formations of metanilpotent groups are always saturated.)
To justify our assertion, let H = SL(2, 3), and let V be the natural F3 H-module of
dimension 2; obviously V is absolutely irreducible. Let G denote the semidirect
product [F]H. Since H has a normal subgroup Q = Quat(8) such that |H/Q| = 3, it
is clear that
593
2. Constructions and examples
HeI(l,3)n33e2S3.
Suppose, by way of contradiction, that the formation g is saturated. Then by IV, 4.6
we have g = LF(f) for a suitable formation function /, and since V = 0,. (G) it
follows that H = G/V e /(3) n g. But then /(3) also contains the group H/Z(H) s
Alt(4), which has a simple module, W say, of dimension 3 over F3; in fact, if V denotes
a non-trivial 1-dimensional module over F3 for the normal subgroup N Z2 x. Z2)
of Alt(4), then W s gAI1’41, and W is absolutely irreducible (see Theorem B, 7.4).
Regarding W as an H-module by inflation, we then observe that the semidirect
product [IT]H belongs to G3/(3) <= LF(f) = g but clearly not to T(l, 3). This
contradiction proves our initial assertion.
(b) We now describe an example to show that, in general, the class X(r, p) need
not be Rc-closed when r > 2.
Let £ be an extraspecial group of order 37 and exponent 3. By A, 20.13 the group
Aut(£) contains an element a of order 7 which acts irreducibly on E/Z(E) and
centralizes Z(£). Put A = <a> and H = [£]/. Let z e Z(E), z A 1, and let 2 be an
element of order 3 in F7*. Then by B, 9.16 there exists for i = 1, 2 a 27-dimensional
faithful, absolutely irreducible £-module Ц over F, such that nz = 2‘v for all v e Ц.
Furthermore, by B, 7.12 we can extend F, and V2 to F7H-modules.
Let {i, j} = {1, 2}. Then z is represented on the module ® by the scalar action
2J, and it follows that all the composition factors of the H-module Ц ® are faithful
for H because Z(E) is the unique minimal normal subgroup of H. Therefore by B,
8.3 and B, 9.16 these composition factors all have dimension 27. On the other hand,
the kernel of H on P, ® V2 contains Z(E) but not £. Since the absolutely irreducible
representations of H/Z(E} (s£(7/3)) over F7 are either trivial, or else, by an easy
application of Clifford's Theorem В, 7.3. have dimension 7, we conclude that
(Fj ® F2) ® F7 has composition factors of dimension 7. Hence, if G = [Ц ф F2]H, it
follows that G/F; e T(2, 7) for i = 1, 2, whereas G ф T(2, 7). Consequently 3(2, 7) is
not Ro-closed.
(c) It is clear from the definition that 3(r, p) 2 T(r + 1. p) for all r e hl. Let Koo, p)
denote the class Q" , 3(r, p). Then we assert that
(2.k) s n к co, p) = e„.epep.,
the class of soluble groups of p-length 1.
Open Question. We have been unable to decide whether 3(cc, p) = GpGr(fp.-
We will now justify Assertion (2л). First, let r e hl, let M,,..., Mr e Chiefp(G) oj (Fp)c,
and let M be a composition factor of ® • • ® Mr ® Fp. Since O„..p(G) centnihzes
all p-chief factors of G, we have OP,P(G) < Ker(G on M). Therefore, if G e 6^6,6,
then G/Ker(G on M) is a p'-group, and so by B, 7.14 the Fp-d,mension of M is a
p'-number. This proves that Gp epGp e2(°°,P)-
Now lei C
«x - o/k«(O о. «
594
IX Fitting classes—examples and properties related to injectors
О (X) = 1 by B, 3.12. We claim that X has an epimorphic image Y which satisfies the
hypotheses of B, 10.11, namely that У is a primitive group, that Soc(Y) is a q-group
for some q * p, and that Op( Y/Soc(Y)) * 1. Let R = С'-^Х), the GpSp.-residual of
X. Since 0p(X) = 1 and X ф top., it follows that R + 1 and hence that X has a q-chief
factor of the form R/S for some prime q * p. Without loss of generality suppose that
S = 1, and let N/R = OP(X/R) e Sylp(X/R). The definition of R implies that N is
not nilpotent, and therefore Op(X/Cx(R}) 1. Moreover, if Re Sylp(N), then NX(P)
complements R in G by the Frattini argument. By A, 15.5 the group Y =
X/(Nx(P)ri CX(R)) is a primitive epimorphic image of X, its socle is X-isomorphic
with R, and Y/Soc( Y) X/Cx(R). From B, 10.11 we can therefore deduce that Y, and
by inflation X, has a simple module V over Fp whose dimension is divisible by p.
Let M* be a composition factor of M ® Fp and note that Ker(G on M*) =
Ker(G on M) by B, 5.26(a). Therefore by Steinberg’s Theorem B, 10.13 the module V
appears as a composition factor of M M ® "• ® M ® Fp) for some
sufficiently large integer s, and it follows that G ф T(s, p). Consequently the class
(6nJ(oo, p))\Zp-&p&p- is empty, and Assertion 2.x is justified.
(d) (Cossey and Ormerod [2]). Here we exploit the idea behind Construction F
to describe a family of examples of Fitting subclasses of'Ji3 which are Schunck classes
but not formations. (In XI, 1.8 and XI, 2.16 we shall show that a Fitting subclass of
9l2 is a Schunck class if and only if it is a formation.)
The examples in question are parametrized by a prime p. Define Sj(p) to be the
class of groups G in 'Ji3 whose complemented p-chief factors V all have the property
that the composition factors of V ® Fp belong to ®lp(G), in other words, that the
absolutely irreducible constituents of V have p'-dimension. Fix a prime p and write
$5 for S>(p). The definition of in terms of properties of complemented chief factors
obviously determines the possible primitive epimorphic images of the groups in £>
and thus ensures that & is a Schunck class. In fact, it is not difficult to see that the
basis of J) consists of all primitive groups of nilpotent length at most 3 except those
groups В of length 3 with Soc(B) a p-group and p dividing the dimensions of the
composition factors of Soc(B) ® Fp. On the other hand, if & were a formation, then
it would be saturated and therefore locally defined. But the argument of Example (a)
above, which shows that §(3) is not locally defined, can easily be modified to show
that S(p) is not locally defined for any prime p (see Exercise 4 below). Thus is not
a formation.
It remains to show that § is a Fitting class. First we prove that & is s„-closed.
Suppose, by way of contradiction, that § is not s„-closed, and let G be a group of
minimal order such that 6e S and s„(G) £ £>. This choice of G ensures that G has
a maximal normal subgroup M such that M ф in particular, M 1. Let N be a
minimal normal subgroup of G contained in M. Since G/N eof) = f>, we have
S„(G/N) £ Sjby the choice of G; in particular, the absolute ranks of the complemented
p-chief factors on M/N are p'-numbers. From this it follows that N is a p-group and
that N is the unique minimal normal subgroup of G contained in M; therefore,
in particular, F(M) is a p-group. Furthermore, p divides the dimensions of the
composition factors of the completely reducible FpNf-module (N ® Fp)M (they all have
the same dimension); and therefore, appealing to B, 7.1, we see that p divides the
degrees of the G-composition factors of N ® Fp. Since G e §, it follows that N < Ф( G),
2. Constructions and examples
595
and furthermore, because M/N e & and M * § = e.& we have N £ ф(М). Therefore
N n Ф(М) _ 1 because N n ф(М) < G. Hence we conclude that Ф(М) = 1 and con-
”‘*y ,tha‘ f *Л/) 1S an elementary abelian p-group. Since M e 9l3, the group
is therefore a metanilpotent group with O(M/F(M)) = 1 and hence
M/F(M} is p-nilpotent. Because N has a complement in M but not in G. we deduce
from A, 11.1 that |G: M | = p and hence that G/F(M) is also p-nilpotent. Now F(M)
may be regarded as an (MJ)-module with a unique minimal submodule N So
viewed, F(M) is indecomposable, and by B, 4.24 all its composition factors are
therefore isomorphic. Hence the G-chief factors below F(1W) are G-isomorphic with
N, and since G e Sy it follows that F(M) < ф(С).
Suppose that, if possible, F(M) < M, and let R/F(M) be a chief factor of G below
M. Since КФ(С)/Ф(С) (R/(R n Ф(С)) is a nilpotent normal subgroup of С/Ф(С), we
have R < F(G) by A, 9.3(c). In particular, R e 91, and therefore R < F(M). This
contradiction shows that M = F(M) and hence that, in particular, M is a p-group.
But then we can conclude that M belongs to Sy and this is our final contradiction.
Therefore Si is indeed s„-closed.
Finally, we show that § is N0-closed. Let G = A, N2, where Nt and N2 are maximal
normal subgroups of G belonging to Sy Certainly G e n0913 = 913. Therefore by II,
2.11 (b) it will suffice to show that G e Sy Thus, if H/K denotes a complemented p-chief
factor of G, we must show that the composition factors of the G-module (H/K) ® Fp
have p'-dimension. Since G/(Ni n N2) is abelian, by A, 9.13 we may suppose without
loss of generality that H < n N2. Since by Clifford’s theorem H/K is completely
reducible as an Armodule (i = 1, 2), we can write H/K = J,/K x x JJK, where
Jj/K is a chief factor of A, for j = 1, ..., t. If L is a complement to H/K in G,
it is straightforward to verify that (Ln А()П*#М's a complement to J2/K in A;.
Thus the dimensions of the composition factors of the A--modiile (J2/K) ® Fp are
p'-numbers because A, e Sy and therefore, if V is a direct summand of (H/K) ® Fp,
it follows that the composition factors of belong to 9K'’(Ai). Since W satisfies
Axiom FF4 by (2.20), we conclude that lAWfC) and hence that Ge f>, as
required.
Next we describe another type of Fitting class based on the procedure of Construc-
tion F. Thus, once more, we will first produce a Fitting family of modules and then
appeal to Theorem 2.18 to obtain the corresponding Fitting class. This example is
based on work of Haberl and Heineken [1] and brings to mind the construction of
saturated formations using rank functions described in Chapter VII, Section 2.
(2.22) Definition. Let q = pm, where p is a prime. A dimension set for q is a set Rq of
natural numbers satisfying the following four conditions.
DS1: If a, b e Rq, then ab e Rq;
DS2: If ab e Rq. then a, be Rq;
DS3: If n e R„ and a|q" — 1, then a e R,;
DS4: If R. contains a prime f, then Rq contains the order of q modulo t
(It should be noted that this concept is not vacuous. It turns out (see Exercise 6 below)
that for a given prime power q there exist exactly three dimension sets: Rq 0,
Rq = N, and one further (non-trivial) example.)
596
IX. Fitting classes—examples and properties related to injectors
(2.23) Definition. Let I? be a set of natural numbers, let G be a finite soluble group,
and let К be a field. We define an associated class ®1R(G) of simple KG-modules thus:
®iR(G) = (F e IrrR(G): DimK(F) e R).
(Thus H1r(G) = 3K',(G) when R consists of all p'-numbers and К = Fp.)
(2.24) Proposition. If q is a prime power, R = Rq is a dimension set over q, and К = F,
then the class
®iR = u
GeS
is a Fitting family of modules over F, in the universe of finite soluble groups.
Proof. If R = 0, we obtain the empty Fitting family for ffllR. Therefore suppose that
R # 0, so that 1 e R by DS2. Since dimensions are unchanged when modules are
deflated or inflated, Axioms FF1 and FF2 are obviously satisfied here. Moreover, if
N 4бе6, V e ®1r(G), and U is a direct summand of KN, then DimR(G) divides
DimR( K) by В, 7.1. Therefore V e 1DlR(N) by DS2, and so Axiom FF3 is fulfilled. Thus
the burden of the proof is the verification of Axiom FF4.
We argue by contradiction, supposing that ®1R fails to satisfy Axiom FF4. Then
there exists a soluble group G and a simple KG-module V such that the following
four properties are satisfied:
(i) G has maximal normal subgroups N and N* with G = NN*;
. I (ii) If G| VN, then DimR(G) e R;
’ ’ | (iii) If 17*| VN., then DimR(G*) e R;
(iv) DimK(F)£R.
Among such counterexamples choose a pair (G, V) with |G| as small as possible, and
note that G 1. Let L = NnN*, and let r = |N: L| = |G: N*|. Since G is soluble,
r is a prime. Let U be a simple submodule of VN, and note that U < V by (2.2) (ii) and
(iv).
First suppose that UL is reducible; then it has a proper simple submodule, W say.
We distinguish two cases:
(a) UL is inhomogeneous. Since L<-N, in this case |N: L|DimR(lT) =
DimK(U) e R, and therefore r = |N: L| e R by DS2.
(b) UL is homogeneous. Let
UL = IF® ф W (s copies, s > 1),
and let |HomRt(W')| = qe. Since DimR(L) is divisible by s and also by e (because of
B, 8.5(i)), we have e, s e R by (2.2)(ii) and DS2, and therefore es e R by DS1. By B,
8.5(H) the prime r divides qes — 1, and therefore by DS3 we have r e R, as before. In
order to show that the assumption that UL is reducible leads to a contradiction, we
again consider two possibilities:
2. Constructions and examples
597
(A) is inhomogeneous. In this case we have DimK(k) = r DirnUU*! which
belongs to R by DS1. This contradicts (2.z)(iv).
(B) VN. is homogeneous. Let s0 denote the composition length of VN. If s = 1
certainly s0 e R. Suppose that s0 > 1. Then by B, 8.5(vi) either s0 = r or s0 divides
o(<j)(mod r). Since we have shown in (a) and (b) that r e R, it follows from DS4 that
R contains o(<j)(mod r) and hence from DS2 that s0 e R. But then DimK(P) =
s0 DimK(G*) e R by DS1, which again contradicts (2J.)(iv).
Thus we have proved that
(I) UL and Lf are simple.
Since the N-submodule V of V remains irreducible when restricted to L, it follows
that N stabilizes one, and hence all, of the simple submodules of lj. (Here we are
appealing to the conjugacy of stabilizers according to Clifford’s Theorem B, 7.3(c).)
Similarly N* is contained in these stabilizers, and therefore G (= NN*) stabilizes
each simple submodule of VL. Consequently
(II) VL is homogeneous.
Next suppose that Ij,. is homogeneous. Since U* is a simple L-submodule but not
an N-submodule of Ij,,, we can apply B, 8.4 to conclude that r = |N:L| divides
DimK(lf) and therefore belongs to R. But then we can reapply the arguments of Cases
(A) and (B) to get contradictions once again. Therefore
(III) If and are inhomogeneous.
In this case Dim^P) = |G: N| DimK(Gj = |G: N*| DimK(U*), and because U and
V* are isomorphic simple submodules of VL, it follows that |G : N| = |G: N*| (= r).
Thus VL is homogeneous with composition length r, and so r divides (qer - V)/(q‘ — I)
by B, 8.5(ii). Hence r divides qe — 1 by B, 8.5(iv), and since e e R, we conclude that re R
by DS3. But then Dim^F) = r DimK(lf) e R by DS1, and we have a final contradic-
tion. Therefore ЭДК satisfies Axiom FF4 and is a Fitting family in the universe G. □
Proposition 2.24, together with Theorem 2.18, yields the following result.
(2.25) Theorem. Let r be a natural number and q a prime power. Let Rq be a dimension
set over q, and put К = IF,. Then the class T(r, 9JiF) n G i.s a Q-:losed Fitting class and
is even a Fitting formation when r = 1.
(2.26) Remarks.
(a) Theorem 2.25 was first proved by Haberl and Heineken [1] in the case when
K = F (pa prime) and r = 1.
(b) If q = 2 then R2 = {1} is the only non-trivial dimension set for q. The corre-
sponding Fitting formation is 9i'21, the class of 2-nilpotent groups, which is s-closed
and saturated. However, if q > 2 and 0 Rq * N, it is not difficult to see that the
corresponding Fitting formation is neither saturated nor s-closed.
(c) If {9Ji,}.,. is a directed set of Fitting families of modules over R, in other
words, if it has the property that for each 2, p e A there exists are sue t at
9Jl.(G) и 9И (G) s 9Ji(G) for all G in the appropriate universe, then it is straight-
forward to verify that (J*eл9J!> « a Fitting family. In the present context lei! p be: a
fixed prime, let R„ denote the unique non-trivial dimension set for p and let W)
denote the Fitting family associated with Rpr. Then/JR(r)}ref<j is a i
598
IX. Fitting classes—examples and properties related to injectors
if ЭД = (J,f\ЭД(4 then, in a soluble universe, 1(1, ЭД) coincides with the class
1(1, ЭДР) defined earlier, comprising the soluble groups all of whose p-chief factors
have absolutely irreducible constituents of p'-degree.
The two types of Fitting families described in Propositions 2.20 and 2.24 in fact
turn out to be special cases of Fitting families defined by a much more general
procedure. This far-reaching approach is joint work of Cossey and Kanes [1].
Because a full treatment of the background representation theory would take up a
disproportionate amount of space, we simply state the bare facts without proof. The
underlying ideas first arose in the study of characters of complex representations of
л-separable groups by Gajendragadkhar [1] and Isaacs [3]. Some care is needed to
adapt their work to a corresponding theory for modules over a field of characteristic
p > 0.
(2.27) Definitions. Let К be an algebraically closed field of characteristic p > 0, let
n be a set of primes, and let G be a л-separable group (that is to say, a group whose
composition factors are either л-groups or n'-groups).
(a) A simple KG-module К is called n-special if
(i) DimK(F) is a л-number, and
(ii) whenever L sn G and U is a composition factor of VL, then Det(x on G) = 1
for all л'-elements x of L.
(b) A simple KG-module К is called n-factorable if there exist a л-special module
V and a л'-special module W such that V = U ® W.
It turns out that if V and W are respectively л-special and n'-special simple
KG-modules, then the product U ® W is also simple and hence n-factorable. More-
over, such a factorization is unique, in the following sense: if U ® W U* ® W*
with U* and W* respectively n-special and n'special, then V = U* and W IV*.
This concept of n-factorable generalizes to a partition & = {лл}лел of P into
pairwise-disjoint sets пд of primes. A module is said to be ^-factorable if it is
isomorphic with a tensor product of the form G, ® ••• ® U, of simple KG-modules
Ц-, where each Gf is Пд.-special and Л,- # f when i ф j. Here the groups under
consideration should be nA-separable for all A e A, and, of course, this is guaranteed
in the universe <5. A similar uniqueness theorem holds for ^-factorable modules. The
key step in checking Axiom FF4 for the Fitting families of Cossey and Kanes is the
following.
(2.28) Theorem. Let G be the product of normal subgroups M and N, and let U be a
simple KG-module. If all the composition factors of UM and UK are ^-factorable, then
U is .^-factorable.
The Fitting families in question are described in the following theorem.
(2.29) Theorem (Cossey and Kanes [1]). Let S’ = {пл}1бЛ, where пл s P and
— 0 when A yt p. For each A e Л let 3E(A) be a Fitting formation. Let К be an
algebraically-closed field cf characteristic p > 0, and, for each soluble group G, let
ЭД(С) denote the class of all simple KG-modules V such that V = Vl ® • • ® G„ where
2. Consiructions and examples
599
(а) Ц is nA-special and =£ for 1 < i j < ц
(b) G/Ker(G on Ц) g Х(Л,).
Then ®1(K, S’ X) = (Jc 6 e 991(67) is a Fitting family of modules over К in the universe
o.
According to Theorem 2.18, we obtain from the Fitting families ®i(K 0> X) a
collection of Fitting classes 3(r, ®l(K, 9, X)) parametrised by the natural number r
a prime p (the characteristic of K), a partition & = {n2}2eAof P, and a set {X(2)L ’
of known Fitting formations.
(2.30) Remarks, (a) It is not difficult to see that T(r, 9Ji(K, X)) = T(r, ЭД(Г , X)),
in other words, the class does not depend on which algebraically-closed’field of
characteristic p is used in the definition. We therefore denote it by T(r, p, 0>, X).
(b) We also know from Theorem 2.18 that 1(1, p, X) is a Fitting formation. It
is not known, and it seems very hard to decide, whether every Fitting formation can
be expressed in the form
A 3(1,р,^,ЭЕ,)
PE P
for suitable choices of the partitions S’ and sets {Хр(А)}ЛеЛ of Fitting formations.
This would be a kind of “local definition” for Fitting classes.
(c) Let p e P. If we takes’ = {n,, n2} with r., = {p} and n2 = {p}, -S(l) = (l)and
X(2) = G, then we obtain T(r, p, S’ X) = T(r, 9Л₽); in other words, the Fitting class
described in (2.21) is a special case of the Cossey-Kanes procedure.
(d) Another special case is the example due to Berger and Cossey [2]. This was
the first known example of a non-saturated Fitting formation consisting of soluble
groups of p-length 1 for all primes p; it can be described in the form T(l, p, X) by
taking S’ = {и,, n2] with it, = {q} and n2 = {q'}, 3E2 = {1} and X2 = where
q is a prime distinct from p.
(e) L.G. Kovacs has characterized those Fitting formations among the
T( 1, p, S’ X) which are saturated. His description appears in the cited paper of Cossey
and Kanes.
(f) In a subsequent paper Cossey [5] has modified his work with Kanes to
accomodate a situation where the field К is not necessarily algebraically closed.
Theorem 2.29 still holds, provided the universe is restricted to soluble <7-groups, where
<7 is a set of primes depending on the field К and the partition in fact, <т — P when
К is algebraically closed. The Fitting classes arising from Cossey’s modification
include those of Haberl and Heineken mentioned in (2.26)(a).
Exercises
1. (Cusack [1]). Let g, ©be Fitting classes in G. Then (G e G: G - G,G, Jis a Fitting
class if and only if there exists a set of primes n such that g S © G and © S gto,-.
2. Let p, q distinct primes, let g = (G: SocjG) < Z(G)), and let 6 = (G: Soc.(G) <
Z(G)). Then (G e <3: G = GaG,J is not a Fitting class
3. Let и be a set of primes, and let X denote the class of all p-soluble groups G whose
600
IX. Fitting classes—examples and properties related to injectors
p-chief factors all have absolutely irreducible constituents of л-dimension. Show
that although X is s,-closed, it is not in general a Fitting class. (See Construction F.)
4. (Berger and Cossey [2]). Let p and q be primes, and let К = if,. If G e <3, let SDi(G)
denote the class of all irreducible KG-modules M such that (i) p|DimK(M), (ii)
G/Ker(G on M) is p-nilpotent, and (iii) if D is the representation of G afforded by
M, then Det(Dp) is a p'-root of unity for all g e G. Show that the class
(G e to: Ti(G) <= UR(G))
is a non-saturated Fitting formation. (See Construction F.)
5. If К is a field and 5 a Fitting formation, define
URr(G) = (Fe IrrK(G)): G/Ker(G on F) e 5)
for each finite group G. Show that (JCOTS(G) is a Fitting family of modules.
6. Let q = p' be a prime power and define a subset R of M recursively as follows:
1 e R, and if {1,..., n} n R is known, then n + 1 e R if and only if either
(a) n + 1 has a factorization n + 1 = ab with a, be {1,..., n) nR or
(b) n + 1 is a prime different from p and the order of q(mod n + 1) belongs to R.
Prove that R is well-defined and is the unique non-trivial dimension set for q.
7. Justify Remark 2.26(b).
8. Justify Remark 2.26(c).
3. Fischer classes, normally embedded, and permutable Fitting classes
A possible response to an intractable question about general Fitting classes is to
impose extra conditions on the classes under consideration; for example, one can try
initially to answer the question for Fitting classes which satisfy additional closure
properties, e.g. one or more of s, q, r„, еф. In this section we increase the availability
of such test situations by defining and investigating a sequence of special properties
of Fitting classes, related to the behaviour of their injectors and Fischer subgroups.
Therefore, in order to guarantee the universal existence of injectors, we confine
ourselves throughout this section to the universe <3, except in (3.3)(a) and (3.5) where
the universe is G.
We begin by recalling the definition of the subgroups originally studied by Fischer
as the natural duals of Gaschiitz’s covering subgroups. (Compare the following
definition with the Conditions CS1 and CS2 of III, 3.6(b).)
(3.1) Definition. Let g be a Fitting class. A Fischer ^-subgroup of a group G is a
subgroup £ of G satisfying the following two conditions:
FS1: £ e 3, and
FS2: if £ < £ < G, then £s < £.
Since Trs(G) is a Fitting set of G by VIII, 2.2(a), the Fischer ^-subgroups of G
obviously coincide with the Fischer TrB(G)-subgroups of G in the sense of VIII, 4.1.
3. Fischer classes, normally embedded, and
permuiable Fitting classes
601
As we remarked in Section 1, the g-injectors
coincide, and therefore by VIII, 4.2 we have the
and the Tr^(G)-injectors of G also
following observation.
(3.2) Remark. If g is a Fitting class and G a group, then an ft-injector of G is a Fischer
^-subgroup.
Each of the three properties of Fitting classes which we are about to define arises
naturally in a different context. The first involves a sharpening of the requirement of
«„-closure, and is of historical interest because for such classes Fischer [1] was able
to show in his original investigation that each finite soluble group has a unique
conjugacy class of Fischer subgroups, in other words that the Fischer subgroups are
in fact injectors. (In (5.19) of this chapter Dark’s construction will be used to show that
this is not true for general Fitting classes.) The coincidence of Fischer subgroups with
injectors was then found to extend to Fitting classes with the second property, that
of having normally embedded injectors, itself an important embedding property in
the general context of soluble groups. And the third property of having permutable
injectors, although not always easy to establish for a given class, gives useful control
over the injectors for related classes.
(3.3) Definitions. For the first definition only, the universe is G.
(a) A class g of arbitrary finite groups is called a Fischer class if
(i) g = Ncg 0, and
(ii) if К < G e g and H/K is a nilpotent subgroup of G/K, then H e g.
We note that in view of Condition (i) it is sufficient to require only that Condition
(ii) holds whenever H/K has prime power order. We also remark that a Fischer class
is obviously a Fitting class, and that an s-closed Fitting class is a Fischer class.
(b) A Fitting class g is said to be normally embedded if for all G e S the g-injectors
of G are normally embedded subgroups of G. Since by (1.5) (a) the g-injectors of a
group are САР-subgroups, by I, 7.12, (g)=>(a), it is sufficient to require that the
g-injectors be locally pronormal in this definition.
(c) A Fitting class g is said to be permutable if for all G e <5 the g-injectors of G
are system permutable subgroups of G.
If g is a Fischer class and Gee, then Trs(G) is evidently a Fischer set of G in the
sense of VIII, 4.3. Therefore from VIII, 4.5 and VIII, 4.7 we can immediately deduce
Parts (a) and (b) of the following theorem; Part (c) is a consequence of I, 7.10.
(3.4) Theorem, (a) (Fischer) A Fischer class is a normally embedded Fitting class.
(b) (Anderson [2]) If g is a normally embedded Fitting class, then the Fischer
^-subgroups cf a finite soluble group G are just the ^-injectors of G.
(c) (Lockett [2]) A normally embedded Fitting class is permutable.
In the course of this section we shall investigate the properties of the three types
of Fitting class just introduced, in particular giving examples to show that the
inclusions described in the preceding theorem are proper. Thus we have
602
IX. Fitting classes—examples and properties related to injectors
{Fischerclasses} {normally embedded Fitting classes}
{permutable Fitting classes}.
(In (5.19) we shall also see. with the help of the Dark construction, that not every
Fitting class of finite soluble groups is permutable.) We shall show that each of the
three types of Fitting class is preserved by forming Fitting products (see (3.8) (b), (3.13),
and (3.10)), and by the Lockett operator Ln( ) (see (3.8)(a), (3.11), and (3.9)(a)). The
intersection of an arbitrary collection of Fischer classes is obviously again a Fischer
class, and therefore it is not surprising that Fischer classes can be described by means
of closure operations; this we shall show next. It is an open question whether the
intersection of two normally embedded Fitting classes is again normally embedded.
For the following result only, the universe is enlarged to G.
(3.5) Proposition (Hawkes [11]). If JE is a class of finite groups, let
sFJE = (H: H < G e JE and Н'л sn G).
Then
(a) sF is a closure operation;
(b) S„ < Sf < s;
(c) A class 5 satisfies Condition (ii) of Definition 3.3(a) if and only if it is Sf-closed;
in particular, 5 is a Fischer class if and only if 5 = <sF, N„>g.
Proof, (a) It is clear that sF is expanding and monotonic. To show that it is idem-
potent, let L e sF(sFJE). Then we can find a group G e JE and subgroups L< H < G
such that L^snH and H’snG. In this case we have L/(LrlH*)~LH,i/H* <
H/H91 e 91, and consequently L* < H91. Since L91 sn H, it follows that L91 sn H91 sn G,
and therefore that L e sFJE. Hence sF is a closure operation.
(b) It is clear from the definition that s„ < sF < s. In (3.7)(b) below we give an
example of a Fischer class which is not s-closed, and in (3.7) (a) an example of a Fitting
class which is not a Fischer class. Therefore both inequalities are strict.
(c) First suppose that 5 satisfies Condition (ii) of (3.3)(a). To show that 5 is
sF-closed, let H < G e 5 with H91 sn G. Then there exists a subnormal chain of the
form
H91 = L, < L,_j < • = G.
By using the distinguished subnormal chain described in A, 14.5 and A, 14.7, we can
additionally suppose that the subgroups Lf in this chain are all H-invariant. We now
show by induction on i that L,H e g for i = 1,2,.... t. This is true by hypothesis when
i = l. Suppose that LkH e g for some к > 1. Then Lk+lH/Lkll = H/(H n Lk+l) e
(jfH/H'11) s 91, and because Lk+1 < LkH e g, we have Lk+1 H e JV by the supposition
that Condition (ii) holds. This completes the induction step. Therefore H = L,H e Й,
and consequently g = sFg.
3. Fischer classes, normally embedded, and permutable Fitting classes 603
Conversely suppose that g = s,g, let К 3 G e g. and let H/K be a nilpotent
subgroup of G/K. Then №* <3 K, and so H'11 sn G. Hence H e sFg = g, and gthere-
fore satisfies Condition (ii) of (3.3)(a). '
Next we show that under suitable hypotheses Construction C of Section 2 yields
Fischer classes; also that for a saturated formation the property of being a Fischer
class is reflected in the local definition.
(3.6) Proposition, (a) Let X and '/) be Fitting classes with 3E £ 9), let R(G) = Gv/Gx,
and set л — {p e P. p11R(G)| for some G e £}. Let f be a formation function, and
assume that f(p) is a (non-empty) Fischer class for all p e n (or possibly the empty class
in the special case where Y = Q and X = (1)). Then the class
HR(f, R) = (G e <3: R(G) is f-hypercentral in G)
is a Fischer class.
(b) Let g = LF(F). Then g is a Fischer class if and only only if F(p) is a Fischer
class for all p e Char(g). (We recall the convention that F denotes the full and
integrated local definition of g.)
Proof (a) The class HR(f, R) is certainly Nc-closed by (2.4). Let G e HR(f, R), let
Л' <! G, and let H/N be a p-subgroup of G/N. We have to show that H e HR(f, R).
Since HVN/N S Hv/Nv and since H/N is a p-group, it follows that Hv/Nv is
hypercentral in H/N9. If the p-group H<B/N^HX is non-trivial, then p e n. If f(p) were
empty, by hypothesis we should have R(G) = G; but then G would be /-hypercentral
in G, which is impossible because p 11 G|. Therefore 1 e f(p), and in any case HV/NVHX
is /-hypercentral in H/N<BHX. It remains to show that NVHX/HX is also an f-
hypercentral section of H. Since
(3.a) N<BHX/HX Nv/Nx,
we consider first a chief factor R/S of G in the normal section N<B/NX of G. Let
R/S be an r-group. Since Nv/Nx = (G^ n IVJGj/Gj < G^/Gj, it follows that
AutG(R/S) e f(r) because GeHR(f, R). Then, since AutN(R/S) < Autc(R/S) and
Aut„(R/S)/AutN(R/S) e S„, it follows that Aut„(R/S) e s, f(r) = f(r) by hypothesis.
From the Q-closure of f(r) we then conclude that R/S is /-hypercentral under the
action of H, and so it follows from (З.ос) that NVHX/HX is /-hypercentral in H.
Therefore H e HR(f, R), which is consequently a Fischer class.
(b) The sufficiency of the condition follows at once from Part (a) (with V) = £ and
ЗЕ = (I)). To prove the necessity, let g be a Fischer class, let p 6 Char(g). and let
G e F(p). Let N <| G and suppose thai H/N is a q-subgroup of G/N. We must show
that H e F(p). Put W = Z/brte G and denote the base group of fV by В Since
W E e F(p) = F(p) cz g and since BN < W with BH/BN e it follows that BH e
SFg = g Ken became 0,(BH) = 1 and g £ ^F(p). conclude that BH e F(p)
and hence that H e <jF(p) = F(p).
604 IX. Fitting classes—examples and properties related to injectors
(3.7) Examples, (a) The Fitting class J3 =(Ge£: Soc3(G) < Z(G)) is not a Fischer
class. In fact, it is not even normally embedded as the following example shows. Let M
be the natural permutation module for the group S = Sym(4) over F3, with per-
mutation basis {m,, m2, m3,m4}. Then the subspace Л' with basis {пц — m4,m2 — m4,
m3 — m4} is clearly a faithful submodule of M, and if V = <( 12)(34), (13)(24)>, the
normal four-subgroup of S, it is easy to verify that the restricted module Nv is a sum
of three simple submodules, on which the kernels of V are the three distinct subgroups
of V of order 2. It then follows from Clifford’s Theorem B, 7.3 that N is a simple
F3S-module. If L denotes the stabilizer of the point 4, then L s Sym(3), and evidently
N is isomorphic to the natural permutation module for Sym(3). Hence CN(L) =
<ni! + m2 + m3> and |CN(L)| = 3.
In (4.19) we shall characterize the 3p-injectors of a group G as follows: Let N be
the p-socle of the Jp-radical of G. Then a subgroup V is a Jp-injector of G if and only
if there exists a Sylow p-subgroup P of G such that V = Cc(CN(P)). In this example
we take G = [N]S. Then N is clearly the J3-radical of G and at the same time the
3-socle of this radical. If P e Syl3(L), we have NP = F3P and hence CN(P) = CN(L);
we therefore conclude that NL = Cc(CN(P)) is a J3-injector of G. But, if T e Syl2(L),
we have <TC> = G, and therefore NL is not 2-normally embedded in G.
(b) The class (G e G: O„(G) < Z„(G)) is a Fischer class by (3.6)(a). However, it is
not s-closed as we pointed out in (2.5)(a).
(c) Let ft be a Fitting class, and let n £ P. Then the class ftG„ G„. is a Fischer class.
Proof. Certainly gG„G„. = ft 0 (£,£,.) is a Fitting class. Let N<Ge gGPGP.,
and let H/N be a p-subgroup of G/N. We must show that H e gG„G„.. If p e л',
this is obviously true. Therefore suppose that pen, and let P e Sylp(H). Since
P < GRS-, we have [Л/, P] < N n — R, where R = N^^. It follows that
RP < NP = H and that RP e gGPGp = gG„. Since N e ftG„G„., we conclude that
H/RP s N/R(N rP}e G„., and therefore that H e ftG„G„., as desired.
We draw two conclusions from this example:
(1) If p s P, and if F(r) is a Fitting class for all rep, then the class
(З-Д) П F(r)QrSr.
rep is
is a Fischer class. Fitting classes of the form (3./J) may be regarded as the duals of
local formations (see Criterion (b) of IV, 3.2). However, no counterpart of the descrip-
tion of local formations by the groups of automorphisms induced on chief factors
has yet been found for these ‘local’ Fitting classes, and they have as yet played no
special role in the theory (see Hartley [1], D’Arcy [4], Schnackenberg [1]).
(2) If ft is a Fitting class, then ft91 = Qpe p gGpGp. is a Fischer class.
(d) We complete this discussion of examples by showing that not every Fischer
class has the form of (3./J). Let q e P, and let ft = (G e G: O,(G) < Z,,(G)), which
is a Fischer class as pointed out in Example (b) above. We suppose that ft =
Prep flOSrSp and derive a contradiction. First we assert that our supposition
implies that each F(r) contains 91. If not, for some r e p the class F(r) has characteristic
я # P, and then ft s G„GrGr. G. Let G e G\G„GrGr.. and for p e P\{<?} form
the wreath product W = ZprLiIC(j G. Since O,( W) = 1, we have W e ft, and therefore
3. Fischer classes, normally embedded, and permurable Fitting classes 605
follow^ tha?9?'t OWhiH Jr “ntra^t,0n- Hcnce 91 £ F(r) for each r e p> and it
follows that JI ( }repF(r)<5r<Zr. = g. But obviously F.(p/q\ e 9l2\g and we aeain
have a contradiction. Therefore the Fitting class g is noXaF in the sfn^ ofTe
preceding example. c
Next we show that the property of being a Fischer class is preserved by the Lockett
operator L„( ) and by the Fitting class product.
(3.8) Theorem (Lockett [1]). Let g and ffi be Fischer classes, and let n s P. Then
(a) G„(g) is a Fischer class, and
(b) go® is a Fischer class.
Proof, (a) Certainly L„(g) is a Fitting class by (1.15)(a). Let N < G e L„(g), and let
H/N be a p-subgroup of G/N. We must show that H e L„(g). By (1.15)(b) we have
G„(g) = L„(g) o Thus we can assume that p e n. Let S e Hall„(H). Since G e
G„(5), there exists an g-injector U of G containing S. We then have U n N 3 U e g
and (Li n H)/(U c. N)^(U n H)N/N e Qp, and consequently U n H e sfg = g. Since
S < U n H and p e л, it follows that Ce ll covers H/N; moreover, (U n H) r> N =
It n N e Injs(N). Hence we conclude from (1.6)(b) that Ur^He InjR(H), and there-
fore H e L„(g).
(b) Since g 0 © is a Fitting class by (1.12)(a), it will suffice to show that if
N < G e g 0 © and if H/N is a p-subgroup of G/N, then H e g 0 ©. Since HGg/NGg
is a p-group and NGg/Gg < G/GB g G, we have HGg/Gg e s, (f> = ©. Consequently
(1) H/(HoGs)e®.
Furthermore, we have Л'г 6й < HoGs and N n GR < GR e g. Because
(H n GS)/(N n G5) is a p-group, it follows that H n GR e sf g = g, and therefore
H n GB < Hs. We conclude that
(2) n (H n Gg)N = (Hr, Gg)(Hg r^N) = (Ho Gg)Nv = H nG„.
Finally, since H/(H n G„)N is a p-factor of H/(H n Gs) e ©, we conclude from (1.9)
that
(3) H/(HnGR)Ne©.
In view of Steps (1) (3) we can apply the quasi-R0-lemma (1.13) and conclude that
H/H-, e ©. Therefore He g 0 ®, as desired. □
Remark. In contrast to Part (a) of the above theorem, the operator LJ ) does not in
general preserve the s-closure of a Fitting class. For example, whereas 91 = s91, the
class L3(91) is not s-closed because Sym(4) e L3.(9l) and Sym(3) ф L3.(9l).
We now turn our attention to normal embedding and permutability. For the sake
of the logical development of this material, these two properties of Fitting classes are
investigated together. There are various reasons for studying normally embedded
Fitting classes. Not only do the Fischer subgroups and injectors coincide for such
classes, but the property is generally more tractable to work with (for example by
VIII 2.21(b) the injectors of a normally embedded Fitting class are characterized by
their cover-avoidance properties). Another motivation is the analogy with normally
embedded Schunck classes, for which a complete description is known (see Chapter
606 IX. Fitting classes—examples and properties related to injectors
VI, Section 4). However, there is clearly no hope of listing the normally embedded
Fitting classes in the same way, because even the determination of all Fischer classes
is well beyond our reach in the present state of knowledge; and then one must add
to the ranks of the normally embedded classes all the 3-normal Fitting classes (see
Definition 2.13(b)), which in their own right form a large and interesting class, being
in one-to-one correspondence with the subgroups of a certain infinite abelian group
as we shall show in Section 4 of Chapter X. Although the family of permutable Fitting
classes is even larger, there one still retains sufficient control over the injectors and
their position in the subgroup lattice to be able to draw conclusions that are not
available in the general case. The behaviour of the operator L,( ) with respect to
permutable Fitting classes is particularly good, as the following result shows.
(3.9) Theorem (Lockett [1]). Let 5 be a permutable Fitting class jet V be an ^-injector
of a group G, and let E = {G„: a £ P} be a Hall system of G which reduces into V. Let
n, and n2 denote sets of primes. Then the following statements are true.
(a) VG„- is an Ln(fi)-injector of G, and L,(ft) is permutable.
(b) FG,; n FG,; = FG(,1V,21- is an LlniUn2,(B)-injector of G, and L(,iU„;|(ft) =
(c) FGMFG„i= FG,,,^). is an Llni r.^)(?fFmjector of G, and L„2(L„JS)) =
Proof, (a) By definition of a permutable Fitting class and by I, 6.7 we have FG„. =
G,. F, and then from (1.16) it follows that the subgroup FG,. is an L„(ft)-injector of
G. Because G,. is E-permutable, from 1,4.29 we conclude that FG,. is E-permutable.
Therefore Statement (a) is true.
(b) Since G„; n Gn-2 = , = G(,1V,2,., it follows from A, 1.2 that FG,; n FG,; =
VG(n,vn2)- Therefore by Statement (a) the subgroup FG,„^„2). is an L(„iU„2|(ft)-injector
of G. Furthermore, each such injector is the intersection of an L,t(ft)-injector with
an L,2(ft)-injector, and from this observation the truth of the second assertion of
Statement (b) now follows.
(c) Because F permutes with E, we have FG,; FG„; = VVG^G^ = FG(„.vlli) =
VGi*,r.n2y- Therefore by Part (a) the subgroup FG„; FG,; is an L(,ir,2l(ft)-injector of
G. Since FG„; FG,. = (FG,;)G,-2, a double application of Statement (a) then shows
that an L|,r.,2|(ft)-injector is an L„2(L,i(g))-injector of G. Therefore L,2(L„i(ft)) =
and by a symmetrical argument we likewise obtain the remaining conclu-
sion of Statement (c). □
(3.10) Theorem (Lockett [1]). If 5 and (5 are permutable Fitting classes, then 5 0 ®
is permutable.
Proof. Let л = Char((f>), and denote L,(ft) by £. Let E be a Hall system of a group
G and F an ft-injector of GL. into which E reduces. Since F is pronormal in G, it
follows from I, 6.8 that E reduces into TVG(F). Because G = Gj.Ac(F) by the Frattini
argument, and because |Gt.: F| is a л'-number, the Hall л-subgroup S in E is
contained in 1VC(F). Let V/V be a (5-injector of FS/F. Then U is an ft 0 (5-injector
of G by (1.22) (a), and since E reduces into FS by 1,4.22(b), we can choose U so that
3. Fischer classes, normally embedded, and permutable Fitting classes 607
S also reduces into U. By I, 4.26 it will suffice to prove that if P e Svl (G) r> £ then
= PU. ByI 4.17(a) the Hall system ^reduces into
©-injector ofG/Ge by (1.22)(b). Since © is permutable, it follows that UG.P is a
subgroup of G by I, 6.7. Furthermore, since £ reduces into F and g is permutable
it follows that UnG^V permutes with G,oP. Then by 1, 6.10 the subgroup
F(Gt.nP) is pronormal in G. Since F(Gt.nP) has (nup)'-index in the normal
subgroup Gv of LGVP, the Frattini argument shows that l (Gv n P) is normalized by
a Hall (л и {p})-subgroup of GGt,P, and since by 1,4.22(b) the system L reduces into
UGeP. it follows from 1,6.8 that F(Gt. n P) is normalized by GGt.P n SP; in particu-
lar, F(LGVP n SP) = F(Gt. n P)(GGt.P n SP) is a subgroup of G. Then by repeated
application of the Dedekind law we have
UP = UP((f, n S)(GC n P) = LPF(Gt. n SP)
= GP(G£ n FSP) = GGVP n FSP
= IfUG^PrSP}.
Thus UP is a subgroup of G.
□
(3.11) Theorem. Let g be a normally embedded Fitting class, and let n be a set cf
primes. Then L„(g) is normally embedded.
Proof. Let V be a (normally embedded) g-i njector of a group G. Since g is permutable
by (3.4)(c), there is a Hall л'-subgroup S of G which permutes with F, and by (3.9)(a)
the subgroup FS is an L„(g)-injector of G. If P 6 Sylp(FS), either p e я', in which case
P e Sylp(G), or pen and P is conjugate to a Sylow p-subgroup of V. In either case
we have P ne G, and so FS ne G. Thus L„(g) is a normally embedded Fitting class.
□
We show next that the Fitting class product also preserves the property of
normal embedding; the following result shows that this is even true locally at each
prime p.
(3.12) Proposition (Lockett [1]). Let g and © be Fitting classes, and let pbea prime.
Assume that in each finite soluble group G the injectors for both g and © are p-normally
embedded subgroups. Then the g 0 ^-injectors of G are also p-normally embedded
subgroups.
Proof. Let n = Char(©), set £ = L„(g), and denote go© by &. Let U be an
Ъ-injector of a group Ge S. write R for Gc, and set F = U n R. Then V e Inj6(R),
and by (1.21) we have Fe InjS(R).
First suppose that pen, let U„ e Syl„(G), and put Fp = U„ c. R e Sylp(F). By defini-
tion of £ we have
(3-У)
рЛЯ : VP\-
1
608 IX Fitting classes—examples and properties related to injectors
If Л' = A'6(lj, by the Frattini argument we have G = NR, and by (1.22)(b) the
quotient 1//F is a ©-injector of N/V. The hypothesis that ffi-injectors are p-normally
embedded implies that GpF/F e Sylp«(l/p F/F)N/k>). Applying to this statement the
homomorphism from N/V to NR/R = G/R defined by nF -> nR (n e N), and writing
К = (Up), we then obtain UpR/R e Sylp(KR/R). Hence
(3.<5) pi\KR:UpR\.
By the usual rules for calculating indices we have
IК : l/p| = | К : UP(K n R)| | Lp(R n R): l/p|
= |K:KnGpR||KnR: UpnR|
= |KR:UpR||KnR:Fp|.
Hence |К: Lp| is a p'-number by (3.y) and (3.5), and therefore Up e Sylp«l/“>).
It remains to consider the possibility that p ф it. Since U/V is a p'-group, we have
Up = Vp. The set of all Sylow p-subgroups of the injectors of a group obviously form
a characteristic conjugacy class, and therefore by the Frattini argument G = Nc( fj,)R,
from which it follows that (Up) = (Vp) = <FpR>. But the fact that the 5-injectors
of R are by hypothesis p-normally embedded in R implies that Fp e Sylp«FpR>), and
so we can again conclude that Up e Sylp« Up ». □
The following theorem is an immediate consequence.
(3.13) Theorem (Lockett [ 1 ]). //8 and ® are normally embedded Fitting classes, then
8 0 © is also normally embedded.
(3.14) Remarks, (a) For any prime p and Fitting class 8. the Fitting class Lp(8) <s
permutable.
(b) The intersection of two permutable Fitting classes is not, in general,
permutable.
Proof, (a) By (1.15)(d) an Lp(8)-injector V of a group G has p-power index in G, and
it follows from 1,4.20(a) that V permutes with any Hall system of G which contains
a Hall p'-subgroup of V.
(b) Suppose that the intersection of two permutable Fitting classes is always
permutable. We shall show that this leads to the conclusion that every Fitting class
is permutable, a possibility that is ruled out by the construction of a non-permutable
Fitting class in Section 5 of this chapter. Clearly this supposition implies by an
obvious induction argument that the intersection of any finite collection of permuta-
ble Fitting classes is again permutable. Let A be an index set, and let {8/ 2 e A} be
a set of permutable Fitting classes. Set 8 = f | !8л: 2 e A}. If G is a finite soluble
group, let {Hly..., be the set of all subgroups of G which do not belong to 8-
For each i = 1,..., n there exists a 2, e A such that И, ф 8л,; let ® = Q"=1 Вл,, and
3. Fischer classes, normally embedded, and permutable Fitting classes 609
note that © is permutable. Since we evidently have TUG) = Trre(G) the «- and
©-injectors of G coincide, and therefore the «-injectors of G are permutablf Since
G was chosen arbitrarily, it follows that « is a permutable Fitting class. But by
*^el"ar .,a) above any FlttIng class « can be expressed as an intersection of permuta-
ble Fitting classes thus:
8=П{МЭ):реР},
from which we can make the (absurd) deduction that every Fitting class is permutable.
□
(3.15) Example. Let « = 33 = (G e <5: Soc3(G) < Z(G)). Then the Fitting class
G2(S) is permutable by (3.14) (a) but, as we shall now show, is not normally embedded.
Fet H = GL(2, 3), and let TV = <yi,, n2> be the natural F3H-module. Let x and у
denote the following elements of H:
1
0
1
1
and
O'
1
Since x3 = y2 = 1 and хг = x \ it is clear that the group L = <x, y> is isomorphic
with Sym(3). Let G = [N]H, and let X = <x>. Then NX e Syl3(G), N = Soc3(G) =
Cc(Soc3(G)),and CK(NX) = <п2У Furthermore, by inspection one checks that NL <
CG(n2). But L has a normal complement Q in H, and every non-identity element of
Q acts fixed-point-freely on N. Therefore NL = Cc(n2), and from an analysis similar
to that of (3.7)(a) we conclude from (4.19) that NL e Inj^fG). However, NL contains
the Hall 2'-complement NX of G. and therefore NL is an L2(«)-injector of G by (1.16).
Let Z = < — Iff Then Z = Z(H), and because Z is the unique minimal normal
subgroup of H, every normal subgroup of G of even order contains Z. But Z is not
contained in any conjugate of NL, and therefore NL is not normally embedded in
G. Observe too that by (3.11) the class 3’ is also not normally embedded.
Our next objective is a characterization of permutable Fitting classes due to
Lockett [2], and by way of motivation we begin with the following result of Fischer,
which will also be applied elsewhere.
(3.16) Theorem (Fischer [2]). Let ft he a Fitting class, and let К be a normal subgroup
of a group G with G/K e 91. 4ssume that the subgroup W = K9 is ^-maximal in K,
and let V be an ^-maximal subgroup of G such that V > W. If L is a Hall system of
G reducing into V and if D = Nc(£), then V = (LBjt3.
Remark. By (1.6)(a) the subgroup V in the statement of this theorem is an «-injector
of G. This result may be regarded as a sharpening of the observation that V - ( )g
with C a Carter subgroup of G.
Proof. Since V e Injg(G), we have VK/K pr G/K, and therefore VK <i byT 6.3(d).
If V e Syl (F), it follows that FpK < G because G/K e 91 If
we have L e Injg(LK), and consequently, if LK < G, it follows that L pr G by VIII,
610 IX. Fitting classes—examples and properties related to injectors
2.14(a). By taking L = Vp W, we conclude that V/W is locally pronormal in G/W. Since
(F/H/)r'i(K/H/) = 1, we deduce from I, 6.17 that V/W < NG/H,{1W/W), which equals
DW/W by I, 5.8. Since V is pronormal, subnormal, and g-maximal in DW, the
conclusion V = (£>1V)B is now clear. □
Let 1 be a Hall system of a group G which reduces into some g-injector F of G, and
let D = NC(Z); further let К <! G with G/K e 91. If V n К < G, then (3.16) applies and
V < (kn K)D. In general, however, we only know that V n К pr G (see VIII, 2.14(a)).
But by I, 6.8 even then the system normalizer D normalizes V r> K, and therefore
D(V c K) is a subgroup of G. Thus we are led to ask: Under what circumstances is
it true that V < D(V с К)? The answer is: Precisely when g is permutable. This is
one of the reasons that permutable Fitting classes are easier to deal with, because
their injectors are ‘controlled’ by system normalizers. To prove this result we need
two preliminary lemmas.
(3.17) Lemma. Let g be a Fitting class, and let Ibea Hall system of a group G which
reduces into an ^-injector V of G; let D = NC(Z). Further, let К < G with G/K e Sp,
and let S el c Hallp(G). Then the following statements are equivalent:
(a) FIS;
(b) (FnK)lSandF<D(FnK).
Proof, (a) => (b): Since (Fr, K}S = VS с К < G, the subgroup F с К permutes with
S. Let Gpelr\ Sylp(G). Then the subgroup Vp = F c Gp is a Sylow p-subgroup of
F, and so FpeSylp(SF). Let P = VpcNsl,(S}. By I, 6.8 we have P e Syl„(Nsr(S)),
and by the Frattini argument Nst,(S) covers SF/(SF с K}. Since SF/(SF rK) =
SVK/K e Sp, we see that P(SV с K) = SF, and with the help of the Dedekind law
we then have
F = VcSV = VcP(SVcK)
= VcP(ScK)(VcK) = (VcPS)(VcK)
= P(V c K)(V c S) = P(Vc K).
But by 1,5.4(a) we have Gp c NG(S) e Sylp(D), and since P < Gpc NG(S), we conclude
that V<D(VcK).
(b) => (a): By 1,6.8 the system normalizer D normalizes both S and F с K. There-
fore D(VcK} normalizes S(FrT), and since V<D(VcK), we have SF =
S(Fr> K)V, which is a subgroup of G. Hence SF = FS. □
(3.18) Lemma. Let g be a Fitting class, and let 1 be a Hall system of a group G which
reduces into an ^-injector V of G. Further, let n s P, let H e 1 c Hall„(G), and let
К < G with G/K e S„. Then the following statements are equivalent:
(a) F1H;
(b) (УсК)ЦНсК).
3. Fischer classes, normally embedded, and permutable Fitting classes 611
Proof, (a) (b): The Dedekind law yields (l'r, KUH r, K\ = VIH r, r K
V(H n K) is a subgroup by A, 1.6(c), so also is (Fr, K)(H n K).
(b)=>(a): The Hall system E reduces into FnК and Ho K, which are both
pronormal subgroups of G. Therefore the subgroup T = (V c\ KUH io Kl is uro
by I, 6.8. But by the Frattini argument AC(T) contains a Hall л-subgroup of G, and
therefore H normalizes T. Consequently (V n K}H = (Vo,K)(Hc. K)H = TH < G.
Since X reduces into F, the subgroup H contains a Hall л-subgroup of F, and therefore
V < (F n K)H because | V: (V to K)| divides |G: K|. which is a л-number by hypo-
thesis. Thus VH = (F n K)H, and VH is therefore a subgroup of G. □
(3.19) Theorem (Lockett [2]). Let g be a Fitting class. Then the following statements
are equivalent in pairs:
(a) g is permutable;
(b) If E is a Hall system of a soluble group G, if V is an ^-injector of G into which
E reduces, and if К < G with G/K e 91, then V < Nc(E)(Fr> K);
(c) If E is a Hall system of a soluble group G reducing into an ^-injector V of a
maximal normal subgroup К of G, and if V is an ^-injector of FAC(E), then V e Inj B(G).
Proof. As usual we prove a circle of implications.
(a) => (b): Suppose that g is permutable, and put D = NC(E). Let p e P and
S e E n Hallp.(G). Then SK < G, and since FS = SF, we can apply (3.17) (with SK in
place of the К of that lemma) to conclude that F < D[V r> SK). Thus D covers the
factor F/(F n SK). Since this holds for all p e P, it follows that D covers F/(F n K);
in other words, F<D(FnK), as desired.
Since the implication: (b) => (c) is obvious, it remains to prove that
(c) => (a): We proceed by induction on |G|. Let E be a Hall system of G, and suppose
that E reduces into F e Inj 5(G). Then we assert that V permutes with E. We assume
inductively that this has already been proved for all groups X with | A' | < |G|. Let К
be a maximal normal subgroup of G, and let p = |G:K|. Since E reduces into
Fr-, К e Injj(K), by induction we have
(i) (F n K) 1 S for S e E n Hall,,.(G), and
(ii) (F n K) 1 (H n K) for all n s P with p e n and H e E n Hall„(G).
Then by (3.18) it follows from (ii) that F1H.
Now let D = NC(E), recall from I, 6.8 that D normalizes F n K, and put U =
D(Fri K) Since E \ D and E \ F n K. by I, 4.22(b) we have E A V, and so there
exists an g-injector. F* say, of U such that E ч F*. Since F* contains the normal
g-subgroup VtoK of V and since Ln К is g-maximal in K, we have Fr,K-
F* К Hence by (1.6)(a) the subgroup F* is contained in some g-injector. Fo say,
of G. Observe that F* n К = Fo K. Since | F0/(F0 to K)| divides |G : K| = p either
F* = F or p* = К = FnK. If F*= Fo, then F* = FbecauseE\.F,F*. On
the other hand, if V* = VtoK, then by hypothesis VtoK is an g-mjector of G,
whence F n К = F and again V* = V. In any case we have F < U, and therefore, in
view of (i), it follows from the implication: (b) => (a) of (3.17) that F± S’. Hence by I,
4.26 (Condition (a)) we conclude that V is E-permutable. This completes the induction
step.
612
IX. Fitting classes—examples and properties related to injectors
The remainder of this section is devoted to connections between the partial order of
strong containment defined in (1.17) on the one hand and properties of normal
embedding and permutability on the other. For this purpose we shall need the
concepts of "boundary’ and ‘avoidance class’ for Fitting classes. We saw in Section 1
of Chapter VI how these concepts control strong containment for Schunck classes,
and we shall now see that in favourable circumstances the same is true for Fitting
classes. At this stage we pursue these ideas only as far as our immediate needs de-
mand; further results about strong containment for Fitting classes are to be found in
Chapter X.
The following definitions are straight analogues of the corresponding concepts for
Schunck classes (see III, 2.1 (c) and III, 4.15), except that here the universe is restricted
to S.
(3.20) Definitions. Let 8 be a Fitting class of finite soluble groups.
(a) The boundary b(8) of 8 is the class
b(8) = (G e 6\0r: if G * К sn G, then К e 8).
(b) The avoidance class u(8) of 8 consists of all single-headed soluble groups G
with the property that the maximal normal subgroup of G contains an 8-injector of G.
(3.21) Remarks. Let 8 be a Fitting class.
(a) It follows directly from the definition that each group G in b(8) is single-headed
and that the unique maximal normal subgroup of G is the 8-tnjector of G. Therefore,
in particular, we have b(8) E o(8)-
(b) Of course, our decision to use the same notation for two quite distinct concepts
of boundary introduces considerable ambiguity. However, we shall rely on the
context to make the meaning clear, and, in any case, for the rest of this chapter and
throughout Chapters X and XI we ordain that b(8) and o(8) shall have the meanings
ascribed to them by Definitions 3.20(a) and (b).
(c) As in the case of Schunck classes, it is possible to define the boundary of a
Fitting class with respect to an arbitrary universe, and then, of course, its meaning
will change from one universe to another. In the sequel, however, we shall use the
Fitting class boundary mainly with the universe G in mind.
The next result shows that the boundary and avoidance class control strong
containment, at least for Fitting classes that are either permutable or dominant (for
the meaning of a 'dominant’ Fitting class we refer the reader forward to Definition
4.1 in the next section).
13.22) Theorem (Doerk and Porta [1]). Let 8 ar>d ® be Fitting classes, and assume
that either 8 is permutable or <5 is dominant. Then the following statements are equivalent.
(a) 8 «(5;
(b) b(6) c fl(8).
Proof. The implication: (a)=>(b) is obvious and holds without any additional as-
sumptions about 3 or <f). To prove the opposite implication, assume that Statement
3. Fischer classes, normally embedded, and permutable Filling classes
613
(b) holds. We shall now derive a contradiction by supposing that Statement (a) is
false. Let G be a group of minimal order with the property that an R-injector of G is
not contained in a (5-injector. Let X be a Hall system of G, and let V and W denote
respectively the R- and (5-injectors of G into which L reduces. Let M be a maximal
normal subgroup of G. Then i'r, M and W n M are respectively the R- and №-
injectors of M into which XnM reduces. By the choice of G, a conjugate of the
pronormal subgroup fn M of M is contained in W n M, and therefore by 1,6.6(b)
we have i'rM < K7 M;in particular, V is not contained in M. Suppose that G
has a second maximal normal subgroup N / M. Then, as before, we have V n N <
(Tn N. If (V n N)(F n M) were equal to F, we should have V < W, contrary to the
choice of G; since the groups V/[V n N) and V/(Vn M), which are isomorphic with
G/N and G/M respectively, have prime order, it follows that l-'riN = VnM. Let
R = M n N. By order considerations we then have VR < G, and since G/R is abelian,
we conclude that FR a G. But we have already seen that V is not contained in a
maximal normal subgroup of G. Therefore M is the unique maximal normal subgroup
of G, and G is single-headed.
If Glf, F = G, then because Glf) and F n M are subgroups of W ri M e Inj l6(M), it
follows that M = Ge Fo M = Ge(F n M) < Wn M e 6. Since G ft, we therefore
have M = G(r, = W and G e b(6) £ a(R), and consequently F < IF, a contradiction.
Thus we may assume that Ge F < G. But now by the minimality of G, we have F <
Wj e Inj(f,(G(f,F), and so GeF = W} e (5.
If now (5 is dominant, then there exists % e Inj ffi(G) with Wo > Wj > F, a con-
tradiction. On the other hand, if R is permutable, let D = NG(L). By I, 6.8, both F
and W are normalized by D, and from (3.19), (a)=>(b), we have F<(Fr, M)D <
( W c. M)D < WD. If WD < G, then the minimal choice of G yields the contradiction
F < W. Thus WD = G, and so W < G and W = Glfl, But now IFF e 6 from above,
and so by the ft-maximality of IF we conclude that F < IF, a final contradiction. Q
We shall draw two conclusions from the preceding theorem, the first being that the
equivalence of its Statements (a) and (b) in fact characterizes permutable Fitting
classes R. This depends on the following elementary observation, which will also be
used elsewhere.
(3.23) Lemma. If R is a Fitting class and n a set of primes, then
b(LJS)) s «(8)-
Proof. Write £ = L„(R), let G e b(£), and let Fe InJs(G). Then by definition of the
boundary, Gc is the unique maximal normal subgroup of G. Since the subgroup
F n Gv is an R-injector of the £-group Gt, it contains a Hall л-subgroup of G If
F < G„ then FGV = G, and therefore F contains a Hall л-subgroup of G. But then
G 6 £, contrary to the assumption that G e b(£). Consequently F is contained in G*.
and we have G e a(R).
(3.24) Theorem. A necessary and sufficient condition for a ^ting class' 8 £
permutable is that for all Fitting classes (5 the statements: (a) R « © and (b) h(6) -
a(R) are equivalent.
614
IX. Fitting classes—examples and properties related to injectors
Proof. The necessity is the content of (3.22). To prove the sufficiency, let n be an
arbitrary set of primes. Then 6(L„(8)) £ a(8) by (3.23), and therefore by the assump-
tion that Statement (b) implies (a) we have 8 « L„(8)- It then follows easily from
(1.18)(b) and I, 4.26 that 5 is a permutable Fitting class. □
The second easy consequence of (3.22) is the following proposition.
(3.25) Proposition. Let 8, 6, and § be Fitting classes, and assume that 8 is permutable
or that Cbr.^iis dominant. If 8 « (5 and 8 « §, then 8 « © n
Proof. By (3.22) it is sufficient to prove that fc(© л §) £ a(8). Lei G e fc(© r> §), and
let V be an S-injector of G. Then either G^y, = Glf) or G(r„1S, = Gs, and G,,-, ^ is either
a ©-injector or an f>-injector of G. Since 8 « © and 8 « it follows that in any
case V < Gen&, and therefore G e a(8)- □
Remark. It is not known whether the conclusion of (3.25) holds for general Fitting
classes.
As we have already remarked in the proof of (3.24), Theorem 1.18(b) implies an
obvious characterization of permutable Fitting classes by strong containment. We
make this explicit in the next theorem, and then go on to show in the following
theorem that normally embedded Fitting classes can be characterized in a compara-
ble fashion.
(3.26) Theorem. A Fitting class ft is permutable if and only if ft « Ln(ft)foralln £ P.
(3.27) Theorem (Doerk and Porta [1]). Let ft be a Fitting class. Any two of the
following four statements are equivalent.
(a) 8 ,s normally embedded;
(b) 8 is strongly contained in a Fitting class © if and only if Lp(ft} n ®SP £ ® for
all primes p;
(c) For all primes p and all Fitting classes © we have 8 « (Fp(8) О ©)<=„.;
(d) For all primes p we have 8 « h(ft)QpQp:
Proof. We shall prove a circle of implications, beginning with
(a) => (b): Assume that 8 is normally embedded. Then 8 is certainly permutable
by (3.4) (c), and therefore 8 « © if and only if b(®) £ a(8) by (3.22). Let © be a Fitting
class which strongly contains 8- Suppose, by way of contradiction, that the class
(f-p(8) ©Gp)\© is not empty, and let G be a group of minimal order in this class.
Then every proper normal subgroup of G belongs to ©. Consequently GK is a
maximal normal subgroup of G, and is therefore the unique ©-injector of G. Since
ft « ©, it follows that G([| contains an 8-injector V of G, and therefore that |G : G(r,|
divides the index |G : F|, which is a p'-number because G e Lp(ft). On the other hand,
G e ®Gp = © О Sp, and therefore |G : Ge| = p. This contradiction shows that
(3-e) f-p(8) r> ®S„ s © for all primes p.
1
3. Fischer classes, normally embedded, and permutable Fitting classes 615
Now suppose that (3.e) holds. Let G 6b(6), and let F be an g-injector of G In
order to show that g « 6, by (3.22) it will suffice to show that Л connedI in the
unique maximal normal subgroup G6 of G. If not, then F/( V n G6) s G/GB which
/ hZ? Л к ‘hen P * C- and since G’s single^
Z 7. Л >|“‘ byihypothesis 8 is normally embedded, and therefore
P e Syl,(G). It then follows that G e Lp(g) n 6S, and hence by supposition that
G e 6, in contradiction to the fact that G e b(6). We therefore deduce that V < GK
and hence that g « (Vj. ~
1 Let a Fitting ClaSS’ let p 6 P’ an<j put 1 = ° ®)Sp- Then
Lp(g) - L„(g) с I. Furthermore, if q / p, we have I = IS, and therefore
L,(g) n X®, s I. The supposition that Statement (b) holds therefore implies that
g«X.
Since it is obvious that (c) => (d), it only remains to prove that
(d) => (a): Let p e P, let P be a Sylow p-subgroup of an g-injector V of a group G,
and put I = L„(g)Spep.. Since by supposition we have g « I, there is an I-injcctor
X of G with V < X. Let R denote the L„(g)-radical of G, and let T/R = OP(G/R). Then
T is the Lp(g)Sp-radical of G, and from (1.23) it follows that X/T is a p'-group. Since
(V r, T)R sn G and (F n T)R e L„(g), we deduce that V n T < GLp(g, = R. But X/T
is a p'-group and V < X, and so we have P < V n T < R. By definition of L„(g) it
follows that P is a Sylow p-subgroup of the normal subgroup R of G, and therefore
V is p-normally embedded in G. Since p and G were chosen arbitrarily, we conclude
that g is a normally embedded Fitting class. □
Concluding Remarks. In Example 5.19 of this chapter we shall exploit the Dark
construction to make an example of a Fitting class which is not permutable. It is our
impression that, in the present state of knowledge, permutability is the property which
marks a sharp dividing line between amenable and intractable behaviour of Fitting
classes. The few examples of non-permutable Fitting classes that are presently known
indicate a difficult passage beyond this frontier. Even within the domain of permuta-
ble and normally embedded Fitting classes many basic questions remain unanswered.
If g is a normally embedded Fitting class, then g n 6, is a'so normally embedded
(see Exercise 5 below). However, the corresponding question for a permutable g
is open. It is also unknown whether in general the intersection of two normally
embedded Fitting classes is again normally embedded, although in this case the
corresponding question for permutable Fitting classes has a negative answer (see
Remark 3.14(b)). A shortage of suitable examples has hindered progress with such
fundamental problems. Most of the known examples of normally embedded Fitting
classes are derived from Fischer classes and normal Fitting classes by means of the
operations L,( ) and the Fitting class product (cf. Theorems 3.11 and 3.13). Moreover,
it often seems difficult to decide, even with the help of Lockett’s criterion (3 19),
whether Fitting classes which are known not to be normally embedded are, in fact,
permutable. For example, the following question was first raised by Lockett [1 ] m
his thesis:
Open Question. Is the soluble Fitting class 3' (of groups with p-socle central) per-
mutable? (See Example 2.9(a).)
616 IX. Fitting classes—examples and properties related to injectors
A negative answer here would settle the question whether there exist dominant Fitting
classes which are not permutable. It is also unknown whether examples of the type
described in Construction F of Section 2 are permutable; they certainly need not be
normally embedded (see Exercise 11 below).
Exercises
1. (Lockett [1]) Let 5 and (5 be Fitting classes. Let W be a (5-injector of an
g-injector F of a group G. If W p-ne V p-ne G, show that W p-ne G.
2. (Porta [1]) Let g and 6 be Fitting classes. If g is normally embedded, show that
g « (5 if and only if Lp(g) £ Lp(6) for all p e P.
3. (Porta [1]) Let g and 6 be Fitting classes which both strongly contain Sp. for
some p e P. If g is normally embedded, prove that g « 6 if and only if g £ (5.
4. (Porta [1]) Prove that a Fitting class g is normally embedded if and only if g is
permutable and there exist distinct primes p and q such that Lp-(g) and L,.(g)
are normally embedded Fitting classes.
5. (Porta [1]) Let g and 6 be normally embedded Fitting classes, and let л be a set
of primes. Prove that (a) 5 r\ is normally embedded and (b) if (5-injectors of
g-groups are always in g and if g n (5 « g, then g n 6 is normally embedded.
6. (Porta [1]) Show that a Fitting class g does not strongly contain 91 if and only
if there is a prime p and a group G in b(g)\a(9f) such that the following four
conditions hold: (i) |G/GR| = p; (ii) {PF(G)p.: P e Sylp(G)} is the set of 9i-injectors
of G; (iii) F(G)p. < Z(G); (iv) If 91 £ g, then the Sylow p-subgroups of G are not
abelian.
7. (Porta [1]) Let Яр denote the Fitting class
= (G e 6: F(G)P. < ZJG)).
If g is a Fitting class, show that any two of the following conditions are
equivalent:
(a) 91« g;
(b) « Lp(g) for all p e P;
(с) Яр £ Lp(g) for all p e P;
(d) 9i « Я, n g for all p e P.
8. (Porta [1]) Let JE denote the smallest Fitting class containing (J rtf'Show
that (a) JE is not normal in S, and (b) if g is a Fitting class with JE £ g, then 9i « g.
9. (Lockett [1]) Prove that a Fitting class g is permutable if and only if for all
groups G the following condition holds: If E is a Hall system of G reducing into an
g-injector of G, then V < G^Ng(Y r, Grv1j).
10. (Porta [1]) If g and (5 are Fitting classes of finite soluble groups and if g is
permutable, prove that g « (5 if and only if Lp(g) « Lp((5) for all p e P. (Com-
pare with Exercise 2 above.)
11. Show that the class 3(1,2) described in Construction F (see (2.19) and (2.21) for
its definition.) is not 2-normally embedded.
12. Let g be a Fitting class of finite soluble groups, and let л £ P. Show that
g«L„(g)oa(Lp(g))Su(g).
4. Dominance and
some characterizations of injectors
617
Open Question. If g and 6 are Fitting classes
that g « ft if and only if n(ft) c a(g)?
of finite soluble groups, is it true
4. Dominance and some characterizations of injectors
The main aim of this section is to find simple descriptions of the injectors for certain
Fitting classes. A useful property of Fitting classes in this connection is that of
dominance, which, as we shall show, is satisfied by a variety of examples arising from
the constructions of Section 2. At the end of the section we also examine some Fitting
classes which satisfy the stronger property of‘normality’ introduced in (2.13). Al-
though many of the results are proved only for a soluble universe, we assume
throughout this section that, unless otherwise stated, the universe is tt.
The fruitful investigation in Chapter VI, Section 3 of Schunck classes with the
so-called D-property leads one naturally to look for a suitable dualization. In a
soluble universe, a direct translation of this concept from the projectors of a Schunck
class to the injectors of a Fitting class leads only to a characterization of the classes
S„. For, if g is a soluble Fitting class of characteristic n, then by (1.9) we have
91„ £ S £ e,.
Therefore, if the g-injectors of a group contain every g-subgroup, they must be Hall
л-subgroups, and so the classes {S„: л £ P} are the only Fitting classes with the
£>-property in a soluble universe. However, the following concept of a ‘dominant’
Fitting class, which was first studied by Lockett [1], seems to offer a more interesting
variation on this theme.
(4.1) Definition. Let X = s,X £ tt, and let g be a Fitting dass. We say that g is
dominant in X if (i) g £ X, and (ii) for all G e X any two g-maximal subgroups of G
containing GB are conjugate in G.
Remarks, (a) We shall say simply “g is dominant” and omit the qualification “m X”
when X is the universe under consideration in a given context.
(b) Obviously Fitting classes which are normal in X in the sense of Definition 2.13
are examples of dominant Fitting classes. However, from the wide variety of examples
described below, it will be evident the dominance is a considerably more general
concept than normality.
(4 2) Lemma. Let %bea Fitting class which is dominant in X — s„X. Then every group
G in X has a unique conjugacy class of ^-injectors, namely the Q-maxtmal subgroup of
G containing GB.
proof Let G e it. and let Г be an 3-maxintal «Я*™»», C contains C,. We
M«t»ion|G|io М Hsan8-inJ.e.ot<TG.Tb.s««™»tn»*C.l.
618
IX. Fitting classes—examples and properties related to injectors
Let К <i- G. Then К e X and = Ga n К < V n K. We assert that I'r.K is
g-maximal in K. Let Vn К < V < К with U eg. Since [Ga, K] <K9< U, we have
GgG e Nog = g, and by the definition of dominance in I it follows that GaU < Vе
for some ge G. Then U < Vе c\ К = (Vn K)", and by order considerations we there-
fore have U = V r> K, thereby justifying our assertion. Consequently by induction
Vn К is an g-injector of K. Since V is g-maximal in G, we conclude from (1.3)(c)
that V e Injg(G). □
(4.3) Proposition (Lockett [1]). Let g be a Fitting class which is dominant in S. Then
either 'Ji £ g or g = S, for some n £ P.
Proof Let л = Char(g), and suppose that there exists a prime p £ n. Let H e S„, and
let G = Zprlircs H. If В is the base group of G, then В = CG(B). If В n Gs > 1, then
p| |Ga|, and by (1.9) we have pen, contrary to supposition. Therefore В n Gg = 1,
and consequently Ga < Cc(B) = B; hence Gg = 1. Let L be an g-maximal subgroup
of G, let qen, and let Q e Syl,(H). Since Q e 91„ £ g, by definition of dominance we
have Q < Ls for some g e G. It follows that |H| | |L|, and since Lr В < G. = 1, we
conclude that G = LB and therefore that H = G/B = LB/В = Le^. Thus we have
shown that S„ £ g, and it now follows from (1.9) that g = S„. □
The following example shows that the necessary condition for dominance in S
given in (4.3) is not a sufficient condition. We also use it to show that the intersection
of two dominant Fitting classes need not be dominant, even if both contain 91.
(4.4) Example. Let g denote the Fitting class 91 О S3. We assert that g is not
dominant in S, although clearly g contains 91. To see this, let S = Sym(3), let M be
a faithful irreducible S-module over F5, and let T = [M]S. Next, let N be a faithful
irreducible T-module over F2 (note that the existence of such modules M and N is
ensured by B, 10.7), and finally set G = [NJ T. Since F(G) = N, a 2-group, the
91-injectors of G (as 9l-maximal subgroups containing F(G)—see (4.12) below) are the
Sylow 2-subgroups of G; consequently MF(G) is the L(3|(91)-radical of G. It follows
from (1.22) that the set of g-injectors of G is precisely {F(G)P: P e Syl3(G)}. However,
a Sylow 2-subgroup of G is an g-subgroup containing Ga = F(G) and is not contained
in any g-injector of G; therefore g is not dominant in <5.
To justify the second claim for this example, let (5 = 9l|2 5)S{2t5).. It is straight-
forward to verify that F(G) is the ©-radical of G and that the g-injectors coincide
with the (5-injectors of G. Consequently (9 is also not dominant in S. However,
clearly (5 = S2 S2. n S 5 S 5., and, as we shall see in the next result, classes of the form
S„S„- are always dominant in S. Therefore the intersection of two dominant Fitting
classes is not in general dominant.
We shall see in (4.13) that 91 is dominant in (S. Since by Sylow’s theorem the class
S3 is obviously dominant in any larger class, the preceding example also shows that
neither the class product nor the Fitting product of two dominant Fitting classes is
in general dominant. However, there are favourable circumstances under which the
Fitting product preserves dominance, as the next result shows.
4. Dominance and some characterizations of injectors
619
(4.5) Proposition In a soluble universe, let g and 6 be Fitting classes, let n = Chart©)
and assumethat% - If if, [s dominant in S, then so also is %0®.In particular
the class g9l is dominant for all Fitting classes g.
Proof Let Ge®, and let X be an g о ©-subgroup of G containing G^„ B. Let
R — Gs and T/R — F(G/R). Since g = gS„. by hypothesis, we have T/R e 91 £ ©
and hence TegO© because Ts = R. Consequently T < X, and therefore from
(1.20) we deduce that = R. It follows that X/R is a ©-subgroup of G/R containing
the ©-radical GRo lf,/R of G/R, and therefore by hypothesis X/R is contained in a
©-injector U/R of G/R. But by (1.23) the subgroup C is an g о ©-injector of G, and
this proves that g о © is dominant.
The final sentence of the Proposition rests on the fact that 91 is dominant, which
is proved subsequently from first principles in (4.12) below. ’ □
After this brief skirmish with the question of when the Fitting product preserves
dominance, we turn the question round and ask whether anything can be said about
g or © when g О © is dominant. Not surprisingly, in general very little can be said.
For example, even when g and g 0 6 are both dominant, there need be no restriction
on ©. For, if g is any Fitting class normal in S, we shall see in X, 3.17 that g о 6 is
again normal (and hence dominant) in S for all Fitting classes 6. However, the next
theorem shows that information about 6 is indeed forthcoming, provided that go©
is assumed to be dominant for sufficiently many Fitting classes g. The equivalence
of Statements (a) and (c) in this theorem is due to Blessenohl [3].
(4.6) Theorem. For the universe S any two of the following statements about a Fitting
class © are equivalent:
(a) g О © is dominant for all Fitting classes g;
(b) go© is dominant for all Fitting classes g such that (I) / g £ 9l2;
(c) © is dominant and 91 £ ©.
In the course of the proof of this theorem we shall need the following fact, whose
proof is postponed until the end.
(4.7) Lemma. Let p e P. let r e P\{p}, and let g = SrSp n 3. where 3 denotes the
class of all groups with central socle. Then the Fitting class g 0 £„• is not dominant
in S.
Proof of (4 6) Of the three statements in the theorem, it is obvious that (a) implies
(b), and it follows at once from (4.5) that (c) implies (a). Therefore the burden of the
proof is to show that (b) implies (c). ,
Assume that Statement (b) holds. We first suppose that 6 is not dominant and
derive a contradiction. On this supposition there exists a group G whrc i has a
©-subgroup H containing G6 and contained in no ©-injector of G. Let p be a prime
which does not divide |G|, let W = G, and let F de"otetbeb^ group of
Since V e Svl (IF) it follows from (1.22) that the set of Sp 0 © J
V e Sg)}. On the other hand, V = ОДИН), and therefore FH.san £, 0 6-
620 IX. Fitting classes—examples and properties related to injectors
subgroup of W-, furthermore, VH contains the Sp О ©-radical VGa of W. Since VH
is clearly not contained in any subgroup of the form VU with U e Injffi(G), we
conclude that Sp О © is not dominant. But since this contradicts our assumption
that Statement (b) holds, it follows that © is dominant.
Let it = Char(©). It remains to prove that л = P. Suppose that л / P. Then
© = S„ by (4.3). If |я'| = 1, then © = Sp- for some p e P. But this possibility is ruled
out by (4.7), which shows the existence of a non-trivial Fitting class 8 £ 9l2 such that
8 0 Sp. is not dominant. If |л'| > 2 and p e л', then by assumption Sp 0 S„ is a
dominant Fitting class of characteristic {p} cj n / P. Therefore once more by (4.3)
we have бр О S,= S(pK,„, which can only happen if n = 0. But if n = 0, then
© = (1), and our assumption implies that 8 = 8 0 © is dominant for all 8 £ 9l2,
which contradicts the evidence of Example 4.4. Hence Char(©) = P, and the proof
of Theorem 4.6 is complete, subject to the validity of Lemma 4.7.
Proof of (4.7). To show that (SrSpn3) 0 Sp. is not dominant, choose a prime
q ф {p, r}, and let £ = E(p/q). By B, 10.7 there exists a faithful irreducible £-module
M over Fr; let L = [M]£. Next let T = E(p/r), set
G=T(\jeL,
and let R = O,(G). From A, 18.5(a) we know that the subgroup R is minimal normal
in G, and so clearly Soc(G) = R. Then by (2.9) (a), (2) we have Gj = CG(R) = R, and,
recalling that 8 = SrSp n 3,we therefore conclude that Gs = R. Furthermore, since
Op.(G/£) = 1, the subgroup R is also the 8 0 Sp.-radical of G. Because M is a
transversal to £ in L, we have RM = Or(T) Qjreg M, and in consequence Z(RM) is
the intersection of R with the diagonal subgroup of the base group of G. It follows
that Z(RM) < Z(RL). Furthermore, since Z(RM) has exponent r and RM is an
r-group, we have Z(RM) = Soc(RM). Let X < L, and let TV be a minimal normal
subgroup of RMX. Since R = CG(R), it follows that N < R. Since RM is an r-group,
we have N n Z(RM) / 1 and therefore N < Z(RM) = Z(RMX). Hence we conclude
that Z(RM) = Soc(RMX).
Consequently, if P e Sylp(L), we obviously have RMP e 8- Let В denote the base
group of G, and let Q e SylJL). From the fact that RM is evidently an 8-injector of
both BM and BMQ we deduce that BMQ ф and hence that BM is the Lp (8)*
radical of G. It follows from (1.22) that RMQ is an 8 0 Sp--injector of G. Since no
conjugate of RMQ contains RMP, which is an 8 0 Sp.-subgroup of G containing
the 8 0 Sp -radical, we conclude that 8 0 Sp. is not dominant. □
Remark. In the context of (4.6) we refer the reader to Exercise 8 in Section 2 of Chapter
X, where there is described a sufficient condition for the dominance of a Fitting class
© to be a consequence of the dominance of 8 0 ©•
Open Question. Let л be a set of primes, and let 8 be a Fitting class which is dominant
in 6. Does it necessarily follow that L„(8) is also dominant in S? (See (4.9) and (4.10)
for sufficient conditions for the dominance of L„(8) )
4. Dominance and some characterizations of injectors 621
A positive answer to this question would imply that dominant Fitting classes are
7 ’in P%ticu,ar’that is permutable (since it will be shown
in (4. 9) that 3 is dominant). To justify this assertion we observe that if L (ft) is
dominant, we can conclude from (3.22) and (3.23) that ft « L„(g), and then from 126)
that g is permutable. v '
(4.8) Proposition. Let peP, and let's be a p-normally embedded Fitting class of finite
soluble groups. Then g 0 Gp. is dominant in <S.
Proof By (3.12) the ft о Gp.-injectors of each finite soluble group are p-normally
embedded, and therefore we may assume without loss of generality that ft = ft о Gp..
Let G e G. If R denotes the ft-radical of G and if T/R = F(G/R), this assumption
implies that T/R is a p-group. Let R < H < G with Heft. Since R < Я n T e s.g =
ft and because R = Ts is g-maximal in T, it follows that H r> T = R e Injg(T) and
hence from (1.6)(b) that Я e Inj S(HT). Let P e Sylp(tf). By hypothesis P ne ЯТ, and
therefore PR/R ne HT/R by I, 7.3(b); in particular, PR/R is pronormal and subnor-
mal, and hence normal, in the p-group PT/R. It follows that [P, T] < PR n T <
Я n T = R, and so by A, 10.6(a) we have P < P r> Cc(T/R) < P r> T < R. Thus H/R
is a p'-group, and we have Я < RQ for some Q e Hallp.(G). Since RQ e gGp. = g,
we therefore conclude that {RQ: Q e Hallp.(G)} is the unique conjugacy class of
g-maximal subgroups of G containing R = Gg; hence ft (= ft 0 Gp.) is dominant.
□
If ft has p-normally embedded injectors, then so does Lp(g) by (1.19)(d). Therefore,
substituting Lp(g) for ft in (4.8) and appealing to the fact that Lp(g)Gp. = Lp(g) by
(1.15)(b), we obtain the following corollary to Proposition 4.8.
(4.9) Corollary. Let ft be a Fitting class whose injectors are always p-normally em-
bedded. Then LP(S} is dominant in G.
Remark. Proposition 4.8 has been proved by Blessenohl [3] under the stronger
hypothesis that ft is a Fischer class. We mention in passing that there seems to be
no obvious connection between Fischer classes and the property of dominance.
Example 4.4 shows that even s-closed Fitting classes need not be dominant, and on
the other hand, the class 33 described in (3.7) (a) is an example of a dominant Fitting
class (see (4 19)) which is not a Fischer class. Among the G-normal Fitting classes,
which of course are dominant in 6, the class G itself is the only one which is a Fischer
class (see X, 1.25 and X, 3.7).
(4.10) Theorem (Doerk and Porta [1]). Let n be a set of primes, and let % be a Fating
class which is dominant in G. Assume further that the ^-injectors of all groups in G
are p-normallv embedded for all p 6 я. Let H be a subgroup of afimte soluble group
G such that G*<He L„(g). Then H is contained in an L„(B)-‘h/ector of G. In
particular, we have
(a) L.(ft) is dominant in G, and
(b) if is a Fitting class satisfying ft £ S S L„(g), then « „(ft).
622 IX. Fitting classes—examples and properties related to injectors
Proof. If n = 0, then L„(8) = S.an<f the statement of the theorem is certainly true.
Next we deal with the special case where |n| = 1. Let n = {p}, and let V* be an
8-injector of H. Then \H: P*| is a p'-number because H e Lp(8)- Since Gs < H, we
have Gr < V*, and therefore, because 8 is dominant, there exists an 8-injector V of
G which contains V*. Let L denote the Lp(8)-radical of G. By (1.19)(a) the index
\VL: L| is a p'-number. Since \HL: V*L( = \H:(Hn F*L)|, which divides |Я : V*\,
it follows that HL/L is a p'-group and hence that HL/L < QL/L for some Q e Hallp.(G).
Since QL is an Lp(g)-injector of G by (1.19)(b), the conclusions of the theorem hold
in this case also.
Finally we handle the general case. Let pen. Then H e Lp(^), and by the case
| л| = 1 there exists an Lp(8)-injector Wp of G containing H. By (1.19)(c) and (1.18)(b)
we have Wp = PGp. for suitable V e InjB(G) and Gp. e Hallp(G). Let Wn = p]pe„ Wp.
By A, 1.6(c) we have Wn = G„ V = KG, with G,. = Qpe, Gp- e Hall„.(G). Then by
(1.16) the subgroup W„ is an L,(8)-injector of G containing H. □
Remarks (a) Let p and r be distinct primes, and let 8 denote the class (2,GFn 3) О £p.
By (4.7) the Fitting class 5 is not dominant in S. But we have 8 = g 0 Sp. = Lp(8)
by (1.15)(d), and therefore by (3.14)(a) the Fitting class 8 is permutable. Thus the
conclusions of (4.8) and (4.9) no longer hold when the p-normal embedding of
injectors in the hypotheses is weakened to permutability.
(b) Let n be a proper subset of P containing at least two primes, p and q say, and
let r e P\n. Let 8 = 91S„. and put
G = Zprb(Z,rb(ZrrbZp))>
where the wreath products are all regular. Let В be the base group of G, let R e Syl, (G)
and P e Sylp(G). Then it is straightforward to verify that В = 0p(G) = F(G), and hence
that BR is an g-injector of G. Since В < P e 8, it follows that 8 = L„(8) is not
dominant in S. Since 8 is obviously normally embedded by (3.13), we conclude that,
when |тг| > 2, the assumption that 8 is dominant cannot be omitted from the
hypotheses of Theorem 4.10.
(4.11) Theorem. In a soluble universe a Fitting class 8 is normally embedded if and
only if Lpffi} is p-normally embedded for all primes p.
Proof. The necessity of the condition is proved in (1.19)(d). To prove the sufficiency,
assume that for all primes p the Lp(g)-injectors are p-normally embedded in every
finite soluble group. Let p e P. By (4.8) the Fitting class Lp(8) 0 Sp. is dominant, and
therefore Lp(8) is dominant by (1.15)(b). Since h(Lp(8)) £ a(8) by (3.23), we deduce
from (3.22) that 8 « Lp(8) for all primes p, and hence from (1.18) that 8 is permutable.
By (3.9) there exists a Ve InjB(G) and an S e Hallp.(G) such that KS is an Lp(8)-
injector of G. If P e Sylp(K), then clearly P e Sylp(KS), and so by assumption P p-ne
G. Since this is true for all p e P, it follows that V ne G. □
Remark. No analogue of (4.11) holds for permutable Fitting classes. For, if 8 is any
soluble Fitting class, it follows from (3.14)(a) that Lp(8) is permutable.
4. Dominance and some characterizations of injectors 623
The rest of this section is devoted to showing that certain special cases of Construc-
tions C and D described in Section 2 give rise to dominant Fitting classes. First we
consider Construction C. Let I and '5 be Fitting classes of finite groups with Is?)
and let R(G) = G^/G* for all G e (£. Let 3 be a subclass of the class 3 of all finite
simple groups, and define a Baer function f as follows:
(4.a)
(1)
/(J)=^D0(J)
for J e 3 r> ?I
for J e 3\?I
for J e 3\3.
e
We recall that HR(f, R) denotes the class of groups G for which R(G) is /-hypercentral
and that it is a Fitting class by (2.4). Then Blessenohl and Laue [2] prove that
HR(f, R) is dominant (ordentlich) in 6. (In fact, they prove slightly more by allowing
f(J) to lie between D0(J) and Dos„(Aut(J)) for J e 3\9I.) We shall be content to prove
two important special cases of this theorem. The first of these includes the theorem
that 91 is dominant in S (originally proved by Fischer [1]) and also contains a
description of the 9l-injectors of a group which is due to Dade.
(4.12) Theorem (Dade; Mann [4]). Let G be a finite group with the property that
Cc(F(G)) < F(G). Let a denote the set of prime divisors of |F(G)|, and for each pea
put F(G) = Fp x Fp., where Fp and Fp. denote respectively the Sylow p-subgroup and
the Hall p'-subgroup of F(G). For each pea let Vp be a Sylow p-subgroup of Cc(Fp.)
for each pea. Then
(a) [Pp, Ц1 = 1 for Mel! with P ^4> and
(b) the subgroup <Pp:peo> = )(pf„ Vp is a nilpotent subgroup of G containing
F(G).
Let V(G) denote the set of all such subgroups <Vp: pea) obtained from the various
choices of Vp e Sylp(Cc(Fp.)). Then
(c) if W is a nilpotent subgroup of G containing F(G), then W < V for some
V e V(G),
(d) T(G} is a conjugacy class of G, and
(e) each V e <(G) is an 9l-injector of G.
Proof, (a) First observe that [Fp, Cc(F,.)] < Cc(Fp.)r> Cc(F,.) < Cc(F(G)) < F(G)
by hypothesis. Hence Cc(Fe.) normalizes FpF(G). Because Fp < Op(Cc(Fp.)) < Vp, we
have VpF(G) = Vp x Fp., and therefore Fp char FpF(G). Consequently
(4.Д)
Cc(F,.) normalizes Vp,
and, in particular, Vq normalizes Vp. By a similar argument Vp normalizes Vq, and it
follows that [Vp, Ц] < Vp nК - /у peo> is the direct product of its
(b) We deduce at once from Part (a) tnat vp.p to/
Sylow subgroups Vp and also that F(G) = \pC„ „ < X₽6» ₽ '
.. i rt nea Since We91, the Sylow p-subgroup Wp of W centralizes 0„(Ю>
~ “"X1 * ’ r
subgroup, Vp say, of Cc(Fp ). Hence И = X₽-
624 IX. Fitting classes—examples and properties related to injectors
(d) Let V = p6 „ Vp and V = X p6 „ Vp be two typical elements of P(G). By Sylow’s
theorem we can find, for each pea, an element x(p) e Cc(Fp.) such that Vp = (V'p)xip>.
Let .x = ПгегМрк where the product may be taken in any order. If q / p, by (4.JB)
the element x(</) normalizes each conjugate of Vp, and therefore Vх = у pep(Ij,)* =
(e) Since we have now shown that G has a unique conjugacy class of maximal
nilpotent subgroups containing F(G), we conclude that 91 is dominant in the following
class:
X = (GeE: Cc(F(G)) < F(G)).
Of course X contains the class S by A, 10.6(a). We assert that X is s„-closed. If
К -a G e X, then F(K) =Kr F(G), and therefore the group CK(F(K)) is normal in К
and centralizes both F(G)/F(K) and F(K). Since the centralizer of F(G) in CK(F(K))
is clearly К r> Z(F(G)), it follows from IV, 6.9 that the automorphism group
Ck(F(K))/K r> Z(F(G)) induced on F(G) is nilpotent. Since К n Z(F(G)) < Z(F(K)),
we conclude that CK(F(K)) is itself nilpotent and is therefore contained in F(K). Thus
X = s„X, and so by (4.2) the groups V in 'F'(G) are 92-injectors of G. □
Remark. In fact, the class X = (G e £: Cc(F(G)) £ F(G)) is a Fitting class strictly
larger than S (Perez Monasor [1]; Iranzo, Perez Monasor [1]).
(4.13) Corollary (Fischer [1]). The class of nilpotent groups is dominant in S.
We recall from (2.6) and the subsequent discussion that the class SB of generalized
nilpotent groups consists of all finite groups which induce only inner automorphisms
on their chief factors; that SB is a Fitting class; and that the SB-radical F*(G) of a group
G is characterized thus:
F*(G)/F(G) = Soc(Cc(F(G))F(G)/F(G)).
Our next goal is to prove that SB is dominant in (£. From the definition of the class
SB in (2.5)(c) we see that it is an example of the class HR(f, R) defined by (4.a) on page
623 and so this result is also a special case of the theorem of Blessenohl and Laue
cited above.
(4.14) Theorem (Blessenohl and Laue [2]). Let SB = (G e (£: G = F*(G)), the class of
generalized nilpotent groups. Let G be a finite group, and let R = Cc(F*(G)/F(G)). Then
(a) CK(F(«)) < F(«) = F(G);
(b) If F*(G) < U еШ, then there exists an9l-injector V of Rsuchthat U < VF*(G);
(c) If V is an 9l-injector of R, then VF*(G) e SB.
Proof, (a) We recall from A, 10.6(a) that Cc(F(G))F(G)/F(G) has no nontrivial abe-
lian normal subgroups; therefore F*(G)/F(G) is a direct product of non-abelian simple
groups and, in particular, has trivial centre. Hence R r> F*(G) < F(G). Put C =
4. Dominance and some characterizations of injectors
625
Cr(F(R)). Since FIG) < К < G, we have F(R) = F(G), and therefore C = C„(F(G))
f ’ i P°STble>C * F(G)’and kt M/F(G) be a minimal normal subgroup
°f h/f (C^nS ,n.CF(G)/f(G)- Then by A’ 413<c) and by definition of F*(G)
г ™ C°nsequent,y f F*(G) = F(G), which is absurd.
1 hus C < F (G) (= F(R}\
(b) Since[F*(G), F(G)] < F*(G) n F(G) = F(G), it follows that F(G)is a nilpotent
subgroup of R containing F(G) = F(F). Therefore by Part (a) we can apply (4.12) to
deduce that R possesses an 9l-injector V which contains F(L'). To complete the proof
of Part (b) we now show that
(4-У)
U = F(G)F*(G).
Let T = Cg(F(G))F(G)/F(G) and C* = CT(F*(G)/F(G)). By definition F*(G)/F(G)
is the socle of T, and C* n (F*(G)/F(G)) < Z(F*(G)/F(G)) = 1; therefore, because
C*^ T, it follows that C* = 1. Now let S denote the subgroup CV(F(G))F(G)/F(G)
of T. Since F(U) < R, we have СГ([;1(Р(С)) < F(G) by Part (a), and consequent-
ly Cu(F(G))F(G) r\ F(L7) = F(G). Obviously CV(F(G)) < CV(F(G)), and therefore, as
U e ®, we have U = CV(F(U))F(U) = CV(F(G))F(U). It then follows that U/F(U) =
Cc,(F(G))F(G)F(L')/F(Lj S CV(F(G))F(G)/(CV(F(G)}F(G} n F(L)) = S, and because
U = F*(U), we conclude that S is a direct product of non-abelian simple groups
containing F*(G)/F(G) as a normal subgroup. By A, 4.13(b) the subgroup F*(G)/F(G)
is a direct factor of S, and since CS(F*(G)/F(G)) < C* = 1, this implies that
F*(G)/F(G) = S. Hence U = F(G)Cc,(F(G))F(G) = F(L')F*(G), and (4.y) is
satisfied.
(c) Put W = PF*(G). Since V < R = Cc(F*(G)/F(G)), we have W/F(G) =
F*(G)/F(G) x V/F(G}. Therefore V = F(H'). Let A = Cf.(C)(F(G)) and К = CA(V).
Since A centralizes V/F(G) and F(G), from IV, 6.9 we know that A/К is nilpotent,
and because A/(A r> F(G)) S F*(G)/F(G), which is perfect, we conclude that A =
K(AnF(G)). Therefore KF(G) = F*(G), and in consequence we have W>
FC^V) > FF(G)K = FF*(G) = W. Hence W = F(B,)C„,(F(BZ)), and it is now clear
that W e !B. □
Let G e (E, and let R = Cc(F*(G)/F(G)). It follows from Statements (b) and (c) of the
preceding theorem that the subgroups
{VF*(G): PeInj«(F)}
are the only maximal generalized nilpotent subgroups of G containing F‘(G). By
(4 121(d) (e) and (1.3)(b) the 9Linjectors of the normal subgroup R of G form a
characteristic conjugacy class of R and are therefore conjugate in G. This observation,
together with (4.2). leads to the following theorem.
(4 15) Theorem (Blessenohl and Laue [2]). The class S of generalized nilpotent
groups is dominant inb-Etery finite
and this consists of those subgroups U of G which are maximal with respect to the
condition F*(G) <U = F*(U)-
626
IX. Fitting classes—examples and properties related to injectors
The Fitting classes that arise from Construction C include, of course, all primitive
saturated formations, and these are by no means all dominant (see Example 4.4). In
fact, the determination of all primitive saturated formations which are dominant
might provide some helpful insight into the property. We end our study of special
cases of Construction C by characterizing the injectors for the class described in
Example 2.5(b), restricted to a soluble universe. This class turns out to be dominant
in S and provides an example which is not covered by the hypotheses of the cited
theorem of Blessenohl and Laue.
We recall from (2.5)(b) that for each it £ P we obtain a Fitting class D", which, in
a soluble universe, may be described as follows:
O" = (Ge6:G/Cc(0,(G))e6J
(4.16) Theorem (Lockett [1]). Let G e S. The conjugacy class of subgroups
= {Cc(O„(G))H: H e Hall„(G)}
of G is the set of ©“-injectors of G. Each subgroup U of G satisfying 0„(G) < U e ©"
is contained in an Cn-injector of G; in particular, the class ©" is dominant in S, and if
§ is a Fitting class with £ F> £ ©", then § « C".
Proof. Let H e Hall„(G), and put C = Cc(O„(G)) and S = CH e y’(G). Then we have
[O„(.Sj, C] < O„(S)nC = 0„(C) = O„(G)n C = Z(O„(G)), and therefore [O„(.Sj, С, C] = 1.
If L e Hall„ (C), it follows from A, 12.4(b) that [0„(S), L] = 1, and since L e Hall„.(S),
we conclude that Cs(O„(S)) has л-index in S. Therefore S e ©". Moreover, it is clear
that У(С) is a conjugacy class of ©’-subgroups of G.
Let U be an ©"-subgroup of G containing 0„(G). Since 0„(G) < 0„(U), it
follows that CV(O„(G)) < C. Thus, if Ho e Hall„(U) and Ho < H e Hall„(G), we have
V = Cv(O„(U)}Ho < CH e .(C(G}. Since S„ £ ©", an ©"-subgroup containing the ©"-
radical of G also contains 0„(G) and is therefore contained in a subgroup in У(С).
Hence ©" is dominant in S, and from (4.2) we conclude that У(С) coincides with the
set of ©“-injectors of G. □
Remarks, (a) It is easy to see that the ©"-radical R of a group G may be described
by the equation
R/Cg(O„(G)) = O„(G/Cc(O„(G)).
In general 0„(G) < R, and so the class ©" satisfies a stronger condition than that
required for dominance.
(b) In Exercise 9 at the end of this section a generalization of Theorem 4.16 is
described in which the role of 0„(G) is taken over by the radical of an arbitrary Fitting
class of characteristic it.
We bring this section to a close by discussing the dominance of some Fitting classes
that arise from Construction D of Section 2, and, where possible, we characterize their
4. Dominance and
some characterizations of injectors
627
injectors. First we consider the class 3₽ in a soluble universe, recalling that for л c p
3" = (G e 6: SocJG) < Z(G)).
The following lemma is required for the characterization of the 3₽-iniectors of a
group.
(4.17) Lemma. Let N be an elementary abelian normal p-subgroup of a group H eV
and let P e Syl„(H). Then CN(P) < Z(H).
Proof. We proceed by induction on | H). If N = H, the lemma is certainly true
Suppose that N < M a- H. Since M e s,3₽ = 3" by (2.8) and since P r, M e Syl (M),
by induction we have CN(P r> M) < Z(M). Let NB = CN(P nM). ” ’
Case 1: |H: M\ = p. Since CN(P) < No and [N(„ M] = 1, we have CN(P) = CN (P) =
CWo(A7P) = CNo(H) < Z(H), as required.
Case 2: \H: M\ = q p. In this case we have P < M, No = CN(P), and evidently
No = Z(1W) r> N. Therefore No < H, and Na may be regarded as an Fp(H/M)-module.
Since No is then completely reducible by Maschke’s theorem, it follows that No <
Soc(H), and therefore from the hypothesis that H e 3₽ we conclude that No < Z(H).
□
In order to formulate our main result about 3f we first prepare some notation.
(4.18) Notation. Let C denote the normal subgroup CG(Socp(G)) of a group GeG.
We saw in Paragraph (2) of Example 2.9(a) that C is the 3₽-radical of G, and so if
TV = Socp(Q, we have N < Z(C). For each P e Sylp(G), define
D(P) = CG(CN(P)).
Thus D(P) is a subgroup of G containing CP.
(4.19) Theorem (Lockett [1], Frantz [1]). Let Gbea group in the universe S. The set
of subgroups {D(P): P e Sylp(G)} defined in (4.18) is the conjugacy class of ^-injectors
of G. If О (G) < H < G and H e 3₽, then H is contained in a 3p-injector of G. In
particular, 3₽ is dominant in G, and we have g « 3" for all Fitting classes g such that
£ g £ 3₽-
Proof. Let Я be a 3₽-subgroup of G containing 0p(G), and let P* be a Sylow
p-subgroup of H contained in a Sylow p-subgroup, P say, of G. The subgroup
N = Soc (C) is characteristic in C (=Cc(Soc„(G))) and is therefore normal m G.
Hence N< 0„(G} < H. Clearly CN(P) < CN(P*l and by (4.17) we have CN(P ) <
Z(H). Therefore H < Cc(CN(P)) = D(P}. „„mim «nd
Next we show that D(P) e 3₽- Let К be a minimal norma p-subgroup of D(P},and
suppose that К n C = 1. Then [К, C] = 1, and since Socp(G) < C (because G e S),
628
ГХ. Fitting classes—examples and properties related to injectors
we have К < Cc(Socp(G)) = C, a contradiction. Therefore К n С ф 1, and so К < C
because К r> C < D(P). Since К n N < D(P) and clearly К n N / 1, by the definition
of К we have К r> N = К < N. Since К < P, we have К r> Z(P) / 1 by A, 8.3, and
consequently 1 / CK(P) < Q(P), which is a subgroup of Z(D(P)) by definition of
D(P). Therefore К Z(D(P)) / 1, and it follows that К < Z{D(P)). Thus the p-socle
of D(P) is central, and £>(P) e 3₽-
If Sp £ 8 s 3₽. then an 8-subgroup V of G containing Gs contains 0p(G), and by
what we have shown above, V is contained in one of the 3₽-subgroups in the
conjugacy class {D(P): P e Sylp(G)}. By considering the special case 8 = 3P, we see
at once that 3₽ is dominant in S, and hence from (4.2) that the 3₽-injectors of G are
just the subgroups D(P). Finally, for general such 8> we take V to be an 8-injector
of G and conclude that 8 « 3₽- □
Using the preceding result, we can now prove more generally that 3" 's dominant
in S for all л £ P.
(4.20) Theorem (Blessenohl [3]), In a soluble universe, let G be a group and H a
subgroup of G with the properties'. 0„(F(G)) <He 3”- Then H is contained in a
^-injector of G. In particular, 3” is dominant in S, and for all Fitting classes 8 such
that 9t„ £ 8 — 3" we have 8 « 3“-
Proof. We prove by induction on |G| that a subgroup H of G with the stated
properties is contained in a 3”-injector of G. Since 3" = ПреяЗ₽. we infer from (4.19)
that for each pen the subgroup И is contained in some 3₽-injector, Vp say, of G. If
Vp = G for all pen, then G e 3“ and the conclusion of the theorem certainly holds.
Therefore suppose that H < Vp < G for some pen. Since a 3”-injector W of G
contains the З’'-radical Cc(Soc„(G)), which in turn contains Op(G), we deduce from
(4.19) that W is contained in a conjugate of Vp, hence Injj.(Fj,) £ Injj.(G). Let
F = F(Vp) and L = HF. We assert that L e 3”. If this is true, it will follow by
induction that L, and therefore H, is contained in a 3“-injector of Vp and hence in
one of G. To prove this assertion, let qen, and let К be a minimal normal «/-subgroup
of L. Since F(G) < Cc(Socp(G)) = GJP < Vp, we have F(G) < F. If К n F(G) = 1, we
have [K, F(G)] = 1 and hence К < Cc(F(G)) < F(G) by A, 10.6(a). This contradic-
tion implies that К n F(G) / 1, whence it follows that К < 0„(F(G)) < H. Since
H e 3"> we have К n Z(H) / 1, and because F < F(L), we can deduce from A, 10.6(b)
that F centralizes K. Therefore 1 Ф К r> Z(H) < CH(HF) < Z(L), and it follows that
К < Z(L). Hence L e 3”. and the proof that H is contained in а З'^гцесЮг of G is
complete. The rest of the statement of the theorem is then clear. □
Comparison of the statements of (4.19) and (4.20) reveals that we have not been
able to generalize the description of S'-injectors to the class 3"-
Open Question. Does there exist a characterization of the 3”-injectors of a finite
soluble group which specializes to the characterization of (4.19) when n = {p}?
4. Dominance and
some characterizations of injectors
Exercises
1. (Blessenohl [3]) Prove that £p£, is dominant in £ if and only if one of the three
sets | p} vj t, p vj t, (p[ u t' coincides with P.
2, (Blessenohl [3]) Let p e P, and let g be a Fitting subclass of £ If № = g о 9i
prove that (5 о £p. is dominant in £.
3. (Blessenohl [3]) Let g be a Fitting class. Prove that 91 о g is dominant in £ if
and only if either (a) g = (1), or (b) g = £p. for some p e P, or possibly (c) 91 s g
and g is dominant in £.
4. (Lockett [1]) Let {nf}ie/ be a set of pairwise disjoint subsets of P, and let
7t=Uici".- Let g = ni6,S„S„. where it* = n\n,. Prove that the Fitting
class g is dominant in £,. Show that for G e £ the set of g-injectors of G is
{П<е/e Halln.(Cc(F(G)B,))}.
5. Let К < G e £ with К < Ф(С). If V/K is an 9l-injector of G/K, prove that V is
an 9l-injector of G.
6. (Bialostoki [1]) Let G be a (soluble) group of odd order. Prove that a subgroup
V of G is an 9l-injector of G if and only if V is a nilpotent subgroup of G of
maximal order.
7.
8.
9.
10.
11.
12.
(Arad and Chillag [1]) Let V < G e £. Prove that V is an 9l-injector of every
proper subgroup of G containing V if and only if V is either an 91-injector of G or
a nilpotent maximal subgroup of G.
(Mann [4]) Let Get Prove that Cc(F(G)) < F(G) if and only if Cc(Gs) < Gs.
In a soluble universe let § be a Fitting class of characteristic n, and let g =
(G e £: G/Cc(G6) e £„) Prove that g is a Fitting class, which is dominant in £,
and that {G„Cc(G£,): G, e Hall,(G)} is the set of g-injectors of G.
Prove that if(l) Z g « 3” in the universe £, then 9I„ £ g £ 3” (see Theorem
4.20).
(Blessenohl [3]) Prove that, although 3" = Пр^З", a 3"-injector cannot in
general be expressed as the intersection of a suitable set of 3₽-injectors as p runs
through n.
(Blessenohl and Laue [2]) Let I £ 9) be Fitting classes. Let 3 be a set of finite
simple groups, and let f be the corresponding Baer function defined by Equation
4.a on page 623. For G e C let S(G) = (Gv/Gx) c SocfG/Gj), and let g denote the
Fitting class HS(f. S) defined by Equation 2.e on page 581 (see Theorem 2.8). Let
л = {p e P: Zp e 3}. and let denote the class of л-soluble groups. Verify the
following statements:
(a) g n is dominant in 'B„.
(b) If n / 0, then g is not dominant in <£.
(c) In the case where 3 = (Zp) denote the class g by g and put T —
((G^^GJnSoCpK^/GJ. Prove that if Ge'Bp and PeSylp(G), then
Cc(Cr(PGK)) is an g -injector of G. j,
(d) Let 3 £ SI and G 6 Let G„ < - < G, < Go be a subgroup chain such
(i) for each i e {1,.. , 11} there exists ap, e л such that G, is an gP1-injector of
Gl_l, and
(">G„eg.
Then prove that G„ is an g-injector of G.
630
IX. Fitting classes—examples and properties related to injectors
13. Let g denote the class of finite groups all of whose composition factors are
isomorphic with Alt(5). Is g dominant in C?
5. Dark’s construction—the theme
In 1972 Dark [2] published the first example of a Fitting class g and a soluble group
G such that the Fischer g-subgroups of G do not form a conjugacy class of G. Thus
the question whether the Fischer g-subgroups of a group always coincide with its
g-injectors, first raised by Fischer [1] in his Habilitationschrift and again by Fischer,
Gaschiitz, and Hartley in [1], was settled negatively. Since then many authors have
used variations of Dark’s method to construct a variety of Fitting classes of special
kinds, often with “awkward” properties, tailored to the needs of counterexamples.
The procedure we describe here is based directly on Dark’s original example, which
we will present in full at the end of this section. In the following section we will deal
with some of the variations.
The basic strategy is the following: first identify a certain “key section” of each
group in the universe under consideration; then specify a certain class £ of groups;
next define an associated class g by requiring that a group belongs to g if and only
if its key section belongs to £; and finally, find conditions on £, which in applications
usually has a special form, to ensure that g is a Fitting class. In all the cases we
consider, the key section is determined by two sets of primes. Unless otherwise stated,
the universe is (E.
(5.1) Definitions. Let it and т be sets of primes.
(a) Let G be a finite group. The key section of G is defined thus:
k(G) = O'(G/O,(G)).
(b) (The class D’(3£)). If £ is a class of finite groups, an associated class D’(X) is
defined as follows:
D,"(£) = (6 e (E: k(G) e £).
(c) (The class O’). Finally we define
E,” = (Ge(E: O”(G) = G and Ot(G) = 1).
Remarks, (a) The class Ci" is the subclass of Q" comprising groups with no non-
trivial normal т-subgroups. Clearly G e C" if and only if G = k(G); furthermore
k(G) e C’ for all 6 elf.
(b) It is obvious from the definition that In £; e D’(X).
(c) It is also clear that D"(£) = D"(£ r> C"), and so there will be no loss of general-
ity in making the hypothesis that £ £ C".
5. Dark’s construction—the theme
631
In outline, the pattern for the rest of this section is as follows. First we describe
some fairly general requirements for the class X which ensure that the associated class
D”(X) is a Fitting class; these are contained in Hypotheses 5.3(a) and (b), and amount
to a weak form of s„- and N0-closure for X. Next, a very special form for the class X
is established. Its ingredients are a fixed group Y and a certain set .₽/ of subgroups of
Aut(F). For a given pair (T, .₽/), the associated class ЗЕ = Х(У, л/) consists, roughly
speaking, of groups X which contain a central product of normal subgroups, each
isomorphic with Y and each having a group of automorphisms in si induced by X.
Then, for classes X of this special form, we seek general conditions on the pair (Y, si)
which ensure that X fulfils the weak s„-closure of Hypothesis 5.3(a). Subsequently
we do the same in respect of weak No-closure, although here the situation is more
complicated, and we confine ourselves to the case where Y has trivial centre. Finally,
we look for more concrete specifications for the structure of Y and its set si of
distinguished groups of automorphisms which guarantee that the more general
conditions are fulfilled.
Thus we end up with various sets of conditions for the pair (У, si) which are
sufficient to ensure that the class X = X(Y, si) satisfies Hypotheses 5.3 and which
therefore force D"(X) to be a Fitting class. In the course of the section we apply these
results to some explicit constructions, which show, inter alia, that
(i) there exist non-permutable Fitting classes, and that
(ii) Fischer (^-subgroups need not be g-injectors.
In Chapter X, Section 6 the construction is again brought into play to show that the
Lockett conjecture is false and that the Lockett star operation does not respect Fitting
class products.
General requirements for X
Our first observation concerns the preservation of the key section of a group in
certain subgroups and quotient groups.
(5.2) Lemma. Let X be a class of finite groups, and let T denote the associated class
Df(X) defined in (5.1)(b). Let G be a finite group.
(a) If К < Gand К < Ot(G), then k(G/K) S k(G); in particular, G e T> if and only
if G/K e I. ,
(b) Let Lbea subgroup of G containing O"(G). Then k(L) S k(G) and, in particular,
G e T> if and only if LeT>.
Proof, (a) Since O,(G/K)= O,(G)/K, it follows that 04(G/K)/0JG/K))s0’'(G/0t(G)).
Therefore k(G/K) s k(G), and Assertion (a) is clear.
(b) Since L/O"(G) is a n-group, we have O”(L) < 0"(6). However, t e group
O”(G)/O"(L), as a subgroup of L/O’(L), is a тг-perfect n-group and is therefore trivial;
invariance of O”( ) and the radical property of O,( ), we av^ oio’tL) r> О (L)) ~
O”(G)/(O^G)cR) = O”(G)/O,(O”(G)) = O’(L)M(O’(L)) = О (L)/(O (L)r^O,(L)) _
632 IX. Fitting classes—examples and properties related to injectors
O’lLJOJLyOJL) = O'(L/OT(L)). Thus k(L) s k(G), and Assertion (b) clearly
holds. □
(5.3) Hypotheses. Let it and r be sets of primes, and let I be a non-empty class of
finite groups. We make the following hypotheses:
(a) If N < X e I and N e Q", then N el;
(b) IfAj and N2 are normal X-subgroups of G = Nt N2,andifG e Q”,then G e X.
Remarks
(a) It is obvious that (5.3)(a) is fulfilled when I = s„X and that (5.3)(b) is fulfilled
when X = n0X. Therefore these hypotheses represent weakened forms of s„- and
n0-closure for the class I.
(b) Hypothesis 5.3(a) implies that X contains groups of order 1 and hence that
G,G„ s РДХ).
(5.4) Theorem. Let it and r be sets of primes, and let 0 # X s C".
(a) If X satisfies Hypothesis 5.3(a), then D*(X) is s„-closed.
(b) If X satisfies Hypothesis 5.3(b), then Dx(X) is N„-closed.
Proof. Let T> denote the class D"(X).
(a) Let Ge® and N < G. Denote the key section k(G) by R/O,(G) and the key
section k(N) by R*/OX(N). Because
(i) Ot(N) = N r> O,(G) <NnR, and
(ii) N/(N R) ~ NR/R e G„,
we conclude that R* < N n R < R. Moreover, since R* char N -a G, we have R* < G,
and therefore OT(R*) = O,(G) r> R* < O,(G) r> N = OX(N) < OX(R*). Consequent-
ly O,(R*) = Ox[\'}, and it follows that k(N) = R*/OX(R*) = R*/(OX(G) n R*) =<
R*OX(G)/OX(G) < R/Ot(G) = k(G) e X. Thus the key section k(N), which belongs to
O', is isomorphic with a normal subgroup of an X-group and therefore belongs to
X by Hypothesis 5.3(a). We conclude that N e T> and hence that T> is s„-closed.
(b) We suppose that the class X satisfies Hypothesis 5.3(b) and that T> is not
N0-closed, and from this we derive a contradiction. Let G be a group of minimal order
in n0T\I\ By II, 2.10(b) we can find normal subgroups Nt and N2 of G such that
and N2 belong to T> and G = NlN2.
Let T = OX(G), and first suppose that T + 1. Then G/T = T/T)(N2T/T), and
for i = 1,2 we have 7Vf T/T s Ni/(Ni nT) = NJO^Nf. Since N,/Ox(Ni) e D by (5.2) (a),
it follows that G/T e n0T>; therefore G/T e T by the minimal choice of G. But then
G e T> by (5.2)(a), and we have a contradiction. Hence we conclude that T = 1, and,
in particular, that OJA'J = 1 for i = 1, 2. By II, 2.12 we have O”(G) = «'(AjJO'fA'j),
and since O”(Nf) e T> by (5.2)(b), it follows that 0”(6) e n0T>. Thus, if O”(G) < G, the
choice of G implies that O”(G) e D and hence by (5.2) (b) that G e®, contrary to our
initial supposition that Ge®.
Thus we are forced to the conclusion that G = O”(G) and have therefore shown
that G e £)”. Since OX(NX) = 1, we have O"(A',) = K(Nf e X for i = 1,2, and by II, 2.12
we have G = 0n(Ni)0”(N2). Hence k(G) = G e X by Hypothesis 5.3(b), and it there-
fore follows that Ge I. This contradiction proves that T> = n0T>. □
5. Dark’s construction the theme
633
Remark. The foregoing analysis can obviously be carried out for a more general key
S\ rTea h T K(G) L(G/?ft) ’ Where 5 *S a fomation’ IS a and
where each satisfies suitable additional closure properties.
A special form for the class X
Our next objective is to construct a class X from the following ingredients-
(i) a fixed group Y, and
(ii) a set si of subgroups of Aut(y), each containing Inn(l').
The pair (У, л/) must satisfy the following hypotheses.
(5.5) Hypotheses. Let it be a set of primes, let У be a finite group, and let .я/ be a
non-empty set of subgroups of Aut( У), each of which contains Inn( У). Assume that
(a) if A e si, then 1 / Л/1пп(У) e O’, and
(b) the set { Л/1пп( У): Л e .я/} и {1} is a Fitting set of Aut( У)/1пп( У).
Remark. If si £ <5, Requirements (a) and (b) of these hypotheses clearly imply that
ЛДпп(У) e S„. for all A e si.
The procedure for constructing I which we are about to describe was followed by
Dark in his original paper and may be viewed as an approximation, for a soluble
group, of the fact that о0(У) is a Fitting class when У is a non-abelian simple group.
In order to define I we need the following notation.
(5.6) Notation. Let I be a group which contains a normal subgroup Yt = У. Let
У,-» У be an isomorphism. Then we obtain an induced homomorphism
фр X->Aut(y)
defined as follows: if x e X, then ффх) is the map from У to У which sends the element
у e У to the element ,|'i(x'1ll'1~1(y)x) e F. It is routine to verify that ф,(х)Е АиЦУ),
and that ф, is a homomorphism with kernel Cx( Ij). Furthermore, the subgroup ф,(Х)
of АШ(У) contains 1пп(У); in fact, the inner automorphism of У induced by an
element у е У is the image under of the element ф< 1 (y) of X. We shall use this
notation consistently in the sequel.
(5.7) Definition (The class X associated with the pair (Y, si)}. Let (У, si) be a pair
satisfying Hypotheses 5.5, and let л and т be sets of primes. We define the associated
class Х”(У, si) as the class of all groups X satisfying the condition
(a) X e D”, and . .
(b) either X = 1 or X has a non-empty set of normal subgroups Уп..., У, satisfy-
ing the following conditions:
II: There exists an isomorphism фр У for i = 1,..., t;
12. фДХ) e si for i = 1...t;
13. [X, = 1 for a11 1 f;
It sUuM emphasized tot we make no assumption about the uniqueness of such
634 IX. Fitting classes—examples and properties related to injectors
a set of subgroups {Yt,..., lj}; we merely require that each non-trivial group in X
has at least one such set, which we fix and call the distinguished set.
Notation, (a) We shall denote the product Y, Y2... Y, of the groups in the distin-
guished set of subgroups of X by p(X)\ equally there is no built-in assumption that
p(X) is a uniquely determined subgroup of X.
(b) If I = X*(Y, we shall denote the class Dt"(3E) by D”(Y,
(5.8) Remarks, (a) We make the obvious convention that the integer t = | {Yj,..., X} I
is zero if and only if X = 1.
(b) By Condition 13 of the above definition, the subgroup p(X) is a central product
of the subgroups Yj,..., X. an<^ then by Condition ЭЕ4 and A, 19.8(b) it follows
that
Cx( YIY2...Y,) = Z(YlY2...Y,) = Z(Yt )Z(Y2)... Z(Y,).
(c) If X e X"(Y, з/), set Cf = CX(I<). Since Yj < Ct for 1 < i j < t, by A, 1.9 we
have
Cl Y,C, = p(X).
i=l
Sufficient conditions for D’(Y, л/) to be s,-closed
By (5.4)(a) the class D'( Y, ssl) will be s„-closed if the class X"( Y, ..c/), defined in (5.7),
fulfils Hypothesis 5.3(a). We shall show that the following set of conditions for the
pair (Y,.«/) are sufficient to ensure this.
(5.9) Hypotheses. Let л be a set of primes, and let (Y, з/) be a pair satisfying Hypo-
theses 5.5.
Condition s„I: If A e then [Inn( Y), A] = Inn( Y).
Condition s„II: Y is a soluble r-group.
Condition s,III: Z(Y) < Y'.
Condition s„IV: For all A e з/ the following two statements hold:
(a) [Z(Y), Л] = 1, and
(b) (|Z(Y)|, |Л : Inn(Y)|) = 1.
Remarks, (a) If Z(Y)= 1, then Conditions s„III and s„IV are automatically satisfied.
(b) A typical situation where all four requirements of (5.9) are met is the following:
Y is a л-soluble group with trivial centre; Aut( Y) has a normal subgroup К such that
K/Inn(Y) is a non-trivial л'-group and з/ consists of all subnormal subgroups of К
which contain Inn(Y) as a proper subgroup; every minimal subnormal subgroup of
K/lnn(Y) acts fixed-point-freely on Y/Y'.
(5.10) Proposition. Let (Y, з/) be a pair satisfying Hypotheses 5.5 and 5.9, and let
X = X”(Y, з/) be the associated class defined in (5.7). If N is a n-perfect normal
subgroup of a group X el, then N e X.
5. Dark’s construction—the theme
635
Proof. Since IsO?, we have Ot(X) = 1. Therefore O,(N) = N r> О (X) = 1 and
SX % 6 ,bZ“p*
estabhshed in K M ,C°ndltlOns 11-14 of (5.7). Using the notation
established in (5.6), we may clearly suppose without loss of generality that these
subgroups have been so labelled that ф,(1У) = 1 if and only if s + 1 < i < t for
k S 1{°’*’‘For ' = f set C. = fx(K)i then q = Ker(^) and
" ! m* or [ “ 11OWS tha‘ N ~ Z^Z^'l -Z(Y,l by (5.8)(b), and
therefore N e 91, by Condition s.II. But N = O*(Aj by hypothesis, and so N = 1 e I
We therefore suppose that s > 1.
Next we prove the following assertion:
(5.a) Let Ae sf, and let В be a non-trivial n-peifect normal subgroup of A;
then Be sf.
Since Y is a soluble n-group, so also is Inn(Y); hence В £ Inn(Y). Consequently
Inn(Y) < Inn(Y)B sn A, and therefore Inn( Y)B e sf by Hypothesis 5.5(b). It then
follows from Condition s„I that lnn(Y) = [Inn( Y), Inn(Y)B] = (Inn(Y))'[Inn(Y), В].
Since [Inn(Y), B] < Inn(Y) e S, we conclude that Inn(Y) = [Inn(Y), B], which is a
subgroup of В because Inn( Y) normalizes B. Thus we have В = Inn( Y)B e st, and
Assertion 5.о is justified.
Now let i e {1,..., s}. Then we claim that Yf < TV. Since 1 f ipj(N) < ФДХ) e sf
and ф^) e qQ” = £F, it follows from (5.a) that q(7V) e sf, and, in particular, that
Inn(Y) = [Inn(Y), ipfNff By the first isomorphism theorem the homomorphism
ips X -»Aut(Y) gives rise to an isomorphism 6s X/Ct -»ф,(Х), which maps Ijq/q
to Inn( Y) and NCJCt to ip^N). Applying Gf1 to the equation Inn( Y)= [Inn( Y), ^;(N)],
we then obtain Y,q/q = [Ijq/q, Nq/q](=[lj, AZ] q/CJ. It therefore follows that
Yj = Yi r> YjCj = Yf r> (X АТ]С, = [Ij, N](Ij n CJ = (X TV]Z(IJ; consequently
VIX, W] e И, and,in particular, Yj < [X A/]. Hence, appealing to Condition s,III,
we have Ij = [Ij, AZ]Z(IJ < (X AZ] Y; = [Ij, AZ], and since [Ij, AZ] < N, our claim
is justified. Thus we have shown that N contains a set of normal subgroups Yt,..., Ys
such that Conditions II, 12, and 13 of Definition 5.7 are satisfied. It remains to verify
Condition 14.
Let C = Cn(Yi Y2... Ys). We must show that C < Yj Y2...YS. Letje {s + 1,..., t}.
Since ipj(N) = 1, we have
N r> Yj < q n Yj < Z(Y]),
and therefore [Y, N] < Z(Y2). Since [Z(5<), X] = 1 by Condition s.IV(a), it follows
that N centralizes Y./ZIY,) and Z()<). Therefore, because Yt is a n-group and N
is n-perfect. we deduce from A, 12.4(a) that N centralizes Ys, consequently C <
C„(Y, Y,...Y) = Z(Y,)Z(Y,)...Z(Yr)nN<Z(X)nN<Z(N). Now by A 1.9 we
have Г\“- Y(C r> N) = Y. Y2.. YSC, and since Condition sBIV(b) implies that
N/ Y, y“S . YJ, "and, as such, is a direct factor and hence an ‘4’“™?^
N/(YtY2... YJ. Since N/(Y, Y2... YJ 6 qQ" = O’. conclude that Y,Y2...YSC-
636
IX. Fitting classes—examples and properties related to injectors
YlY2..Ys. This completes the proof that N (together with its distinguished subgroups
y,....Ys) satisfies all the requirements of (5.7). □
Thus we have shown that the class Л' = X*(Y, st) satisfies Hypothesis 5.3(a) when
(Y, .</) satisfies Conditions s„I-sJV. By (5.4)(a) we therefore have:
(5.11) Corollary. If the pair (Y, ..cZ) satisfies Hypotheses 5.5 and 5.9, then the class
D*(Y,.«/) is s„-closed.
Sufficient conditions for D*(Y, st) to be N0-closed when Z(Y) — 1
Compared with the treatment of s„-closure, our approach to N0-closure is less
direct. First, as an intermediate stage in the discussion, we describe a set of fairly
general conditions (n0I-n0III) on the class J = Xf(Y, st), which ensure that I fulfils
the requirements for weak n„-closure described in (5.3)(b). Subsequently we show
that certain sets of rather elaborate properties for the pair (Y, ..cZ) imply that these
conditions are satisfied.
(5.12) Hypotheses. Let n and r be sets of primes, let (Y, .</) be a pair satisfying
Hypotheses 5.5, and let I denote the associated class I = X’(Y, st) defined in (5.7).
Condition n0I: Let Yo, X < Y0X, where Yo = Y and X e I; let Yt,..., Y, denote the
distinguished normal subgroups of X. Then either Yo r> X = 1 or
Yo = Yj for some i e {1,..., t}.
Conditionn„II: Let R, X < RX. If R is a central product Yj Y2... Y, of subgroups
Yj S Y, and if X e X, then Y, < RX for i = 1,..., t.
Condition n0 III: If A e st and ЛДпп(У) has a central p-chief factor, then per.
Recall that if X e X"(Y, st), then p(X) denotes the (central) product of the distin-
guished normal subgroups Yj,..., Y, of X.
(5.13) Remark. If X el = X”(Y, st) and if I satisfies Condition n0I of (5.12), then
Aut(X) permutes the direct components Y,,..., Y, of p(X).
(5.14) Proposition. Let (Y, st) be a pair satisfying Hypotheses 5.5, and let X =
X”(Y, st) denote the associated class defined in (5.7). Assume that Z(Y) = 1 and that
Conditions n0I, n„II, and n0III of (5.12) are satisfied. If X and X* are normal
X-subgroups of a group G = XX* e Q”, then GeX.
Proof. Let p(X) = Y, Y2... Y, and p(X*) = Yt* Y*... YJT in our standard notation. Let
i e {1,..., t}, and put R = p(X). By (5.13) we have R < C, and then Condition n0II
implies that Yj < RX*. Therefore Y, < G, and then by Condition n0I we have either
Y; r>X* = 1 or Yf = Y* for some j e {1,..., t*}. A similar reasoning applied to the
subgroups Yj*,..., Y* shows that without loss of generality we can therefore suppose
that the suffices of the distinguished subgroups { Yt}5=1 and {1J*}have been chosen
so that
(i) for i = 1,..., s we have Yf = Y(*
(ii) Yj r> X* = 1 for i — s + 1,..., t, and
5. Dark’s construction—the theme
637
(iii) Y* n X = 1 for j = s + 1,.... t*.
Let J denote the set {У„ Y„ Y*+l,.... y*}; these will constitute fte
guished normal subgroups of G, required for showing that Gel We note that C
satisfies (5.7)(a) by hypothesis, and that Condition .Hl of (5.7)(b) is also fulfilled. If
Ltisfiedby’thegrouSn? j and it follows that Condition 13 is also
F°r ' = *’' ’f let C‘ = * 1 then
/ ' " 7* !)(r n 'Z Д SmC£ XCi/Cl = X/C*M = whh similar isomor-
phisms for X , It follows that G induces on Ff a group of automorphisms whose image
in Аш(У) is the normal product AA* of ^/-subgroups A and A* of Aut(F) From
the Fitting set requirement of Hypothesis 5.5(b) we have AA* e a/, and so Condition
12 is therefore satisfied for the subgroups Yt,..., ys. On the other hand, for s + 1 <
; < t we have shown that X* < CG( )<); consequently G and X induce on f the same
group of automorphisms, whose image in Aut( У) is A e rf. A similar conclusion holds
for the group of automorphisms induced by G on for s + 1 < к < t*. Hence
Condition 12 of (5.7)(b) is satisfied for all subgroups in the set
It remains to check that Condition 14 holds. Let C = Cc(p(G))^g G, and suppose,
by way of contradiction, that C +1. Since Z(Y)= 1, we have Cr.X<Cx(Y, Y2... y;)=
1, and similarly Co X* = 1. Hence C < CG(XX*) = Z(G), and therefore C is a
т'-group because by hypothesis Ot(G) = 1. Let К = X n X*. Then CK/K is a non-
trivial normal subgroup of (X/K) x (X*/K) which intersects both direct factors
trivially. Let H/K denote the projection of CK/K onto X/K. Then H/K is a non-
trivial, central, normal т'-factor of X/K. Put T = У5+1 У1+2 К Since [}<, X*] <
Yj n X* = 1 for j = s + 1,..., t, it follows that TnK<Tr>X*< Z(X*) = 1, and
hence that T ~ TK/K. Because Z(T) = 1, we conclude that H r> TK = К and there-
fore that TH/TK is a non-trivial, central, normal r'-factor of X/TK. Since У2 Y2 • • • Yt<
K, we have p(X) < TK, and so X/p(X) has a central т'-chief factor. By (5.8)(c) the
group X/p(X) is a subdirect product of the groups X/ljCv(l<), i = 1,..., 1, and so at
least one of these has a central т'-chief factor by IV, 1.3. From the analysis in the
proof of (5.10) we know there are isomorphisms 0f: X/Cx(Y// -»^,(X) = A < Aut(F)
mapping KCx(K)/Cx(X) to 1пп(У), and therefore Х/У(Сх(К) S ЛДпп(У) for some
A e л/. But Condition N0III then implies that X/ljCx(lj) has no central r -chief
factors, and we have reached the desired contradiction. Therefore C = 1, and we have
confirmed that the group G, together with its set & of distinguished normal sub-
groups, satisfies all the requirements of Definition 5.7. Hence Gel. □
The preceding proposition shows that I fulfils Hypothesis 5.3(b), and from (5.4)(b)
we may therefore deduce the following result.
(5.15) Corollary. Let (У, .</) be a pair satisfying Hypotheses 5.5 and assume that
Z(T) = 1. Assume further that (У, and its associated class ХДУ sd} satisfy Condi-
tions^!, n0II,n0III of (5.12). Then the class D/(Y, st) is N0-closed.
Our next objective is to formulate a list of properties, solely in terms of the group
У and the set of automorphism groups of У, which will guarantee that the class
ХДУ, ^) satisfies Conditions n0I-n0III. It is possible to vary this list considerably
(5.7)
638 IX. Fitting classes—examples and properties related to injectors
without affecting the conclusion. Our choice of properties has been governed by the
need to be comprehensive enough to include the subsequent applications and simple
enough to make the verification of the properties relatively easy. However, we have
allowed ourselves one indulgence in the form of an additional condition that is not
strictly required in the sequel; nevertheless, it raises an interesting question and at
the same time indicates how the properties can be modified to enlarge the range of
examples. To formulate this condition we need the following concept.
(5.16) Definition. A finite group G is said to be normally detectable in a direct power
if whenever
f5.fi) G < G, x • • x G„
with G s G, for i = 1,..., n, then G = G} for some j e {1,.... n}. In other words, the
only normal subgroups of the direct power G" which are isomorphic with G are the
obvious direct components.
From the analysis of direct decompositions for the Krull-Remak-Schmidt theorem
in A, 4.9, it is straightforward to verify that the following two conditions are necessary
for G to be normally detectable:
(i) G is directly indecomposable, and
(ii) (|G:G'|,|Z(G)|)= 1.
In fact, by that theorem, these are just the conditions which ensure that the de-
composition of f5.fi) is unique up to permutation of the suffices. This suggests the
following
Open Question. If G satisfies Conditions (i) and (ii) of (5.y), does it follow that G is
normally detectable in a direct power?
In [7] Hauck has made a significant contribution to the resolution of this question.
In particular, he has shown that if G is a group satisfying both conditions of (5.y),
then any one of the following additional conditions is sufficient to ensure that G is
normally detectable:
(1) All automorphisms a of G for which G has a section isomorphic with <a> leave
each conjugacy class of G invariant;
(2) G has only one maximal normal subgroup;
(3) Soc(G) has G-composition length at most two;
(4) FfG) < Soc(G);
(5) For each prime p dividing |G/G'|, the Sylow p-subgroups of G are abelian.
In the same work, Hauck also studies groups which are subnormally detectable in a
direct power (this concept is defined by replacing “ <” by “sn” in f5.fi)), and he obtains
the satisfying criterion that a finite group G is subnormally detectable if and only if
G is directly indecomposable and (| G/G'|, |F(G)|) = 1. Most of Hauck’s work applies
to groups with finite composition series.
5. Dark’s construction - the theme
639
M *7)rZ Ье SetS °f Primes',et -*0 be a pair satisfvina
Hypotheses 5.5. Let a = c(Y),andlet co denote theset of primes r such that t^ereex^
° S ° CenCral faCtOr' C°nSider the f(dlov.ing conditions.'
(2) to c t;
(3) If A e .</ and A < H < NAuim(A), then O„(H} = 1пп(У);
(3 ) If A e з/ and Y <g YA < АиЦУ) with Y s Y, then Y= 1пп(У);
(4) (a) Y has a unique minimal normal subgroup,
(b) [Inn( У), Л] = Inn( Y) for all A e st, and
(c) Z(K)= 1 if 2 e m.
(4') (a) Y is normally detectable in a direct power,
(b) сог><т(У/У') = 0,and
(c) there is a prime p such that the set of minimal normal p-subgroups of Y
contains a unique member M of maximal order, and [M, У'] 1 if 2 e co.
If at least one of the following sets of these conditions is satisfied:
{1,2,3,4}, {1,2,3', 4}, {1,2,3,4'},
then the class X = X"(Y, .sf} satisfies Conditions n0I, n0II, and n0III of (5.12).
Proof. We verify each of the Conditions n0I n0II1 in turn.
Condition n0I: Let Yo, X < G = Y0X, with Yo S Y and X e I. Let p(X) =
Yt x x Y„ and for i = 0, 1,..., t let : Yt -» У be an isomorphism. We suppose
that Уо r> X ± 1 and prove that then Yo = Y} for some i e {1,..., t}. We first deal with
a special case.
Case (a). Assume that Yo < X, in other words that X — G. Let i6 {!,...,t}, let
фр X -»Aut(y) denote the homomorphism induced from ifz, described in (5.6), and
let C; = Сс(У(). Then Ker(^;) = C;, and the isomorphism theorem yields a cor-
responding isomorphism 0;: X[C, -»ф^Х) < Aut(Y') which maps YiCJCl to 1пп(У);
furthermore, since X e I, its image ф{(Х) under 0, belongs to s$. We now deal
separately with the different sets of hypotheses.
Subcase (i). Assume that Hypotheses (1), (2), (3), and (4) of the theorem are satisfied,
and let ie {l,...,t}. If Уоп 1} = L then [Уо, X] = Land Уо< Cf. But Q'i=1 Q = 1
by (5.8)(b) and Hypothesis (1) of the theorem, and therefore Yo r> Yj 1 for at least
one j e {1 t} Let M denote the unique minimal normal subgroup of Y, and let
= ф^’(М)(от i = 0,1,..., t. Then char 1} < X, and so < X. If the subgroup
Уо n У is non-trivial, as a normal subgroup of Yo it contains M„, but then Mo, as a
normal subgroup of K, contains Mt, and consequently Mo = Mr Since Yt n Yt = 1
for 1 < i * i < t we therefore conclude that Уо n Yj is non-tnvial for precisely one
value off, say for’J = 1. It then follows that [Уо, У(] = 1 for i > 2, and hence that YB
is contained in the subgroup Cf = C,. Since Y * a norma o-sub^oup, of X,
we have е((У0С,./С,) < ОД0ДХ/С..)), and therefore
thesis (3); in particular Y0<Y,Ct. Consequently Уо^С, - У^С, nC.) L,
and so Уо = У1, as desired.
640 IX. Fitting classes—examples and properties related to injectors
Subcase (ii). Assume that Hypotheses (1), (2), (3), and (4') of the theorem are satisfied.
As shown in Subcase (i). Hypothesis (3) implies that Yo < Qj=i fjC,, and therefore
f0 P(A1 by (5.8)(c). Since is normal in p(X) and is by hypothesis normally
detectable, it follows by definition that - fj for some j e {1,..., t}.
Subcase (iii). Assume that Hypotheses (1), (2), (3'), and (4) of the theorem are satisfied.
As in Subcase (i) the subgroup f^ has non-trivial intersection with exactly one of the
subgroups fj, Yz,..., X, say with fj, and in this case Mo = Af, and Yo < CJ =
P|'=2 Ct. Suppose that Ct n # 1. Then Ct contains the unique minimal normal
subgroup Mo of fo, and we conclude that Л/, < C, n fj = Z(fj), which contradicts
Hypothesis (1) of the theorem. Therefore C, n Fo = 1, and we have Yo s Y0C1/Ci. It
follows that e^oCi/Ct) is a normal subgroup of ©JA'C^CJ isomorphic with
К and Hypothesis (3') of the theorem then implies that ©Jf^q/CJ = Inn(f) =
0,(У, C,/C,). In particular, we have f0 < f,C,, and therefore f0 < CJ n У, С, = yj,
again yielding the desired conclusion.
Case(b). The general case. Let i e {1,..., t}. If g e G, then Yf < X, and by applying
Case (a) with Yf in the role of fj,, we conclude that Y? = fj for some j e {1,..., t}.
Hence, in particular, Уо permutes the direct components {fj,..., У,} of p(X). If there
were a yo-orbit of length greater than one, then |[p(X), fj,] | would be strictly greater
than | )o| by A, 18.3(c), which is impossible since [p(X), Уо] < fj,. Therefore f0
normalizes each fj, and so Y, < G for i = 0, 1,..., t; in particular, p(X) < G. As in
Case (a), set Cf = CG( fj), and let 0,: G/C, -» Aut( У) be the isomorphism derived from
the homomorphism G -» Aut(f). Since X e 3E, we have 0,(ХС;/С,) e з/ for i =
1,..., t. We suppose that fj, n X / 1. If Yo n Yt = 1 for i = 1,..., t, then Yo n X <
H-i(C,nX) = 1 by (5.8)(b) and Hypothesis (1) of the theorem. Hence without
loss of generality we can further suppose that fj, n fj / 1, and our objective is then
to show that fj, = f j. Put C = CG(p(X)), noting that C < G and that [X, C] = 1
because X n C = Cx(p(X)) = 1.
First we deal with the case where Hypothesis (4) holds and У has a unique minimal
normal subgroup M. Then Mo = and f j, л С, = I by the arguments of Case
(a). With У= e^foCj/Cj) and A = et(XC\/Ct} e .tf, it follows that У is a normal
tr-subgroup of 0,(G/C,) = YA. (Strictly speaking, the fact that X e3E means that
0J(X/(X n С,)) e .в/, where 0J is the isomorphism derived from (ф, )x, but 0! and 0J
each have the same image in Aut(f).) Since Y s У, we conclude that Y = 1пп(У) if
Hypothesis (3') of the theorem holds, and У < O„(YA) = Inn(f) if Hypothesis (3)
holds. In either case we have fj, < fjC,. Since Mo = if YBr\ Yt ф 1, and since
У] n Yt = 1 for i = 2,..., t, it follows that Уо n У, = 1 for i > 2, and hence that
Уо < CJ = Q 1=2 Cj. Consequently Yo < YtCt C? = Yt *. C. Since Y(tr.C = 1, we
have f j C = Y0C by order considerations; hence by Hypothesis (4)(b) of the theorem
fj = [fj, X] = [У,С, X] = [fj,C, X] = [Уо, X] < fo and so fj, = fj, as desired.
Finally, it remains to consider the case where Hypotheses (1), (2), (3), and (4')
of the theorem are satisfied. Let ie {1,..., t}, and let A = 0ДХС(/С() and H =
0,(G/C,). Then Лез/ and A < H < Aut(f’), and therefore the normal tr-subgroup
®i(foCj/C() of H is contained in O„(H) = Inn(f) by Hypothesis (3). Consequently
fj, < A'=, f;q = p(X)C by A, 1.9, and it follows that G = f0X < p(X)CX = CX;
5. Dark’s construction—the theme
641
hence G - C x A'. Next we remark that for any X e X
the central chief factors of X are w-groups
The reasons for this are as follows: For i = 1,.... f the group X/(X n C() is isomorphic
with a group m and therefore by definition of to belongs to the formation of groups
whose central chief factors are all oj-groups (see IV, 1.3). Hence X also belongs to
this formation because Q'=I (X r\ Q = Cx(p(X)) = 1.
Now put W = Yo n X; then W = Уо n p(X)C n X = Yo n p(X). We also note that
C' - [С, С] < [C, L0X] = [С, Уо] < Уо. Since C^CX/X = Y0X/X, we have C/C's
WW X = >o/Oo) W, and it follows that of C/C') £ a(Y/Y'). In view of this, it now
follows from Hypothesis (4')(b) and (5.<5) that
(\Z(X/W)\\(X/W)/(X/W)'\, |C/C'|) = 1.
But G/C'W = (CW/CW) x (C'X/C'W) =: (C/C) x (X/W), and Y0/CW is a direct
complement to C'X/C'W in G/C'W. Therefore by A, 4.10 we have Yo = CW = C x W.
But the hypothesis that X> is normally detectable certainly implies that Ц, is directly
indecomposable, and since W is non-trivial by supposition, it follows that C = 1.
Hence Уо < p(X), and the desired conclusion now follows from Case (a).
Condition n0II: Let G — RX, where R and X are normal subgroups of G, where
А e X, and where R is a central product of subgroups У,,..., У„ each isomorphic
with У. Since Z(Y) = 1 by hypothesis, R is a direct product, and by A, 4.9 its direct
components У, f j are permuted under conjugation by A since У is directly
indecomposable by (4)(a) or (4')(a). We suppose that there is an X-orbit of length
greater than one and derive a contradiction. Without loss of generality let {У1;..., 3S},
s > 2, be the orbit in question, and put D = fj x x Ys', then D is normalized by
R and X and hence by G. Let M denote either the unique minimal normal subgroup
of У if Hypothesis (4) of the theorem holds or else the subgroup described in
Hypothesis (4')(c). For i = 1,..., s let фр X - У be an isomorphism, and let M, =
i///1 (A/). Since M, is obviously a characteristic subgroup oi X- we conclude that the
subgroups Mi,..., Ms are permuted transitively under conjugation by A and hence
that the subgroup
N = MiM2...Ms
is normal in G. Because D and X are normal in G, it follows that [£>, X] is a normal
о-subgroup of G contained in X. ,
Next we remark that is a non-central minimal normal ^bgroup of X by
Hypothesis (1) of the theorem. Furthermore, if s = 2, then |X: Nx(X)l - 2 and from
(56) we conclude that 2 e o, Therefore if s = 2, Hypotheses (4)(a) an1(c), and(4 )(cX
each imply that [M„ X'] * 1 for i = 1, 2. Hence, applying Lemma A, 18.3, we can
draw the following conclusions:
(i) N < D' < [D, X] ^DnX.and
(ii) M2,.... M, are pairwise non-isomorphic [£>, X]-modules.
642
IX. Fitting classes—examples and properties related to injectors
Let p(X) = У* x • • x If N n Yj* = 1 for j = 1,..., t*, then [A, )<♦] = 1, and it
follows that N < Cx(p(X)) = 1, which is absurd. Therefore for some j e {1,..., t*j
the group N ri is a non-trivial normal subgroup of [£>, X]. But then by B, 3.5 we
know that Nr, I'* is a direct product of a suitable subset of Mi,..., M„ and in
particular that Mk^ I/ for some к e {1, s}. It follows that А = <Мкх><<1<*л> =
Г
Now suppose that Y has a unique minimal normal subgroup, and let M* denote
the unique minimal normal subgroup of lj*; then obviously M* < N. The fact that
M* char Yj* < X implies that the subgroup Mf is normal in [£>, X] and hence
contains one of the subgroups Miy 1 < I < s, for the reasons given above. Then by
order considerations we have Mt = M* < X, and consequently N = (M*) =
<(ftf*)x> = M*, which contradicts the fact that |A| = |Af |s > |M| = | Mf |. Therefore
s = 1 and Condition n0II holds if Hypothesis (4) is satisfied.
It remains to deal with the case where Hypotheses (1), (2), (3), and (4') are satisfied.
Let ie {1,..., t*}, let Y* -» Y be an isomorphism, and put Q = CX(Y*)- Let
ф?: X-»Aut( Y) denote the induced homomorphism described in (5.6) and в*: X/С,—»
Aut(Y) the corresponding monomorphism. Since X is an 3E-group, we have lm(0*)e si
and therefore 0*([D, X] C./CJ < O„(Im(6?)) = Inn( У) by Hypothesis (3) of the theo-
rem. Hence [D, X] < QJlj Yf] = p(X) by (5.8)(c). Let T denote the projection of the
normal subgroup [D, X] of Y* x x Y* into Ij*, the component containing N.
Since T and [£>, X] have equivalent actions on the subgroups M,,..., M,,\t follows
that these subgroups may be viewed as pairwise non-isomorphic T-modules. If
{M,denotes a Yj*-orbit, after suitable renumbering, we deduce from B, 7.8
that Mj M2 Mu is a minimal normal subgroup of Yj*. But M, has maximal order
among the minimal normal p-subgroups of Yj* by Hypothesis (4')(c), and therefore
и = 1; in other words, we have shown that M, < Yj* for i = 1,..., s. However, by
hypothesis Yj* has only one minimal normal subgroup of order |Л/,|, and therefore
s = 1. This contradiction again shows that each of the subgroups I j,..., Ys is normal-
ized by X and is therefore normal in G, as required for Condition n0II.
Finally, we remark that Condition n0III is an immediate consequence of Hypo-
thesis (2) of the theorem, so the proof of Theorem 5.17 is complete. □
With an eye to subsequent applications, we now describe a special set of circum-
stances, which, in the light of our earlier results, ensures that D”(Y, si) is a Fitting
class.
(5.18) Corollary. Let я and r be sets of primes, and let Y be a finite soluble r.-group
with Z(Y) = 1. Assume that Aut(Y)/Inn(Y) contains a non-trivial soluble normal n'-
subgroup K/Inn(Y), and let
si = {A: Inn(Y) < A sn K}.
Assume further that the following conditions are satisfied:
(A) For all A e si the orders of the central chief factors of A belong to r;
(B) Either (i) Inn(Y) e Hall„(Aut( Y)), where a = of Y),
Or (ii) If A e si and Y < YA < Aut(Y) with I S Y, then Y = Inn( Y);
5. Dark’s construction—the theme M3
F]71'<7Ue W‘"iWa' n°mal SUb9r0UP M> Which is a and * 21 IK I,
(D) [1пп(У), A] = lnn(Y) for all A e si.
Then D"fi\ si) is a Fitting class containing si.
vtu t ™ 7; и Г zv t"?'1 ",S'' - furthermore, as pointed out in
XnI’n r2 S’ U bet ['4/ nn(} A e -tf) u {1} 'S a Fitting set of Aut( У). Thus the pair
(I, .o') fulfils Hypotheses 5.5. Condition s.I of Hypotheses 5.9 coincides with Hypo-
thesis (D) of this corollary, and Conditions s.II, sJII, and sJV are obviously also
satisfied. Therefore £>"(У, .</) is s.-closed by (5.11).
We now turn to the hypotheses of Theorem 5.17. Hypothesis (1) of that theorem
is included m our hypotheses here and Hypothesis (2) coincides with Hypothesis (A)
of this corollary. Furthermore, Hypotheses (Q and (D) of this corollary imply that
Hypothesis (4) of (5.17) is satisfied, Hypothesis (B)(i) implies Hypothesis (3) of (5.17),
and Hypothesis (B)(ii) is the same as Hypothesis (3') of (5.17). Hence D"(F,//) is
No-closed by (5.17) and (5.15) and is therefore a Fitting class.
Let A e si. By Hypothesis (C) of this Corollary we have О,(У) = 1, and be-
cause Z(T)= 1 and hence САи|(п(1пп(У)) = 1, it follows easily that О,(Л) = 1.
Since ЛДпп(У) is a л'-group, we have 0”(Л)1пп(У) = A, and hence 1пп(У) =
[Inn( У), Inn( У)О"(Л)] by Hypothesis (D). Thus О“(Л) n Inn( У) is a normal supple-
ment to 1пп(У)' in 1пп(У) and therefore coincides with 1пп(У) since 1пп(У) is soluble.
Hence A = О"(Л) = к(Л), and since У = 1пп(У), it follows that A e А ’(У, si) £
£>"(У,з/). □
In our first application we present the example originally used by Dark [2] to
exhibit an g-Fischer subgroup which is not an J-injector. It also yields an example
of a non-permutable Fitting class.
(5.19) Example. Our first objective is to identify the group У. Our next task is then
to compute Aut( У), and having specified the set si of subgroups of Aut(T), to verify
that the pair (У, si) fulfils the hypotheses of Corollary 5.18.
We begin by considering the extended affine group over the Galois field F8 (see B,
12.9). It has the form CBA, where C s Fs+ is a normal subgroup of order 8, and where
В S (F8 is a subgroup of order 7 normalized by the subgroup A of order 3 which is
generated by a field automorphism. The group CBA has a natural doubly transitive
permutation representation v on the field elements, which we label {0, 1,..., 7} so
that BA is the stabilizer of 0. We denote the transitive permutation representation
of BA on the remaining points {1,7} by p. Next we consider the extended affine
group over the field F125, which has the form FED, where F is a minimal normal
subgroup of order 125, and where E is a cyclic subgroup of or er
normalized by a subgroup D generated by a field automorphism of order 3. We then
form the wreath product
W = FE£>4a. Sym(7),
and label the direct factors of the base group of Ж and their corresponding subgroups
644 IX. Fitting classes—examples and properties related to injectors
and elements, with the suffices 1.7. We regard p as an identification of BA with
a subgroup of Sym(7). Then В is generated by a 7-cycle, and the normalizer of В in
Sym(7) has the form B(AT), where the subgroup AT = A x Tis cyclic of order 6 and
may be taken to be the stabilizer in BAT of the point 1. The subgroup E of FED has
the form E = H x L, where Я = Z4 and Ls Z31, and we note that D centralizes Я
because Я corresponds to the multiplicative group of the prime subfield of (F,25. At
this stage we are mainly interested in the structure of the following subgroup of W:
W(, = FLD%BA.
Let A = <a>, and D — <d>, and let d denote the element d, d2... d7 of the base group
of W. Further, let R denote the subgroup FL of FED, and note that R is isomorphic
with the primitive group £(31/5) of order 53.31. Evidently BA centralizes d, and so
В (da) is a subgroup of Wo isomorphic with £(3/7). We set Q = R" = R, x • x R7,
and denote by У and К the following subgroups of Wo:
Y = QB <з К = QB(da).
Obviously the group Y is isomorphic with R rbrcB Z7 and by A, 18.5(b) is primitive
of order 7.(53.31)7. Since Q = O(5 31 j(AZ) char К and since CK(Q) = 1, there exists
a monomorphism from К into Aut((7) which maps the subgroup Q onto Inn(Q).
To simplify the notation we now identify К with its image in Aut(Q); since Q
char У, we can also identify Аш(У) with the normalizer of У in Aut(Q). By A,
18.14 we have Aut(Q) = Aut(R)rlinM Sym(7), and from B, 12.10 we know that
Aut(R) = FED. Therefore Aut(Q) = W. Let U denote the subgroup (H£>)"Sym(7)
( = (H x L>)rLinat Sym(7)) of W. Since HD (sZ4 x Z3) is a complement to R in FED,
the subgroup Я is a complement to Q in Aut(Q). Let Я = </i>, and let h denote
the element hth2...h2 of Я". Since Q is a normal subgroup of И7 contained in У,
we _ then have АЫ(У) =_ЯА.|(0)£У) = QNV(B) = Q((h) x <d> x NSym(7)(B)) =
Q((h) x (d) x BAT) = Y((h) x <d> x <a> x T). Thus АиЦУ)/У is_isomorphic
with Z4 x Z3 x Z3 x Z2, is therefore abelian and contains K/Y = (da)Y/Y as a
normal subgroup of order 3.
We now set л = 3' and т = {3, 7, 31}, and proceed to verify that the groups У and
К fulfil the hypotheses of (5.18) for these sets n and r. Certainly Z(Y) = 1, and
K/Inn( У) = K/Y is a non-trivial soluble л'-subgroup of Aut( У)/У. As we pointed out
above, the subgroup F11 = О5(У) is the unique minimal normal subgroup of У and
hence of K; in particular, Hypothesis (C) of (5.18) is satisfied. Since 31 has order 6
modulo 7, the section = [g/F", У] = [Q/F", B] is a chief factor of У of order 316,
and the centralizer M2 of У (or, equivalently, of B) in Q/F'> is a central chief factor of
У of order 31; furthermore, and M2 are evidently also chief factors of К because
F=, Q, and У are normal subgroups of K. Since the element a obviously centralizes
Vf2 and the element d acts fixed-point-freely on M2 (in fact, on the whole of Q/F"),
it follows that К induces on M2 an automorphism group of order 3. It is therefore
clear that К/ Y is the only central chief factor of K. Since .в/ = {£}, these observations
imply that Hypotheses (A) and (D) of (5.18) are satisfied, and because <т( У) =
5. Dark’s construction - the theme
645
Pl’rn ™'S “ 3i‘group’ U is also clear that Hypothesis (B)(i) is
fulfilled. Therefore by Theorem 5.18 the class D,"( Y .</) is a Fittine class
wepwt> - w.** a
also a Fitting class. Our next goal is to describe a group G possessing an g-injector
V which is not system permutable in G. We shall also show that V is a Fischer
Г-subgroup of G but not a Г-mjector of G. The group in question is the wreath
product
G^FLD^CBA,
a group of order 3.7.8.(3.31.53)8, where v is the representation of CBA on the ele-
ments of F8 labelled 0, 1,..., 7. Maintaining our earlier notation and, in particular,
identifying the group FLD BA considered above with the subgroup
{(FLD). x • • x (FLD)f)BA of G, we set
F = (R0 xR, x x R7)B<da>
and note that V = Ro x K. Since Ro s £(31/5) and O"(K) = K, it is obvious that
O,(F) = 1 and O“(F) = K. Thus x(F) = KesJ, and it follows from (5.18) that V
belongs to Г and hence to g. We now prove one by one the properties claimed for
the subgroup V.
(5 c)
V is an ^-injector of G.
Proof. From now on the base group notation‘t]’ will refer to the wreath product G.
First we show that R" = (FL)" is a Г-maximal subgroup of (F LDf. Since O'(FL) = 1,
we certainly have (FL)" e Г. Let (FL)" < S < (FLD)". Then 31 |S|, and therefore
O3'(S) ф 1, But because F" is self-centralizing in G, we have O,(S) = 1 and hence
k(S) / 1. Since 7f|S|, it therefore follows that S£T, and consequently (FL)" is
Г-maximal in (FLD)", as claimed. Thus (FL)" is a Г-injector of (FLD)", and since
(FL)" e g, we conclude that (FL)" is an g-injector of (FLD)" as well. Now observe that
g is a class of 2'-groups and that therefore an g-injector of (FLD^C is contained in
the Hall 2'-subgroup (FLD)" of (FLDfC. Because V is an g-subgroup supplementing
(FLDfC in G and satisfying VrfFLDfC e Injg((FLD)"C), we then conclude from
(L6)(b) that F e Injg(G). °
Next we prove that
(5 £) у permutes with no Sylow 2-subgroup of G.
Proof Suppose, to the contrary, that FC’ = C’F for some xeG. Let X = FC’,
and let N denote the normal subgroup (FLD)"C* = (FLD)*C of G. Then AnЛ-
C‘(X n(FLDf) = CW. Because C e Syl2(M we may suppose that x e Nandcan
therefore write x = ctr with с e C, t e Do x • • • x D7, and r e Ro x x R, -(FL) -
F n N Thus FC’ = VC" = FC, and so without loss of generality we can suppose
that x 6 Do x • x D7. Since (FL)" < G and since A normalizes C, it follows that A
646 IX. Fitting classes—examples and properties related to injectors
normalizes CX(FLf = X ryN ; furthermore ad e X < NG(X n N). Thus [x, a]d =
(a*)-1 ad normalizes XoN. Because the element a normalizes Do x • x p7
and fixes Do and D, (by definition A stabilizes 0 and 1), it follows that
[x. a] e [Do x - - x D„ a] < D2 x - x D, an^ hence that [x, a]d is a non-trivial
element of (D, x x D,)r Nn(X n N). However, Nn(X aN) = N„(CX(V n N)) =
(NK(C)(Vn N))x by A, 6.4(a). Since C permutes the eight direct components (FED),
of (FED)6 in a regular orbit, we have Nn(C)(Vn N) = C(dod1.. .d7>(Fn N), and
therefore Nn(X n N) = Cx{d(ldt ...||7)х(Ил N)x = Cx(.dodi.. .<i7)(Fn N). Hence
(D, x ... x D,)nNN(X nN) = 1 since the D.are 3-groups, И nN is a {5,31}-group,
and C is a 2-group. This contradiction proves Assertion 5f. □
It follows that V is not system permutable in G and therefore that
(5.1/) The Fitting class g is not permutable.
Our next goal is to show that
(5.0) V is a Fischer T-subgroup of G.
Proof. Let J be a T-subgroup of G normalized by V. Since R6 < V, the subgroup R6J
is a normal product, and consequently R6J e T. To prove that И is a Fischer T-
subgroup of G we must show that J < V, and can therefore suppose without loss of
generality that Rb < J. As shown in the proof of (5.e), the subgroup R6 is T-maximal
in (FLD)6, and so J n (FLD)6 = R6. Consider the subgroup (FLD)6(J n N) of (FLDfC =
N; it is normalized by V and N and hence by VN = G. Since (FLD)6C/(FLD)6 is a
2-chief factor of G, it follows that (FLD)6(J n N) is either (FLD)6 or (FLDfC. If J n N
covers (FLD)6C/(FLD)6, then J nN contains a Sylow 2-subgroup Cx of N; in this case
J n N = R6CX and so V(J nN) = CXV <G. contrary to (5.Q. Hence J n N = R6.
Since G/N is a primitive group of order 21, and since JN < G, then either J < N or
JN contains the subgroup BN. If J < N, then J = R6 < F, as desired. Therefore
suppose that BN JN. Then BN n J is a Hall {5, 7, 31}-subgroup of G normalized
by the {5, 7, 31}-subgroup R6B. Consequently R6B < J, and it follows that R6B <
JnVand that JF/R6Bisa 3-group. Hence J V e NfJ(J, V) £ T>. But JVe <35<33167<33,
and therefore JV belongs to g. Since we have already shown in (5.e) that V is
^-maximal in G, we conclude that J < V. □
Finally we show that
(5.1) V is not a T-injector of G.
Proof. Since O*(R6C) = 1, we have R6C e T. Therefore R° = V n N is not T-maximal
in N, and so V is not a T>-injector of G. □
In the preceding example, the Fischer T-subgroup, although not a T>-injector,
turns out to be an 5*-injector for some Fitting set &. This raises the following
question.
6. Dark s construction—variations
647
IV a FiSCher R-SUb^OUP * finite soluble group G for some
Fitting class Й. Is V an injector for some Fitting set of G? (According to VIII 3 3
this is equivalent to asking whether the set of all subnormal subgroups of the
conjugates of V forms a Fitting set of G.)
6. Dark’s construction—variations
Dark s Fitting class construction, described in the preceding section, has provided
the inspiration for many variations, which have extended and deepened our knowl-
edge of the theory s scope and limitations. We will summarise these examples at the
end of this section, after we have studied several of them in more detail and described
their implications and applications.
The first variation we consider is due to Bryce, Cossey and Ormerod [1]. Entitled
“Fitting classes after Dark”, their paper sheds light on the complicated ideas in
McCann’s dissertation [1] (see also McCann [2]) and yields an example, similar, but
not identical, to McCann’s, from which his theorem about the Fitting class generated
by Sym(4) can be deduced. In fact, Bryce, Cossey and Ormerod describe two distinct
variations, which we will call 1(a) and 1(b). Variation 1(a) is close to Dark’s original
construction but is not covered by the hypotheses of Theorem 5.17. Variation 1(b)
has a different form and needs 1(a) to establish its credentials.
Variation 1(a)
Throughout the discussion of this variation, p will denote a fixed prime, and the key
section will be defined in terms of p and a certain Fitting class Я.
(6.1) Definition. Let p be a prime and Я a Fitting class satisfying Я = ЯЗр. If G is a
finite group, we define an associated key section k0(G) by
k0(G) = O'’(G)/(X'(G)B.
Clearly k0(G) is contained in the class ££, defined in (5.l)(c).
(6 2) Hypotheses. Let К be a finite soluble group, and let a be an automorphism of
У of order p, our fixed prime. We make the following hypotheses:
(i)Z(r)=l;
(ii) У has a unique minimal normal subgroup,
GV) Lei A denote the semidirect product [У] <«> Then, whenever A A2 < G =
A, A2 with А, г A2 S a and Z(G) = I, either A, = A2 or [A„ A2] -
It follows from Hypotheses (i) and (iii) that Сл( У) = MnJ' (УлО
A as a subgroup of Autd); with = {A} ™ саП,,“ V) s о (W
delined in (3.7). Il is easy le see Ibal if
and so an associated class £>S(L A), analogous
648
IX. Fitting classes—examples and properties related to injectors
thus:
(M
Dg'(F, Л) = (G e G: k0(G) e s,D„(H)).
If N < G, it is a routine matter to check that k0(N) is isomorphic with a normal
subgroup of k0(G), and consequently Pg'(K A) is s„-closed. (The modification of the
key section in this variation does not materially affect the arguments used in Section
5 for Dark’s construction; the important condition that k0(G) S DJ is still satisfied.)
The harder task is now to show that DPA(Y, Л) is n0-closed. Unfortunately, we cannot
apply Theorem 5.17 here. For although Hypotheses (1) and (4) of (5.17) are satisfied
if Л has odd order, we cannot deduce (2) and (3) from (6.2). We therefore argue afresh,
closely following Bryce, Cossey and Ormerod and beginning with three preparatory
lemmas.
(6.3) Lemma. Let S be a subnormal subgroup of defect at most 2 in a group H, and
assume that S < O2(H). Let
(6.Д)
К = Ft x x
be a normal subgroup of H, and assume that each Y, (^ 1) is directly indecomposable
and that the direct decomposition is unique, up to the order of the factors. If S = Yj for
some i, then S normalizes Yt.
Proof. Let heH and je {1,..., t}. Our hypotheses imply that (Yff1 = Yr for some
f e {1,..., t}, and on setting n(h): j-*f, we obtain a permutation representation
it: H -> Sym(t). Let s e S, and let 12 be the orbit of n(s) containing i. We must prove
that |I2| = 1. Set
M = [I<, s, s],
and first suppose, by way of contradiction, that |I2| > 3. For у e Yt, we then have
[y, s, s] = yy 2”y’\
and so the subgroup M of R projects onto the direct component Yj of R in (6./?). Thus
|M| > |2j| = |S|. However, since S has subnormal defect at most 2, we have M <
[2^, S, S] < S. Thus S = M < R and, in particular, S normalizes the normal subgroup
Y{ of R, a contradiction. Hence |I2| < 2.
Next suppose that |f2| = 2. In this case
[y, s, s] = (yys2)(y a)2,
and it follows that O2(M) projects onto O2(2Ja). Hence |O2(Af)| > [О2(У?’)1 But
M < S’ implies that O2(M) < O2(S), and since S = F(a, we conclude that
O2(S) = O2(M).
6. Dark’s construction—variations 649
However, since S < 02(Я), which is generated by elements of odd order, n(s) is an
even permutation, and it follows that there is an element j in {1,t}\(i, in(s)}
belonging to a n(s)-orbit fi* of length at least 2. Thus, if M* = [Y s s] by the
above arguments we have O1(Yj) <, O2(M*) and 02(Л/*) < 02(S) = 02(M). Since
= с1еаг1У Л/ Л/* = 1, and consequently 02(Л/*) = 1. Thus Yj is a non-
trivial 2-group, and the decomposition Yt x Yf is not unique by A, 4.10. Since this
contradicts our hypotheses, we conclude that О = {i}. □
(6.4) Lemma. Let R and R* be normal subgroups of a group H admitting direct
decompositions as follows:
(a) R = K] x • x Y„ where each subgroup Yj has a unique minimal normal subgroup
and a trivial centre;
(b) R* = Y* x x Y*,where Yf= - -Y*^Yiforsomeie{l,...,t}andR*^
O2(H).
Let x e NH(Yj*) for j = 1,..., t*, and assume that [Yj, x] <, R*. Then x normal-
izes Yt.
Proof. By A, 4.10 the direct decompositions for R and R* are unique (up to order).
Since each Yj* is a subnormal subgroup of H of defect at most 2, we can deduce from
(6.3) that R* normalizes Yj. We argue by contradiction, supposing that [Yj, x] < R*
and that x normalizes all the Yj* without normalizing Yj. Since x induces by conjuga-
tion a permutation of the set {Y,,..., Yj}, our supposition implies that <Yj, Yjx> =
Yj x Yx. By hypothesis (Yj x Yjx)nR* projects onto each direct factor of Yj x Yjx,
and since Z(Yj) = 1, we have Soc(Yj) < [Yj, Yj] = [[Yj, x], Yj] < R*. But
Soc(Yj) char Yj and R* normalizes Yj; therefore Soc(Yj) < R*. Because each Yj* is
monolithic with trivial centre, the only minimal normal subgroups of R* are
Soc(Yj*),..., Soc(Y„*), and consequently Soc(Yj) = Soc(Yj*) for some je {1,..., u}.
Since x normalizes Yj*, it normalizes its socle and hence normalizes Soc(Yj). But this
obviously contradicts the fact that Yf n Yx = 1; hence our supposition is false, and x
normalizes Yj. □
(6.5) Lemma. Let PandYbe normal subgroups of their product G = PY. Assume that
Pisa p-group and that Y is a non-abelian primitive soluble group. Then P has a subgroup
Po such that G = Po x Y.
Proof. If P n Y = 1, simply take Po = P and the result is clear. We may therefore
suppose that P n Y = N, where N denotes the socle of the primitive group Y. Since
N char Y < G, we have N •< G, and therefore P < Cc(N) by B, 3.12. Let R = O„ „ ( Y).
Since Y is non-abelian, we have N < Rand therefore [N, R] = N. Since N < Z(P),
the quotient R/N is a p'-group of operators for P, centralizing P/N because [P, R] <
PoR = N. Hence by A, 12.5 we have P = [P, R]GP(R) = N x Cr(R) because
CM(R)=L Set Pn = CP(R), and note that P0<G. Thus [Y, Po] < YcP0 =
Y с. P n Po = N n Po = 1. and it follows that G = PY = P0N Y = Po Y = Po x Y, as
desired.
The next result contains the heart of the proof that D$(Y, A) is N0-closed; it will
also be used in establishing N0-closure in Variation 1(b) below.
650 IX. Fitting classes—examples and properties related to injectors
(6.6) Proposition. Assume that Y, a, and A = [У]<а> satisfy Hypotheses 6.2, and if
Op(Y) 1, assume further that У is primitive. Let X denote the class of all groups X
satisfying
(i) OP'(X) = X, and
(ii) OP(X) = У, x x Y„ where Y Yt < X and X/Cx(Yf} S A fori = 1,..., t.
If H = XX*, where X and X* are normal X-subgroups of H, then H eX.
Proof. Write
R = OP(X) = Yt x x X and R* = OP(X*) = Yt* x x Y*,
where each Yt (respectively У<*) is a normal subgroup of X (respectively X*) isomor-
phic with У We show first that У' < H for i = 1,..., t. Suppose, by way of contradic-
tion, that Yt is not normal in Я = XX*. Then X* contains an element x such that
У<х X, and s’nce W permutes the direct components Yj of R (<H), we have
<У<, У<ж> = X- x Y*. The subgroup Я = [У;, x] ( = [У<Х, x'1]) is normal in
Yj x I/, projects onto both direct factors, and therefore contains [У<, [ Yit x]] = Y-
and [УУ, [У;, x]] = (У")х. Consequently N contains У/1 x (У/1)*, and since
[X91 X (У;9гГ, Я] = [У^91, У<] X [(У?)91, У;х] = У^91 х (у;9|)х, it follows that Я91 =
у,<п х But Y = op(Y) by (6.2)(iii), hence (У< x УХ)/(У; x У<л)91 is a p'-group,
and therefore OP(N) = N. Since Я < X*, we can deduce that Я < OP(X*) = R*, and
because X* = OP'(X*) < OP'(H), it follows that R* < O2(H). Thus Lemma 6.4 is
applicable and implies that x normalizes У^. This contradiction shows that Yt < H
for i = 1,..., t, and by symmetry that Y* < Я for j = 1,. ., u.
Assume that Y:r.R* 1. To simplify notation, we suppose that i = 1, and we
adopt the convention that if U is a subgroup of У, then Uj (respectively C*) will denote
the corresponding subgroup of У< (respectively Ук*) under the agreed isomorphism
with У Let T denote the unique minimal normal subgroup of У. Then our assumption
implies that T\ < У] n R*, and since Tt char У] < Я, we certainly have 7j < R*. But
it follows from Hypotheses 6.2(i) and (ii) that T*, ..., T* are the only minimal
normal subgroups of R*, and therefore by order considerations we have Tt = Tf
for some ye {1,..., t*}. Renumbering if necessary, we suppose that j = 1. Hence
У, п(У2* x x Y*) = 1. For if the projection of Yt n R* into Yk* (k > 2) were
non-trivial, it would contain Tk*, and then we should have Tk* = [Tk*, Ук*] <
[У, n R*, Ук*] < yt, and hence 7j < Tk*, which is impossible. Therefore У] n R* =
У] n Y*, and by a symmetrical argument У,* гл R = У, n Yt*.
Let С = С„(У*) < Я. Since T> = 7\* £ Z(Y*), it follows that СпУ, = 1 and
consequently that С < С„(У,). Again by a symmetrical argument we have СН(У]) <
C and therefore С„(У,) = С. Since X*C/C X*!Cx.(Y?) s A, it follows that
XC/C = A and hence that Я/С = (XC/C)(X*C/C) is a normal product of two copies
of A. If zC e Z(H/C), then [ У,, z] < У, n С = I, and so z e C. Thus Z(H/C) = 1, and
since [TjC/C, YfC/CJ > TtC/C * 1, Hypothesis 6.2 (iv) implies that XC = X*C.
Therefore У> C = YfC, and it follows that У, Yt* = У, У>* n У> С = У, (^ Yf n C) and
hence that
(6.У)
У, Yf = Yix(Y1 Y* n C).
6. Dark’s construction—variations
651
Now X п(У, У* n C) = Yl(X n У*) n C = Y,(R n Yt*)r\ C < Y. n C = 1, and so
the normal subgroup У, У* n C centralizes X; it similarly centralizes X‘, and we
conclude that
(6 <5) У, У>* n C < Z(H).
Now relabel the subgroups Yt* so that
for< = l, ...,s and
1 for i = s + 1,u.
If the subgroup Rr\(Yt* x x У/) were non-trivial, it would contain one of the
only minimal normal subgroups T*,..., T* of Yt* x • x УД which is not the case.
Therefore Rr.(Y* x x У/) = 1. On the other hand, for к e {s + 1,..., u}, it
follows from (6.y) and (6.6) (for the appropriate suffices) that RY* < RZ(H). By II,
2.12 we have OP(H) = OP(X)OP(X*) = RR*, and putting these facts together, we
obtain
OP(H) = У, x x Y, x У* x x ys* x L,
where L < Z(H). However
RR* = [K,X][K*,X*] < [RR*, H] < Yi x • •• x Y, x У* x x У/
because [L, H] = 1, and therefore L = 1.
Since OP‘(H) = Op (X)OP'(X*) — XX* = H, to prove that H e JE it is enough to
show that H/CH(Yj) = A = H/CH(Yj*) for i = 1,..., t and j = 1,..., s. To this end,
let ie {1,..., t}, set С, = СН(УД, and first suppose that Yji-^R* = 1. In this case
Y; r\ X* is isomorphic with a subgroup of X*/R* and is therefore a p-group. If
Yj n X* = 1, then X* < Cj, and, in particular,
(6.e) XX* = XCj.
On the other hand, if Yt n X* ± 1, then ОДУ) # 1, and У is primitive by hypothesis.
Hence Yj n X* = Soc(X), and we can apply Lemma 6.5 to the product of the normal
p-subgroup Х*СДС, with the normal subgroup ^C./q (S У), which is a non-abelian,
primitive soluble group. The conclusion of (6.5) implies that X* has a subgroup U
such that
X*YjCj/Cj = (UCj/Cj) x (YjCj/Cj).
It follows that [1<, 17] < YjC\ q = 1 and hence that U < Ct. Consequently X* <
1<C, and again (6.e) is satisfied.
There remains the possibility that IJoR* / 1. But in this case we showed above
(with i = 1) that XCj = X*C, and hence that XX* = XX*Q = XXC, - XCj. Thus,
in any case, (6.c) holds, and we have H/C, = XCJQ = Х/Сх(Ъ = A, as desired A
symmetrical argument shows that H/CH(Y^*) S A for j = 1,..., s, and the proof that
H belongs to JE is complete. LJ
We are now in a position to establish the bona fides of this variation of the Dark class.
652
IX. Fitting classes—examples and properties related to injectors
(6.7) Theorem. (Bryce, Cossey and Ormerod [1]) Let p be a prime, and let Y be a
soluble p'-group. Assume that Y and a (eAut( У)) satisfy Hypotheses 6.2, and let Я be
a Fitting class such that RQp = ST Then the class £>g (Y, A) defined in (6.a) is a Fitting
class.
Proof. We have already indicated after (6.2) why Dg( Y, A) is s„-closed when У is a
p'-group. To prove n0-closure, suppose that G = NN*, where N and N* are normal
subgroups belonging to Dg(K A). Let X = Op (N) and X* = Op (N*). Then
OP'(G) = XX*, and if К = Op'(G)a, the key section k0(G) is equal to the normal
product (XK/K)(X*K/K). Now XK/K S Х/Хл = к0(Х), which belongs to s„d0(4),
and it follows easily that Op(x0(X)) satisfies Condition (ii) of Proposition 6.6; so also
does O₽(x0(X*)), and we can deduce from that result that x0(G) belongs to the class
defined there. But obviously a group H belongs to JE if and only if H/Op(H) e
when У is a p'-group, and since Op(k0(G)) = 1 by the assumption that Я<5р = Я, we
conclude that k0(G) e s„d0(A). Hence G e £>g (У, A). □
Variation 1(b)
This is the example of Bryce, Cossey and Ormerod based on the construction in
McCann’s dissertation (see McCann [1] and [2]); its main ingredients are 3 fixed
primes p, q, r with p q r (although the possibility p = r is allowed), a Fitting class
8 £ G,-, and a certain group X with the special properties described below.
(6.8) Hypotheses. Let p, q and r be primes with p q r, and let 8 be a Fitting class
of ^'-groups. Let X be a finite group with a series of normal subgroups
I < N = X^<U <B < X
such that \X/B\ = p, B/U is a «/-group, and U/N is an r-group. Let B/U = ®(B/U}
and U/N = <b(U/N), and assume that
(a) O« (B) = B,
(b) X/B and B/U are primitive groups, and
(c) the group A = X/U satisfies Hypothesis 6.2(iv).
X
<i N = XB
1
6. Dark’s construction—variations
653
(6.9) Remarks. Assume that X satisfies Hypotheses 6.8.
(a) Let Y denote the normal subgroup B/V of A. Since Y is primitive, its socle V/V
is self-centralizing, and evidently Z( У) = 1. Since V/V is complemented in A, a Sylow
p-subgroup, <a> say, of a complement has order p and is a complement to Y in A. It
follows easily from the definition of В and the fact that X/B s E(p/q) that [У, a] = Y.
Thus Y and a satisfy Conditions (i), (ii), (iii) and (iv) of Hypotheses 6.2; moreover A
is primitive. However, the Y here need not be a p'-group, as it was for Variation 1(a).
(b) If Z/N = Z(X/N), then Z < V; for otherwise ZV/V would be a non-trivial
central normal subgroup of the primitive group A, which has a trivial centre.
We now describe the candidate for the Fitting class of this variation. As usual, we
adopt the convention that G, will denote an isomorphic copy of G and that if H < G,
then Hj will denote the image of H under the given isomorphism.
(6.10) Definition (The McCann class M(^, X)). Assume that the Fitting class and
the finite group X satisfy Hypotheses 6.8; in particular, recall that Y = B/V is a
normal subgroup of index p in A = X/V. Then М(Д, X) is defined to be the class of
all groups G with the property that the subgroup Q = OP'(G) has a (possibly empty)
set of normal subgroups B,,..., B„ each isomorphic with B, satisfying the following
three conditions:
941: [B;, B,] < Q~, for all 1 < i j < t;
942: Let С, = Се(В,/Ц); then Q/Ct S Л;
943: Op(Q} = Bi...B,Op(Qs)
(6.11) Remarks, (a) If t = 0, Conditions 9.41 and 942 are vacuously satisfied, and in
this case Condition ЯПЗ implies that M(g, X) always contains the class SGpGp..
(b) Let F = Cg. Then [B„ F] < B; n F = (B,)s < (X,)g = A, < V„ and so F cen-
tralizes X = В.-/Ц. Thus F < QU, C.-
fc) For 1 < i ± j < t, by 941 we have [B„ В,] < B, n F < Vt as in Remark (b).
Therefore B, < C„ and consequently
(6.0
Пв,с; = в1в2...в/Q q
by(d) We assert that, in the presence of 941 and 9.42, Condition 9.43 is equivalent to
the following:
943*: | Q'i=i С : Uj G,F| is a power of p.
To see this let C denote Г1!,, C,. Since (B,... Д)ДС, ...p,) x x Y„ which,
like У, has trivial centre, it follows that C n (B,... B,) = U.... Ц, and therefore
(6.,,) |B,... B,C: B,... B,F\ = |C: V,... P,F|.
Condition 943 implies that |G: B,... B,F\ is a of p, then certainly so is
IB,... B.C: B,... B,F|, and by (6.0 Condition 943 now follows.
654
IX. Fitting classes- -examples and properties related to injectors
Conversely suppose that Ш13* holds. Since |Q: B,C,| = p by Condition ЗД12, the
index |g: Q'=1 AGIis a power of p. But by (6.0 and (6.^ we have |g: Bj... B,F\ =
|(?: B1...BtC\\BJ...B,C:Bl...B,F\ = ]g: Q;=1 В,С,||С: Uj... U(F|, which is also a
power of p by ®13*. Hence 0p(Q) < Bt... B,F, and by II, 2.12 we conclude that
0p(Q) = O^O'Ce» < Op(Bi)...Op(B,)Op(F)
= BI...B,O’’(F)
since Op(Bj) = Bt (i = 1,..., t) by (6.8)(a). On the other hand, since B; < Q, we have
В/ = Op(Bi) < 0p(Q); hence B, ...B,OP(F) < 0p(Q), and therefore Ш13 holds.
(e) Unlike all previously-discussed examples of Dark’s construction, this class of
McCann’s cannot be formulated purely in terms of the structure of a key section,
which in this case would obviously have to be the section 0p (G)/0p (G)g. For
although ®11 and 9Л2 have the consequence that Q/F contains a central product of
normal subgroups Kt, ..., K, (each isomorphic with B/N) on which Q/F induces
suitable groups of automorphisms, there is no reason why merely postulating the
existence of such a central product in the key section should guarantee that each
central factor Kt must have the form BiF/F for some normal subgroup B; of Q
isomorphic with B, as required in this example.
Our first objective will be to show that the class M(5, X) is s„-closed, and for this
we will need the following lemma.
(6.12) Lemma. Let H be a group whose 91-residual H^1 is a product of subgroups
each normal in H. If К < H, then К* = (Л\ n K*)(N2<-y K*). . .(N,c, K*).
Proof. By definition of the 9l-residual, there exists an integer r such that K9i =
[К,/., KJ; morever, [К91, К,.3.., К] = К 91 for all s > 0. Since [7Vi; К] < N. nK,
it follows that [ty, К,.'.., K] < №*, and therefore К91 = [K91, K.Z., K] <
[H9’,K,Z.,K]<nUi[W1,Kl.,K]<riUi(Ari^K91)<K91. Thus equality
holds, and the lemma is proved. □
(6.13) Proposition. Assume that Hypotheses 6.8 are satisfied. Then the class AF(g, X)
defined in (6.10) is s„-closed.
Proof. Let N S G e M((J, X). We must show that N e M(g, X). Set Q = OP (G), and
recall from Definition 6.10 that Q has normal subgroups Bj,..., B„ each isomorphic
with B, such that Conditions ЯЧ1-9ЛЗ hold. If t = 0, then G e gGpGp., and since
8SPGP- is s.-closed, we have N e 5GpGp. and hence N e Mffi, X) by (6.1 l)(a). There-
fore suppose that t > 1, and set Q* = OP(N), F = Qg and F* = (Q*)g = Q* n F. We
recall that Q denotes the normal subgroup Се(В,/Ц) of Q and that Q/C, s A for
i = 1, ..., t. Since the quotient group (?/ЦС(( = Л/8ос(Л)) has a unique maximal
normal subgroup, which is a (/-group, the normal subgroup (Q* UiCf/tUiCf, if proper,
is a (/-group. Thus for each i e {1, ..., t} just one of the following two situations
arises:
6. Dark's construction—variations
655
Case I. Q*VtCl = Q;
Case 2. (Q* UiC^l^Ci) is a (/-group.
We first suppose that Case 1 arises and show that then
B, < OP(Q*).
Since UjQ/Ci is the unique minimal normal subgroup of Q/Ch it must be contained
in the non-trivial normal subgroup С*С,/С;. Therefore in Case 1 we have
(6.0)
Q = Q*Q.
Now put R = £«(= 0p(Q)) and R* = (C*)91(= 0p(Q*)). Since B, < 2*C, for i = 1,
..., t, by II, 2.12 we obtain
(6-0
B, = 0р(В,) < Op(Q*)Op(Ci) < R*(Q R).
Set F = 0p(F) and note that R = Bt... BtF by Condition Ш13. Since R* < Q, from
Lemma 6.12 we conclude that
(6.x)
R* = (ВпЛ*)П(В;пЛ*),
and since 0p(F) and B, are both subgroups of C, r> 0p(Q) = Qr .R for j i, from (6.1)
and (6.x) it follows that
Bi<(BinB*)(CinR).
Then by the Dedekind law Bt = (B.r. R*)(B,r.Cir. R) = (BtnR*)U„ and since
Ц/Nj < Ф(В^\), we must have B, = (Bf r> R*)N;. However, it then follows that
В,/(В; n R*) S Ni/(Nt r\ В;ъ R*) e Qft £ G,-, and consequently B, r> R* = B( by Hy-
pothesis 6.8(a). Thus B, < R* = 0p(Q*) in Case 1, as claimed.
Since Q*( = 0p(N)) is (/-perfect, in Case 2 we have Q* < CQ. By renumbering
the groups B>, .... B, if necessary, we can suppose that B1? B, are subgroups
of R* and that Case 2 arises for i = s + 1, ..., t, and as R* < R = B,... B,F, we
obtain
R* = Bj... B,(R* n (Bs+1... B,F)).
For t > s + 1, let I(t) denote the intersection
1(0 = П (ЦС,пВ5+1...В,В).
j=s+l
Since R* < VjC, for j = s + 1,..., t, we have R* n Bs+1 ...B,F< l(t). We aim to show
by induction on t that
656
IX. Fitting classes—examples and properties related to injectors
I(t) = Us+,... U,F.
Since F < Ct and Bk < C, for к < t, we have
U,C, о (Bs+i... B.F) = Bs+1... B,-1(VIC, n B,)F
(Us+IF if t = s + 1
(Bs+I...B,_lFUl ift>s+l.
Therefore I(s + 1) = Us+1 F, and we have a starting point for the induction. Moreover,
since I(t — 1) < Bs+1... B, i F, by the Dedekind law we have
I(t) = ( П 4G Bs+1... В,_, F) Ц = i(t - 1) V.,
\J=s+l /
and the induction step is complete. Because R* n (Bs+1... B,F) < I(t), we therefore
have
R* = B1...Bs(R*rr(t/s+1...t/,f)).
Now (l/s+j... U,F}/F is a product of normal r-groups VjF/F( = Vj/Nj), and it follows
that R*/(Bt... BS(R* n F)) is a normal r-section of Q*. Therefore, if r = p, the fact
that R*( = 0p(Q*)) is p-perfect implies that
(6.2)
R* = B, .. BS(R* n F) = B,... Bs(R*)g.
If, on the other hand, r p and s + 1 < j < t, the r-perfect group Q*, being con-
tained in VjCj, centralizes Ц/Uj and therefore centralizes Ц/Nj by A, 12.7. But
then R*/(BX... BS(R* F)) is a central normal r-subgroup of p-power index in
Q*/(B1... BS(R* n F)) and is therefore a direct factor. However, this group, like Q*,
is r-perfect and has no non-trivial r-groups as direct factors, and we can again con-
clude that (6.Л) holds.
We are now in a position to verify that N belongs to Mffi, X). Certainly Q* =
0p (N) has normal subgroups Bt,...,B, satisfying Condition 9)13 because of (6.2) and
the fact that F/(R* r\ F) is a p-group (whence 0p(R* rx F) = OP(F)). Moreover, for
I < i j < s we have [B(, В;] < R*nF = R^, and so 9)11 also holds. Finally, if
1 < i < s, we have Q*/(Q* n C() = C*Cj/C( = Q/C-, by (6.0) and since 9R2 holds for
G, we have С*/Се.(В^Ц) = A. Thus Conditions ®ll 9ЙЗ all hold for the subset
B],..., B5 of g* = OP'(N), and we have proved that N e M (ft, X). □
Our next goal is to show that M(ft, -X) is also No-closed and is therefore a Fitting
class. We begin with a preparatory lemma.
(6.14) Lemma. Let L < H = KL with К sn H. Assume that O*(K) = К and that L e
<S, for some set n of primes. Then К = O“(H).
6. Dark’s construction—variations 657
Proof. We argue by induction | H|. Since there is nothing to prove if К = H we may
suppose that H has a proper normal subgroup Ho containing K. Set L(l =’ Ho n L.
Since |H. Яо| - \H0L: H()| = |L: L0|t which is a л-number by hypothesis we have
0"(H) = O’(H0). Now L0 < Ho = KL0, and К = O’(K)snHo. Stace LoeG„, the
induction hypothesis applied to Ho yields К = O’(H0) and therefore К = 0”(H) as
desired.
(6.15) Proposition. If ft and X fulfil Hypotheses 6.8, then the class M(ft, X) defined
in (6.10) is N0-closed.
Proof. Suppose that G = NN*, where N and N* are normal subgroups of G and
belong to M(ft, X). By II, 2.11 it will suffice to show that G e M(ft, X).
We set Q = Op (N) and R = OP{Q\ and star these symbols to denote the corre-
sponding subgroups of N*. Because N and N* satisfy Condition ®I3 of (6.10) and
because OP(QS) < OP(Q) = Rg, we can write
R = Bi...B,Rs and R* = BJ... B„*(R*)g,
where each Bt (respectively B*) is a normal subgroup of Q (respectively Q*) isomor-
phic with B. If t = и = 0, then G e N0(ftepGp.) =ftSpGp. s Mffi, X) by (6.1 l)(a), as
desired. Therefore suppose that t > 1. Now put Q = OP'(G), R = OP(Q) and T = Rs,
and note that Q = QQ* and R = RR* by II, 2.12. Let
W/T = ®(RT/T).
From this definition it is clear that W < G, and we now look for another descrip-
tion of W. Evidently Rg = Tn R, and since [B„ B7] < Rg by Conditions ®11 and
ЗДЗ, we have [Bj, BJ < T for 1 < iV j < t. Thus RT/T = f]Ui BfT/T is a central
product of the subgroups BfT/Т. Now В, Т/Т B,/(B, n T) = BJN^ and there-
fore LfT/T = Ф(В,Т/Т) by definition of V; consequently t/,... VtT/T < W/T.
However, (RT/T')/(U1...U,T/T) is isomorphic with a central product of the
groups Bj/Gj,..., B,/U„ and since Z(B/U) = 1, this product is in fact direct. Thus
W/(U1...U,T)) = <b((RT)/(U1...U,T)) = 1 by A, 9.4, consequently W = V'...VtT,
and it follows that
RW/W = (Bt W/W) x • • • x (B, W/W)
is isomorphic with a direct product of t copies of Y = B/V = OP(A). Since Y is directly
indecomposable and Z( Y) = 1, we know that this direct decomposition is unique up
to the order of the factors, and so G induces by conjugation a permutation on the set
{Bof direct components.
An important fact for our subsequent reasoning is the following. If g e G, then
(6 (B; W/Wf = Bj W/W if and only if Bf = Bj.
Suppose that (B.W/W)9 = BjW/W, and hence that BfW = BjW. Observing that W =
658
IX. Fitting classes—examples and properties related to injectors
ц... Ц T is a r/'-group (since T e ft s and U e ftG, £ G,.), we can deduce from
(6.14) that
В/ = 0" (B/W) = 0“ (B,W) = Bj.
Since the reverse implication is obvious, (6.p) is justified.
Now let W/T = 4>(R/T). We claim that
(6.v) №'nRT=H<
Since Ф(КТ/Т) < ®(R/T) by A, 9.2(e), we have W < Wr.RT. Because W/T is nil-
potent, we have (WnRT)IT < F(RT/T) = Ui...V,T/T. Now Ul...U,W/W is a
direct product of minimal normal subgroups UjW/W(^ VJUj) of RT/W. which in
turn is a direct product of copies of Y. Since those minimal normal subgroups
are therefore pairwise non-isomorphic as RT/lV-modules, it follows from B, 7.8 that
the sums of their G/W-orbits are simple and hence that the normal section U =
Ut... VtW/W is a semisimple Fr(G/W)-module. If S denotes a Sylow «/-subgroup
of RT/W, we know by the Frattini argument that the subgroup NOIW(S) supplements
U in G/W and hence complements U because evidently Сг(5) = 1. From the fact that
U is complemented and semisimple it follows that <D(G/IF) n U = 1 and hence
that Wn RT < W, thus justifying (6.v).
We are now ready to complete the proof of this proposition. Set V = WW*(< G),
noting that V < W by A, 9.2(e) and hence that V n RT = IF by (6.v). Further, set
К = QV/V and K* = Q* V/V. Now К is isomorphic with QT/(QT n F), and because
RT nF = W, the group 0p(K)(= RV/V) is isomorphic with RT/W. But RT/W is a
direct product of QT-invariant subgroups isomorphic with Y, and on each direct
component QT induces automorphism groups isomorphic with A. Thus К (and
likewise K*) belong to the class X defined in the statement of Proposition 6.6. It
follows from the proposition that KK* eX and, more particularly from its proof,
that fora suitable subset {B,+1,..., B,+s} of {Bf,..., B*} we can write
R(=OP’P(G)) = Bi...B,...Bl+,V,
where OP\G/V)( = QV/V) induces on each Bt V/V a group of automorphisms isomor-
phic with A, and where [B(, B(] < F for 1 < i # j < t + s. Since R = RR* =
(Bj ...B,)(Bf . ,.B*)RB(R*)B, it then follows from (6.14) that Bj...B,+s =
(Bj... BS)(B*... B*) and hence that Condition ®13 of (6.10) holds. Furthermore,
Q/C^B.C,) = (QY/yy/CQy/yiBjV/V) = A, and so Ш12 is also satisfied.
It remains to prove that [В;,В;] < gB for 1 < i A j < t + s. Let S( e Syl,(B,). Since
[R<> Bj] < F e G,-, it follows that B; normalizes BSV and therefore normalizes B, by
(6-14); thus [S;, Bj] < В7п F = Ц- by (6.v). It follows from Sylow’s theorem that S;
normalizes, and therefore centralizes, a Sylow q-subgroup of В3\ moreover, since S, is
a q-group centralizing the r-group (Ц/А,)/Ф(Ц/А,), we conclude from A, 12.7 that
S; centralizes Ц/A,. Hence we have shown that [S„ B,] < Ns< R^(R*)~, < R~„ and
since Bj is generated by its Sylow q-subgroups, we conclude that [Bj, BJ < RB < £>B
6. Dark s construction—variations
659
Therefore Condition ®ll is satisfied, and we have verified that G e MCX X) as
desired. ’
By way of illustration, we now use the above construction to investigate Fit(Sym(4)),
the Fitting class generated by the symmetric group of degree four. Here we will need
the fact that the McCann class M((l), Sym(4)) is a Fitting class, and by (6.13) and
(6.15) this will follow if we can verify that Hypotheses 6.8 are satisfied with X =
Sym(4), p = r = 2, q = 3 and ft = (1). In the notation of (6.8) we take В = Alt(4),
в = v = O2(X), the normal four group ofSym(4), and V = 1. Then certainly we have
O3'(B) = B; furthermore, the groups X/B(s Sym(3)) and B/U(^ Alt(4)) are primi-
tive, and so Conditions (a) and (b) of (6.8) are satisfied. To complete the verification
we must show that A = X/U = X fulfils Hypothesis 6.2 (iv).
(6.16) Lemma. Let At, A2 < G = A2A2 with Ai A2 = Sym(4), and assume that
Z(G) = 1. Then either At = A2 or [Л], Л2] = 1.
Proof. The only normal subgroups of A, are 1, U2, B2 and Ax itself. We consider in
turn these possibilities for At A2.
If/lj n A2 = 1, then [/lj, A2] = 1, as desired. If At n A2 = l/j, then Ut = I72,and
the Л,-action on Ut commutes with the Л2-асйоп. Since If is a simple F2/l1-module,
it follows in this case that A2/U2 is isomorphic with a subgroup of the group of units
of Endx (Ц); but this is impossible because ЕпйЛ1(Ц) is a finite field by B, 3.17,
whereas A2/U2(= Sym(3)) is non-abelian.
Next suppose that At пЛ2 = B,(S Alt(4)), in which case |С/(Л] n Л2)| = 4. How-
ever, since l/j char Alt(4) Aut(Alt(4)) and АШ(Ц) GL(2,2) = Sym(3), it follows
easily that Aut(Alt(4)) S Sym(4) and hence that G/Cg(Ax n A2) s Sym(4). But then
Cg(A ! n A2) is a normal subgroup of G of order 2 and is therefore contained in Z(G),
contradicting the hypothesis that Z(G) = 1.
The remaining possibility that At A2 = Aj yields the desired conclusion At =
Л2. □
(6.17) Lemma. S2S3 n Fit(Sym(4)) = Fit(Alt(4)).
Proof. Let 80 = s„(Sym(4)), and for i > 1 set ft, = №>, where Ло denotes the
class map of forming normal products; thus Л0Х consists of all groups which are
products of normal X-groups. Since the class ft = Ui=o Si *s evidently s„-closed and
closed under forming normal products, it is clear that ft = Fit(Sym(4)).
Since obviously Alt(4) e ftf for all i > 0, to prove the lemma it will suffice to show
that
(6.e> е2е3пЭ^Н1(АИ(4))
for each i > 0. We remark that Fit(Alt(4)) contains all nilpotent {2, 3}-groups by (1.7)
and (1.9). Moreover, <52<53 n g0 = (Z2, Z2 x Z2, Alt(4)) <= Fit(Alt(4)), and we have
a starting point for a proof of (6.«) by induction on i Let i> a and suppose
inductively that <52<S3 r> ftf-i S Fit(Alt(4)). Let HeS2 3^ft,-- en sn
660
IX. Fitting classes—examples and properties related to injectors
A.'1K2...X„ where each K, is a normal g,.,-subgroup of K. Thus HsnA.SjSj,
and since Ke32S3S2, it is straightforward to deduce that =
F(K) fL'=i (К,)г2гэ e n0(«2 8.-1» S Fit(Alt(4)). Hence H e Fit(Alt(4)),
and the induction step is complete. □
We can now prove McCann’s characterization of Fit(Sym(4)) (see Theorem 8.1 of
Bryce. Cossey and Ormerod [I]).
(6.18) Theorem (McCann [1]). A group G belongs to the Fitting class generated by
Sym(4) if and only if G = MN, where M, N < G and
(a) M e Fit(Alt(4)), and
(b) O2(N) is either 1 or a direct product of subgroups У,,..., Y„ each normal in N
and isomorphic with Alt(4), such that N/CN(Yf S Sym(4).
Proof If N satisfies Condition (b) of the theorem with t > 1, then evidently
СЯ(У, x x X)ri(Fi x " x X) = 1. Thus СЛ-(У, x ••• x X) is a 2-group, and N
is the product of O2(N} with a subnormal subdirect product of copies of Sym(4). Thus
groups G having the stated form of the theorem belong to Fit(Sym(4)).
Conversely, let G e Fit(Sym(4)). Then obviously G e S2S3S2 and, in particular,
G = O2(G)O3(G). Let M = O2(G). Then M e e2S3 n Fit(Sym(4)) = Fit(Alt(4)) by
Lemma 6.17. Let N — O3(G). Since Sym(4) belongs to the Fitting class
531 = M((l), Sym(4)), it follows that N e 531, and because N = O2 (TV), it follows from
the definition of 531 that N satisfies Conditions 5311, 5312 and 5313 of Definition 6.10
with 8 = (1)- But these conditions evidently imply that N fulfils Condition (b) of this
theorem, and so G has the stated form. □
Theorem 6.18 reduces the problem of describing Fit(Sym(4)) to the apparently easier
one of describing Fit(Alt(4)). However, finding an illuminating description of a Fitting
class generated by a primitive metanilpotent group appears to be a difficult task, and
even Fit(Sym(3)) has defied analysis so far. The next variation which we will describe
yields, nevertheless, an effective characterization of certain Fitting classes generated
by metanilpotent groups of an extremely special form.
Variation II
Before launching into the details of this construction, we prepare the way with some
technical results about central products.
(6.19) Lemma. Let X = PlP2...Ps, where [Pj, = 1 for all I < i j < s. Assume
that X is nilpotent and that for some n > 2 the group T = K„(X) is abelian. Further
assume that for some integer q > 3 the automorphism group of X contains a subgroup
A which transitively permutes the set of subgroups Pt T,... ,PqTof X. Then K'm([A', Л])
contains the subgroup Кт(РЗ... K„(Pq) for all m > 2.
Proof. Let {i, j, k} be a 3-element subset of {1, 2,..., q}, and let x, у e Pt. Because A
is transitive, we can find elements a, b e A such that x“ e TPt and yb e TPk and can
write x ° — uxj and у b = wyt with u, we T, x}e P2 and yk e Pt. Set g — XjX and
6. Dark s construction—variations ^61
h = yty, and denote K2([X, 4]) by R. Then R contains the element [x»x y'4] =
[uy, wh], and by the standard commutator laws (see A, 7.2) we have
[ug, wh] = [u, h]»[s, w]’[0, w])1'.
Since ue T and T = K^P^^.K^P,) by A, 19.8(a), we have [u, yt] e K„+1(Pt) and
[u, y] e K,ltl(P,), and because K„+fPt) < X, it follows that
[u, h] = [u, y] [u, yj e K^lPdK^fPJ.
Denoting Pt... P, by L, we conclude that [u, /1] e K„+1(L) < K3(L} because n > 2
by hypothesis. Similarly [fl, w] e K3(L). Now [u, w] e Г = 1 by hypothesis, and
K3(L) < A. Therefore
[x"x, y~by] = [s, /1] (mod K3(L)).
But [fl, h] = [x7x, yty] = [x, y], a typical element of P-. Hence RK}(L) contains P[
for i = 1,..., q and therefore contains L'. Thus K2(L) = L' n RK}(L] <(Lr\ R]Ki(L],
and since L/(L n R) is nilpotent, it follows easily that K2(L) < L n R. Since K2(L) =
K2(P1)... K2(PS) by A, 19.8(a), we have proved the lemma for m = 2.
We now prove the general case by induction on m, assuming that КД[Х, 4]) >
Ki(L) for I - 2, ..., m — 1. Let t e Rmt(P:l < Km_,([.X; 4]). Then with x, a, etc.,
defined as in the previous paragraph and R now denoting K„([X, 4]), we see that
К contains [t, x-x] = [t, ug] = [1, g] [1, u]e. Since и e K„(PI)... K„(Pt), by A, 7.8(b)
we have [t, u] e R.m_1+n(Pj), which is contained in Km+1(L) since n > 2. Con-
sequently [t, x] = [t, g] = [t. x“°x](mod Km+1(L)), and it follows that RKm+i(L)
contains [Km JPJ, PJ for i = 1, ..., q. Thus Km(L) < RKm+1(L), whence Km(L) <
(L n R)R.m+1(L), and since L/(L n R) is nilpotent, it follows as before that R contains
K„(L) = R'^IP,). .A.'„(Pe). Thus the induction step is complete, and the lemma is
proved. □
We recall that an automorphism of a group G is called central if it commutes with
all inner automorphisms of G. Thus CAu,(G)(Inn(G)) is the group of all central auto-
morphisms of G and is obviously normal in Aut(G). It is easy to verify that a e Aut(G)
is central if and only if [G, a J < Z(G). The following result appears in Huppert [5]
as Satz I, 12.5(a) (see Satz I, 12.3 for the proof).
(6.20) Theorem. Let G = G, x x Gr = Ht x x Hs he two direct decomposi-
tions of a group G into products of non-trivial indecomposable groups Gt and Hj. Then
r = s andG has a central automorphism 0 such that, with suitable numbering of the
components, Gf = H,.
(6.21) Lemma. Let p be a prime, and let P be a directly indecomposable p-group such
that Z(P) < P'. Let
X = Pi x • x Ps
662
IX. Fitting classes—examples and properties related to injectors
where Pt = P for i = 1, s. If A is a p'-subgroup of Aut(X), then there exists
a central automorphism 0 of X such that the action of A induces a permutation
on the set {P°:i= 1, s}; in particular, A permutes the subgroups in the set
{PiZ(X):i=i,...,s}.
Proof Let X = {(x,,..., x,): x, e P}. If a e Aut(P), let a, be the automorphism of X
defined by
(Xj, ..., Xs)“< = (Xj ,..., xf_!, x?, xi+1,..., xs),
and set At = {af: a e Aut(P)}. Thus clearly A, is a subgroup of Aut(X) isomorphic
with Aut(P). If a e Sym(s), let a* denote the automorphism of X defined by
(A1,...,XS)‘’* = (X1O,...,XSO),
and set X = {a*: a e Sym(s)}. Then X is evidently a subgroup of Aut(X) isomorphic
with Sym(s). Finally, let C denote the normal subgroup of Aut(X) comprising all
central automorphisms of X. Then it follows easily from Theorem 6.20 that
Aut(X) = CL,
where L = (A, x • • • x A,) X (= Aut(P) Qj„al X). Since the hypothesis that Z(P) < P’
is easily seen to imply that Z(X) < X' < Ф(Х), it follows that [X, С] < Ф(Х) and
hence that C is a p-group by A, 12.7. Under the well-known isomorphism from LC/C
(= Aut(X)/C)) to L/(L n C) denote the image of AC/C by B/(L n C), a p'-group by
hypothesis. Theorem A, 11.3 of Schur and Zassenhaus ensures the existence of a
complement Ao to L n C in В and clearly A0C = AC. By the same theorem A = (Л0)е
for some ве C, and since вАв~1 is contained in L, whose action permutes the
groups in the set {fj}’=1, it follows that A itself permutes the subgroups {/? }f=1 of X.
□
The construction of Variation II will follow the familiar pattern: we choose a group
with a carefully specified structure, identify a class I comprising subnormal subdirect
products of this group, and then define the candidate for our Fitting class to consist
of all groups whose “key sections” belong to I. Let us first specify the structural
restrictions to be imposed on the chosen group.
(6.22) Hypotheses. Let p and q be distinct primes with q 2. Let P be a p-group,
and let Q e Syl,(Aut(P)). Assume that each of the following conditions holds:
(a) P/Z(P) is directly indecomposable;
(b) P has class c > 3;
(c) Zt(P) = КС+1_((В) for i = 1,2;
(d) iei = q-,
(e) Q acts fixed-point-freely on P/Z(P)-,
(f) [Z(P), Q] = 1.
6. Dark’s construction—variations 663
We will discuss the existence of p-groups P satisfying Hypotheses 6.22 later The
following elementary observation about such groups will be useful meanwhile.
(6.23) Remark. Let P be a p-group satisfying (6.22), and let Sbea q-group of operators
for P. If [P, 5] # 1, then [P, S] = P and |S/Cs(P)| = q. In any case, [Z(P), S] = 1.
Proof. LetK = CS(P), the kernel ofthe homomorphism from S to Aut(P). If [P, S]
1, then К < S, and by (6.22)(d) we have |S: K| = q. Let .s e S\K, and let a be the
automorphism (of order q) induced by s on P. Since Z(P) = KJP) < K2(P) = F by
(6.22)(b) and (c), and since <a> is conjugate in Aut(P) to the distinguished Sylow
q-subgroup Q of Aut(P) mentioned in (6.22), it follows from (6.22)(e) that a acts
fixed-point-freely on P/P'. Hence P'[P, a] = P, and consequently [P, s] = [P, a] =
P. The final assertion follows from (6.22)(f). □
The class I, which controls the key section in this variation, consists of subnormal
“subdirect” subgroups of central products of copies of the group [P]Q. It is more
conveniently formulated as follows:
(6.24) Definition. Let P be a group satisfying Hypotheses 6.22. The class I consists
of groups of order 1 together with all groups of the form
X = KA,
where
(i) A e Syl,(X),
(ii) К = 0p(X) is a central product of Л-invariant subgroups Pn ..., Ps, each
isomorphic with P and satisfying [Pf, Л] 1, and
(iii) 0q(X) = 1.
Whenever 1 ± X e I and we write X = KA, it is to be understood that the subgroups
К and A have the meanings described in this definition.
To prove that the Fitting class of this variation is s„-closed, we will formulate it in
the terms and notation of Section 5. Set it = q' and r = p, and observe that Hypo-
theses 5.5 are fulfilled with
(6.7Г)
У = P and .с/ = {Q" Inn(P)/Inn(P): a e Aut(P)}.
(Evidently .</ и {1} is the Fitting set of all q-subgroups of Aut(P)/Inn(P).) It follows
from (6.23) that if 1 X e I, then О’(Л') = X and hence by Hypothesis 6.24 (in) that
X e Q”. It also follows from this hypothesis that Сл(Р,... Ps) = C\ (K) - Z(K), and
it is clear that the class A defined in (6.23) coincides with the class X”(Y, sf) defined
in (5.7). Furthermore, the pair (У, .</) also satisfies Hypotheses 5.9. Condition s
follows from (6.23); Condition s.II is obvious; Condition s III is implied by(6.22)(b)
and (c); Condition s.IV(o) is a consequence of (6.23) and Condition s„IV(6) ho
because p/q.
664
IX. Fitting classes—examples and properties related to injectors
Let k(G) denote the key section O’’(G/Op.(G)) of a group G, and let I be the class
defined in (6.24). Then the candidate for the Fitting class of this variation is
(6.p) 8(P) = {G e (E: k(G) e X}.
Thus g(P) is the class £><(!) defined in (5.1 )(b) and subsequently denoted by D^(T, j/),
where Y and .</ are as in (6.л). Since the pair (У, j/) satisfies (5.9), Corollary 5.11
applies and shows that 8(P) is s,-closed.
(6.25) Theorem (Hawkes [ 12]). If P is a p-group satisfying Hypotheses 6.22, then the
class g(P) defined in (6.p) is a Fitting class.
Proof In view of the preceding discussion, it remains to show that g(P) is n„-closed,
and by (5.4) (b) it will therefore suffice to prove the following. If
G = XX* e Elf.
where X and X* are normal I-subgroups of G, then Gel. (Here, and subsequently,
we use the fact, mentioned earlier, that I £ £2’..) By definition of the class I and by
(6.23) we can therefore write X = KA, where К = P,... Ps, a central product of
4-invariant copies of P satisfying [Pf, 4] = Pit and where A e Syl4(A"). We adopt the
convention that for every statement about X there is a corresponding “starred”
version for X*. Let S e Syl,(G). Then S n X e Syl,(A') and S = (S n X)(S n Л-*). Since
the definition of X = KA e I does not depend on which conjugate of A is used,
we may therefore suppose that S = A A *. Since \A: Сл(Р()| = q and C4(Pf) <
0,(Л] = 1, it is clear that A (and likewise 4*) is elementary abelian. We now divide
the proof into convenient steps.
Step 1: Here we aim to show that [Z(K), S] = 1. Because Z(K) = Z(Pt)... Z(P„), we
have [Z(K), 4] = 1 by (6.23). Let 1 # a e A* and i e {1,..., s); then it will suffice to
prove that [Z(Pt), d] = 1, and without loss of generality we take i = 1. Write Z =
Z(K), and let v denote the natural homomorphism from К to K/Z. By A, 19.7 we have
v(K) = v(P1) x • • • x v(Ps),
where v(P;) PJZIP^ and is therefore indecomposable by Hypothesis 6.22(a) for
i = L..., s. Therefore by (6.21) the automorphism a induced in v(K) by conjugation
by a permutes the subgroups of the set {vfPJZfvfK)): i = 1,.... s}. Since Z(v(Pf)) =
Z2(Pf)Z/Z, we have
Z(v(K)) = Ц Z(v(P,)) = ([J Z2(P,))z/Z = Z2(K)Z/Z
by A, 19.8(b). Put T = Z2(K); then T = Kc-t(K) by A, 19.8(a) and Hypothesis 6.22(c),
and therefore appealing to A, 7.8(b) we conclude that [T, T] < K2c_2(K) <
A;+i(K) = 1 since each P, has class c > 3. Thus, to summarize, we have so far shown
6. Dark s construction—variations
665
that opermutes ^dements {P, T: i = 1,..., s) by conjugation and that the subgroup
First suppose that (Ptiy * P,T, and designate the <o>-conjugates of PtT by
{Р1Г, P2T,PqT}, renumbering the P-s if necessary. We now apply Lemma 6.19
with К in place of X, <a> instead of Л, and with n = c — 1, and deduce that
KC([K, <«>]) contains Kc(Pi)...Kc(Pq), which equals Z^)... Z(Pq) by Hypothesis
6.22(c). But [К, <o>] < [К, К*] < к* (Since К* = 0„(Х*) char X* < G), and there-
fore Z(Pt) < КДК*) = Z(K*) by (6.22) (c) again. But A* centralizes Z(A*) by (6.23),
and so in this case we certainly have the desired conclusion that [Z(P,). a] = 1.
Now suppose that (Pt T)° = P,T Since P,T = Р,(П(/.22(Р;)), a central product,
and since each Z2(P;) is abelian, it follows from A, 19.8 that for i = 1,..., s
(6.<r)
(i) ZdlT) = Z(P1)(nz1(PJ)j, and
(ii) К„(Р:Т) = К„(Р;) for и >2.
Since the element a normalizes P, T, it leaves invariant Z(P} T); furthermore, it also
normalizes A because A < S. Hence [Z(P, T), Л] is <a>-invariant. Since A induces
on Pj a non-trivial group of automorphisms, it follows from (6.22)(e) that A acts
fixed-point-freely on Z2(P;)/Z(Pj), and since Z2(Pf) is abelian and A centralizes Z(P;),
by A, 12.5 we have
(6.r)
Z2(P() = Л x Z(Pj),
where Jf = [Z2(Pf), A] for i = 1, ..., s. Denote the Л-invariant normal subgroup
J,... Jj-jJj+j... Js of К by Ih and note that [/,, Л] = It by A, 12.4(b). Using this
notation, we can now rewrite Equation 6.<r(i) thus
Z(PjT) = A x Z,
and it is then clear that [Z(PtT), A] = which is therefore Л<а>-п«апапГ Since
[Р;, Л] = Pj and [T, Л] = Jjfj, we have [Р,Т, Л] = PM = P x 7f. Hence P,7, is
also Л (a>-invariant, and it follows that Л(а> is a ц-group of operators for the
group PJi/Ii = Pt. By (6.23) we conclude that Л<о> acts trivially on Z(P,7,/71) =
ZfP,)/,//,, and so, in particular, [Z(P,), < I,. Then from (6.22)(c) and (6.cr)(ii)
we have
Z(P,)“ = KC(P, T)° = KC((P, TH = f^P, T) = Z(P,),
and finally therefore [Z(P,), <«>] < Z(P,)n 7, = 1, as desired. Since a was an arbi-
trary element of Л’, we have shown that [Z, A*] = 1, and Step 1 is complete. We
also record the fact that, by symmetry, [Z(K*), S] = 1.
Z char К < G, we have Cc(Z) < G.
Cc(Z). But X* is generated by the
Step 2: Next we prove that ZZ* < Z(G). Since
Therefore from Step 1 we infer that <(Л*)С> <
666 IX. Fitting classes—examples and properties related to injectors
^♦-conjugates of its Sylow q-subgroup A* because C'iX*) = X*, and consequently
X* < CG(Z). But A also centralizes Z by (6.23) and hence so does X. Therefore Cc(Z)
contains XX* = G. By a symmetrical argument we have Z* < Z(G).
Step 3: The object of this step is to show that
P,T< G
for i = 1,..., s. As in Step 1, nothing is lost if we confine our attention to the case
i = 1. Let /> # 1 denote a p'-element of X* ( = Х*Л*) and set В = {by. As in Step 1,
for suitable reordering of the set {Р,};ч the conjugates of P, T under the action В
may be taken to have the form
{Р1Т,Р2Т,...,Р,.Т}.
We suppose that r > 1 (which evidently implies that r = q > 3) and derive a con-
tradiction. Since Equation 6.cr(ii) implies that E = X2(PfT) an£l Z(P,) — ХС(Р(Т) and
hence that each of these subgroups is characteristic in PtT, it follows that the sets of
subgroups {Р/}[=1 and {Z(P()}[=1 form B-orbits under conjugation. Thus, if Ko
denotes the central product P2 P2... P„ the subgroup Zo = Z(X0) = Z(Pl)... Z(Pr) is
В-invariant. By A, 19.7 the quotient Ko/Z^ has a direct decomposition
D = (PiZq/Zq) x ••• x (PrZ0/Z0),
and its derived group D' = K'o/Zo has a direct decomposition whose components
f^'Z0/Z0 (s/7/Z(Pf)) are permuted regularly by В and are non-trivial because c > 3.
Hence В centralizes a diagonal subgroup of O'. Let Co = CKb(B). By Step 2 we know
that Co contains Zo, and by A, 12.1 the diagonal subgroup of D' which В centralizes
is covered by Co; hence, in particular Co > Zo.
We now make a small digression to prove that for an arbitrary q-subgroup Q of
X* the following inclusion holds
(M C[K.,cl(C) < Z*.
Suppose that the groups {: i = 1,..., s* } have been numbered so that [P*, Q] 1
if and only if i e {1,..., t}. Recall that P* < X* and therefore [P*, Q] = P* by (6.23)
for i = 1,.... t. Hence [A.*, Q] = P*.. P,*, and as Q acts fixed-point-freely on Pj/Z(P()
for i = 1,..., t and centralizes Z(Pf), we conclude that C(K.,01(Q) = Z(Pt)... Z(Pt) <
Z*, as claimed. Note that this argument yields in particular the fact that СК.(Л*) =
Z*
Returning to the main thread of the proof of Step 3, we note that Lemma 6.19
applies for the group В permuting the groups {PtT,PrT} and deduce that Xf, is
contained in [К, В]. But [К, В] < K* and by A, 12.4(b) it follows that [К, B] =
[К, В, В] < [X*, В]. Hence Co < C[(£. B](B) < Z* by (6.u), and since [Z*, Л] = 1 by
Step 1, we conclude that Co < СКо(Л) = Zo. Since this contradicts the earlier conclu-
sion that Co > Zo, we conclude that r = 1 and hence that P, T is normalized by each
6. Dark s construction—variations
667
p'-element of X*. But X* is generated by its p'-elements, and therefore P, T < XX* =
G, as required.
Step 4: Our goal here is to prove that the Sylow q-subgroups of G are elementary
abelian. Recall that AA* = Se SyljG). It follows from Step 3. Equation 6.<r(ii),and
Hypothesis 6.22(c) that Z2(Pf) is normalized by S. Since [Z(P(), S] = 1 by Step 1, the
term J, in Equation 6.r equals [Z2(P,), S], and, in particular, J, and Z(P£) (= Cz (F ,(S))
are both S-invariant subgroups. 1 '
Let i e {1,..., s}, and recall the notation Ц Jj. Since S normalizes PtT by
Step 3 and also normalizes Л, it normalizes [P/L Л] = and therefore acts as a
q-group of operators on PJ,//,, as in Step 1. Let Q = QP^j/Jj). Since [P(, Л] = P(,
the S-action on PJ./L is non-trivial, and therefore |S/C,| = q by (6.22)(d). Thus, setting
c=ik
i=l
we see that S/C is elementary abelian. However, by A, 12.4(b) we have
[P;T, CJ = [PjT, C„ CJ < [PJ;, CJ < Ii < T.
Hence [К, С] = [(P, T)...(P,7 ), С] < T, and so C centralizes К/Ф( К (because T <
К' < Ф(К). Therefore [К, C] = 1 by A, 12.7, and we conclude that C n C* <
CS(KK*) < Oq(G). But by hypothesis O,(G) = 1, and it follows that S (~S/(C n C*))
is elementary abelian.
Step 5: We know by hypothesis that К is a central product of Л-invariant copies of
P; in this step we show that “Л-invariant” here can be replaced by “S-invariant”.
Before embarking on the proof of this assertion, we introduce some more notation.
First we put J = JlJ2.-.Js and note that J = J, x Ц for i = 1, ..., s. Recall that
q = Cs(Pj7j/7j) for i = 1,..., s, and by suitably indexing the subgroups Pj, suppose
that {Cj, C2,..., Cj} is the set of distinct subgroups in the list C,,...,CS. For 1 < i < I,
let .9J denote the set {j: 1 < j < s and Cj = C,}, and then put
£j = П V’
jG У,
and
Hj= п л-
Since L < R and Jt n Z(K) = 1, it follows that E, = (Па ty x Hi If к £ .S'?, the
definitions of С, and ,9} imply that l(PkIk/Ik), CJ # 1; from Hypotheses 6.22(d) and
(e) we then conclude that Cf induces on PkIk/Ik an automorphism of order q, which
acts fixed-point-freely on its central quotient group and, in particular, on the section
JkZ(Pk)h/Z(Pk}Ik, which is operator-isomorphic with Jk. Since Jk is S-invanant, we
therefore have [Jt, C;] = Jk for к (t .9} and hence [J, Q] = = M, CJ; conse-
quently CJ(1(C,) = 1.
66g IX. Fitting classes—examples and properties related to injectors
Next set Ff = C£,(G)> and observe that Ff n = 1. By a familiar argument we
have [Ff/;, CJ = [f,/., G. G] < [A. G] СЛ G] = Ht. Therefore G centralizes EJH^
and we can deduce from A, 12.1 that £, = F,//,. Thus we have located an S-invariant
complement F, to H, in £,. Let j e .9?. Since C, centralizes EJH^ it certainly normalizes
Р НЬ which is also clearly Л-invariant. Now C, is a normal maximal subgroup of S
and does not contain A; therefore S = AQ and S normalizes PjHj. Let
Q^F^PjH, (=CPjJf|(q)).
Then Qj is evidently an S-invariant complement to H, in PjHh and so Qt = Pj = P. It
also follows easily that
я= П e,
je V,
Furthermore, this product is central because H; < Z(£,) and therefore for j,ke^
and j # к we have [Q7, < [PjHh PkH^ = 1. This means that QjHt = Qt x H, =
Pj x Ht, hence that Q] = (QjHf)' = Р/, and consequently that F- = P-.
Now let 1 < m < n < I, and set W = FmFn. We claim that W is a subgroup of K.
Certainly the subset
п n nV
XjG^ /\кеУ„ J
is a subgroup. Moreover, (£„£„)' = (П;е ym ^)(П*е s>-„П) = G.G is a normal sub-
group of K. Since (£m£j)/(F^F^) is therefore abelian, the subgroups F„F^ and F^F„
are normal and their product W is indeed a subgroup and obviously admits S. Since
[F(, Ct] = 1 if i = к and equals F, otherwise, we have Fm = [F„F„, C„] and hence
Fm < B'S. From this it follows that Cws(Fm) is a normal subgroup of B'S, which
contains C„ and therefore contains <C£">. But F„ = [F„, C„] < <€£"> and so
[Fm, F„] = 1. Thus we have shown that К = П!=1 's a central product of its
subgroups F],..., F(, and we can therefore conclude that
к = Д Qj
is also a central product of its subgroups Qt,..., Qs, each of which is an S-invariant
copy of P. This completes Step 5.
Step 6: In this final step, we will complete the proof by showing that Gel. Without
loss of generality we may now suppose that the subgroups C, have been labelled so
that A* < C,n-aCt and A* £ C, for j = k+ 1, .... I. Let U = FlF2...Fk and
= Fk+i ••• Ff. Since [Fj, C,] = 1 fori = 1.1, we have U < СК(Л*) and U r\ K* <
Gt-И*) = Z*. Moreover, from the fact that [F;, Л*] = Fj for j = к + 1, .... I, it
follows that
V < [К, Л*] < К n 0„(G) r.X* = Kr.K*,
fy. Dark s construction—variations
669
and hence that К - UV < U(K n X*) < K; consequently К = U(K n K*). Since
^UnK^K*W^Ur^*)^Z*V, from Step 2 and the fact that
[I/, V] = 1 we conclude that [I/, К n X*] = 1.
Similarly K* may be expressed as a central product U*(K n X*), where U*.
like U, is a central product of S-invariant copies of P. Evidently [I/ I/*] <
[К, X*] < X. гу К* and [[17, l/*], К су К*] = 1. Hence [U, L'*] < Z(X A K‘) <;
Z(U(K n X*)) = Z(X) < Z(G) by Step 2. Since [I/, A] = U and [L'*, A] = 1, by the
Three Subgroups Lemma (see A, 7.6) we have
[17, I/*] = [I/, Л, I/*] < [A, U*. U] [U*, U, A] = 1.
Now 0„(G) (=KK*) is a Sylow p-subgroup of G and equals K(KcyK*)U* =
KU* — UVU*. Since V < К су К* < CK.(U*), it follows that 0p(G) is a central
product of its subgroups U, V and U*, each of which is a central product of
S-invariant copies Q and P satisfying [Q, S] = Q. Therefore O„(G)S = Gel □
If P is a non-identity p-group, we know that Fit(P) = The following theorem uses
Variation II to characterize the Fitting class generated by a certain non-nilpotent
group.
(6.26) Theorem (Hawkes [12]; see also Corollary 8.8 of Brison [4]). Let P be a
p-group satisfying Hypotheses 6.22, and let 8(F) denote the associated class defined in
(6.p). Further, let S denote the semidirect product [F]Q, where Q e Syl,(Aut(F)), and set
8 = ePe.n8(p).
Then
(a) g = Fit(S),
(b) the class 8/91 = (G/F(G): G e 8) is the class of elementary abelian q-groups,
(c) the class 8” = (G51: G e 8)is contained in cy 91t, and
(d) the class © = (G: G = A x В, A e 9l{p ,r, В e 8) is a Fitting class lying strictly
between 91 and 9191.
Proof (a) It is clear from (6.25) that 8 is a Fitting class, and since 8 obviously
contains S, we have
Fit(S) £ 8-
To prove the reverse inclusion, let G e 8> ahd note that by (1.7) and (1.9) the class
Fit(S) contains all nilpotent {p, q}-groups. Since G e (=„<=,, we have G = O₽(G)O«(G),
where O“(G) is a p-group and therefore belongs to Fit(S). Write
R = O’(G),
and observe that R = O'(G) e 8- To show that G e Fit(S), it will evidently suffice to
show that R e Fit(S). By definition of 8. the group R/Oq(R) belongs to the class I
defined in (6.24) and is therefore a normal subgroup of a central product of copies of
670 IX. Fitting classes -examples and properties related to injectors
S. Hence R/0„(R) e Fit(S). Since R/0„(R) e S, £ Fit(S), by the Quasi-R0 Lemma 1.13
we conclude that R e Fit(S), and it follows that g = Fit(S).
(b) Since OP(R) e Sylp(R), it follows that
(M) Op(R/Oq(R)) = = F(R)/Oq(R),
and because R/Oq(R) e X, it follows that R/F(R) is an elementary abelian q-group;
but O’(G)O,(R) < F(G), and consequently G/F(G) is an elementary abelian q-group.
If D is a direct product of r copies of S, then D e g and D/F(D] is an elementary
abelian q-group of rank r. Assertion (b) is now clear.
(c) Again let G e g and R = OP(G). From (6.<5) and the fact that R/Oq(R) e X, it
follows that Op(R) is a central product of copies of P, which is a p-group of class c.
Since G91 = OP(G) n O’(G) < R n Op(G) < OP(R), we see that G91 is also a p-group of
class at most c, and Assertion (c) is justified.
(d) If gj and g2 are Fitting classes of disjoint characteristics, it is straightforward
to verify that
g, x g2 = (G: G = N, x N2 with N, e gb i = 1, 2)
is also a Fitting class. Thus S = 92(pe )• x g is a Fitting class of full characteristic
and so contains 91. It is clear from Assertion (b) that (5 £ 9191(g) and hence that
9i c (5 c 9191. □
(6.27) Example. We now turn to the question whether any groups P satisfying
Hypotheses 6.22 actually exist. In fact, it seems likely that such groups can be found
for all pairs of odd primes, but we will discuss only one such example (with p = 7
and q = 19); it is analysed in detail by Rex Dark [3] in a different context, and we
will be content to quote his conclusions without proof. The group Dark considers is
the free metabelian 7-group of exponent 7 and class 4 on three generators. Let us call
this group D (although Dark calls it P). The Frattini quotient D/®(D) = D/D' has
order 73, and D admits GL(3, 7) as a group of automorphisms. Let H = £(3/19), the
non-abelian group of order 57. By B, 12.10 the group H is an irreducible subgroup
of GL(3, 7), and so we can regard H as a subgroup of Aut(D), acting faithfully and
irreducibly on D/D'. Dark’s commutator calculations, via the associated Lie algebra,
reveal that H also acts faithfully and irreducibly on the second term D'/K3(D) of the
descending central series of D, and that if Q = H'( ^ Z19)and T = K3(D)/K4(D), then
|7j = 78 and the submodule To = CT(Q) of T (viewed as an F7H-module) has order
72. Since T < Z(D/Kq(D)), we can also regard T as a module for the semidirect
product with D in its kernel and can apply Maschke’s Theorem A, 11.4 to
deduce that T has a complementary submodule, Tt say, to To. Thus, if A is a subgroup
of D containing K4(D) and satisfying
7j < N/K^D) < T,
then the group P = D/N is a 7-group of class 3 which admits Q as an automorphism
group acting irreducibly on P/P and P'/K3(P) and trivially on K3(P)(S T/N * 1).
6. Dark’s construction—variations
671
We claim that Z(P) _ K3(P): for, if not, the H-invariant group Z(P)/K3(P) would
have a non-tnvial projection onto P/P or P?K3(P), both of which are H-simple; in
either case, this would force the conclusion that P < Z(P), against the fact that P
has class 3. Thus Z(P) = K3(P), and by a similar argument Z2(P) = P. If P/Z(P) were
directly decomposable, one of the direct factors would be abelian and hence contained
in Z2(P)/Z(P), then the other direct factor would cover P/P' and would then coincide
with P/Z(P). Thus we have shown that P satisfies Hypotheses (a), (b), and (c) of (6.22).
Let A = Aut(P). By A, 12.7 the group СЛ(Р/Р) is a 7-group, and since А/СЛ(Р/Р)
is isomorphic with a subgroup of |GL(3, 7)|, which has order 26 • 314-19, it follows
that Q is a Sylow 19-subgroup of Aut(P). Therefore Hypotheses (d), (e) and (f) of(6.22)
are also satisfied.
For later reference we need some extra information about Aut(P) which depends
on the choice of the subgroup N above. Dark’s calculations also show that the
H-module To, which has Q in its kernel, is a sum of two non-isomorphic 1-dimensional
submodules. Thus if N corresponds to one of these, the group P = D/N admits H as
a group of automorphisms; on the other hand, we can also choose N/K4(D) to be a
7-dimensional subspace which is not H-invariant, and then P = D/N admits Q, but
not H, as a group of automorphisms. It follows from Huppert [5], II, Satz 7.3(a) in
this case that HAut(P1(Q) = CAut(P1(6) and hence that the subgroup
{Q; Aut(P)} = <[g, a]: a e Aut(P), g and [g, a] e Q)
of Aut(P) is trivial. Thus we obtain the following observations which are due to Brison
[4].
Case 1: If the above subgroup N is chosen to be H-invariant, then {Q; Aut(P)} = Q.
Case 2: The subgroup N can be chosen not to be H-invariant, and in this case
{Q; Aut(P)} = 1.
The existence of a group P satisfying Hypotheses 6.22 with c — 3 shows, according
to Theorem 6.26, that there exists a Fitting class g contained in SpS,, namely
g = Fit(PQ), satisfying
g/9I £ 9I(q) and g91 £ 9I3.
This is in striking contrast to the fact that, if g = Fit(£(q/p)), then g/91 contains all
у-groups and F91 contains groups of arbitrarily large nilpotency class; this will be
proved in Chapter XI, Theorem 3.3. The question of the existence of Fitting classes
g satisfying g91 £ 912 has been settled positively by Rex Dark. In unpublished work,
he has shown the following:
Let p and g be distinct primes with g odd and (g, p - 1) = 1- Let ф denote the da s
consisting of groups of order 1, together with all central products ex^aspec al
p-groups (of exponent p if p is odd). Let X be the class comprts.ng all groups X of the
form X = KH, where
672
IX. Fitting classes—examples and properties related to injectors
(i) К = 0„(X) e 'Э,
(ii) H e Syl,(X), and
(iii) [К, H] = К and [К', H] = 1.
Then the class ® = (G: O’(G) e X) is a Fitting class.
It follows that (®n6,£,)’1 £ 9l2. At the time of writing the following question
seems to be still unresolved.
Open Question. Do there exist primes p and q and a non-nilpotent group T e
such that the Fitting class S = Fit(T) satisfies
1/91 £ 9I(q) and X91 £ 9l2?
Another contribution to the study of Fitting classes generated by a finite group (or
equivalently, by finitely many finite groups) has been made by Bryce in [2]. There he
considers a class U of groups which, for a fixed prime p, have the following properties:
(6-X)
(i) Each group T e 3 is p'-perfect, OP(T) is monolithic with trivial centre,
and every subnormal subgroup of T is comparable with OP(T);
(ii) A p'-perfect subnormal subgroup of a T-group belongs to 3;
(iii) IfO'’-2(T1)isisomorphicwithanon-trivialsectionofO',,2(T2)for7],
T2 e S.then 7j = T2;
(iv) If Я = 7j T2, a product of normal 3-subgroups II and T2, then either
[Ti,T2] = lorHeX
(These conditions are evidently fulfilled when I = ([ У] <a>) and У is a p'-group
satisfying Hypotheses 6.2.)
If g is a Fitting class satisfying = § and 6 is a Fitting formation such that
Sp.6 = 6, Bryce defines a key section x(G) of a group G by
x(G) = (G/Gg)® (sG®/(G®)g),
and goes on to show that the class Dg(S) comprising all finite groups G such that
x(G) is a p'-perfect subnormal subgroup of a direct product of 3-groups is a Fitting
class. He uses this result to describe the groups in Fit(I) in a way analogous to that
of McCann’s Theorem 6.18. Furthermore, if Ro denotes the smallest Fitting subclass
of FitfX) containing the class (PP(T): Tel) and having the same characteristic as
Fit(3), Bryce shows that, when 2 is finite, there are only finitely many Fitting classes
lying in the interval between g0 and Fit(3), thus providing the first evidence support-
ing the possibility of dualizing the theorems of Bryant, Bryce and Hartley, which
show that the (saturated) formation generated by a finite soluble group contains only
finitely many (saturated) subformations (see Theorem VII, 1.6 and Corollary VII, 1.7).
In the cited work, Bryce discusses two concrete example of classes 3 satisfying
Conditions 6/. The first consists of a set of subgroups ЛВ0С,(1 < i < и) of an
extended affine group Г(гр") (see B, 12.9), where AB0 is the primitive group E(q/r)
and C, £ Zpl. The second consists of 2'-perfect subnormal subgroups of a group of
6. Dark s construction—variations 673
the form MES, where S is a generalized quaternion group of order 2”+1 acting
faithfully on an extraspecial g-group £ of order q3 and centralizing Z(£), and where
M is a faithful simple ES module over Fr. Here the primes q and r must satisfy
q = 3 (mod 4), 2" || q + 1, r = 1 (mod q) and r = 3 (mod 4). Both of these examples can
be described in terms of a pair (У,.;/) satisfying Hypotheses 5.5.
A completely different approach to constructing Fitting classes, in particular those
of Dark type , has been developed by Pense in [3]. He generalizes the concept of
an g-Fitting pair (see Definition 2.10(c)) to that of an outer ^-Fitting pair id. A). Here,
A is a group, possibly infinite and non-abelian, and d denotes a family of maps
(dG e Hom(G, A): G e ft) with the property that for each normal embedding
v: N -> G e ft, there exists an inner automorphism a of A such that
dG о v = a о dN
(Of course, when A is abelian, this coincides with the definition of an ft-Fitting pair).
Pense extends the definition of a Fitting set S' to an infinite group by requiring it to
mean a set of finite subgroups closed under conjugation and under the usual opera-
tions of taking normal subgroups and forming finite normal products. He proves
straightforwardly that if (d, A) is an outer ft-Fitting pair and if S' is a Fitting set of
A, then
d '(JJ = (Geg:dcGe
is a Fitting class, and further that the d-1 (^(-radical of a group G e g is the inverse
image of the ^-radical of dcG.
The value of Pense’s method depends on finding actual examples of outer Fitting
pairs (d, A) and Fitting sets of A. One important example arises as follows: Let J be
a finite simple group, and let Dj(G) denote the direct product of all the J-chief factors
(see Definition IV, 4.9(c)) of some chief series of G between Ggl and Ggl, where ft]
and g2(e ft,) are given fixed Fitting classes. Embed Dj(G), viewed as an operator
domain for G, into the countable restricted direct product J1"’ of copies of J as a
“left-hand” summand, that is to say onto the first n summand of J'"’, where n is the
composition length of D/G). If A, denotes the group of those automorphisms of J'”’
which centralize all but finitely many components, the action of G by conjugation on
£>j(G) gives a homomorphism
d®-'»2: G _ Aj.
Pense proves that (d S1'®2, A J is an outer G-Fitting pair, showing on the way that the
definition of d®-'®2 is independent of the chosen chief series “up to equivalence of
outer Fitting pairs”. If J = Z„ then A, is the stable linear group Lp consisting of all
linear transformations of a countably infinite vector space F,» over F, which fix the
elements of a subspace of finite codimension, whereas if J is non-abelian, then A, has
the form Aut(J)4., S„, where S. denotes the restricted symmetric group on N.
To construct Fitting sets of Lp which yield Fitting classes like Darks, Pense
considers an irreducible subgroup U of GL(„, p) and a Fitting set ^(U) of U whose
674 IX. Fitting classes—examples and properties related to injectors
non-identity subgroups are irreducible (for example, the set of subnormal subgroups
of GL(2. 2) has this property). The Fitting set & of Lp generated by :jr(U), where
&(U) is identified with the obvious subgroup of Lp fixing all but the first n basis
elements of gives rise to a Fitting class <1-1 (&) (where d = d®1'®2 and (ds,/®J, Lp)
is the outer Fitting pair described above) which is related to, but somewhat larger
than, the Fitting classes of Dark and McCann.
Pense’s Dissertation [2] contains many original ideas, and at the time of writing
it seems that the full potential of his approach has yet to be realized. Before moving
on, we pick out one more idea for special mention because it could find applications
elsewhere. It depends on the following lemma, derived from a theorem of Wielandt
[1], which states that if S is a perfect single-headed subnormal subgroup of a group
which is the join oftwo subnormal subgroups Sj and S2, then either S < St or S < S2.
(6.28) Lemma (Pense [2]). Let G be a finite group.
(a) Let R/S be a non-abelian composition factor of G. Then the set of subnormal
subgroups supplementing SinR has a unique minimal element (called the anchor of R/S).
(b) Let H/K be a non-abelian chief factor of G. Then the anchors of the minimal
normal subgroups of H/K are conjugate in G. (Thus one can associate with each chief
factor an anchor type, being the isomorphism type of the anchor of any minimal
normal subgroup of H/K).
(c) The anchors (anchors types) are preserved under the Schreier-Zassenhaus corre-
spondence (the correspondence used in the proof of Huppert [5] I, Satz 11.5, which we
cite for the proof of the Jordan-Holder Theorem A, 3.2) between composition (chief)
factors.
Proof, (a) By A, 14.15 a minimal subnormal supplement to S in R is perfect and
single-headed. If S1 and S2 are two such supplements, we can apply Wielandt’s
theorem, cited above, to <S;, S2 > to conclude that St < S2 and S2 < St, thus proving
(a).
(b) Since the minimal normal subgroups of H/K are conjugate in G, it follows that
their (unique) anchors are conjugate.
(c) If Rt/S'(i = 1, 2) are corresponding factors in two composition series, then
(J? i n R2)St = Rt. If Vj is the anchor of RJS^i = 1, 2), Wielandt’s theorem implies
that either V2 < St or Ц < n R2, and since Ц f St, it follows that Ц < R2. Since
К j n S2 < St, we have Vt f S2, and therefore Ц is a subnormal supplement to S2 in
R2; moreover, it is a minimal such supplement because it is single-headed. Thus
Fj = V2 by Part (a).
To obtain the analogous result for chief series, simply refine the chief series to
composition series and observe that the Schreier-Zassenhaus correspondence re-
spects the refinements. □
Pense uses this result to enrich his source of outer Fitting pairs: Those which are
defined in terms of the action of a group G on the product of its J-chief factors in a
given radical section G^JG^ can now be refined by restricting their domain of
definition to the J-chief factors of a given anchor-type. Furthermore, Pense shows
that all such chief factors appear in one normal section of G. Pense’s idea obviously
6. Dark s construction—variations
675
provides a general method of increasing the range and resolving power of Fitting
classes that can be defined in terms of conditions on non-abelian chief factors (see,
for example, Constructions C and D in Section 2 of this chapter).
Unfortunately, Lemma 6.28 fails for abelian chief and composition factors—for
example, if N is a non-cyclic maximal subgroup of D = Dih(8), the minimal sub-
normal supplements of D/N have orders 2 and 4. When this result fails, Pense has a
partial substitute in the shape of the following lemma, which he then uses to cut down
the size of the classes of Dark type that he has constructed.
(6.29) Lemma (Pense [2]). Let Q be a group of operators for G, and let N be an
Q-invariant, Q-hypercentral normal subgroup of G such that [G/N, fi] = G/N. Then
there exists a unique Q-invariant subnormal supplement to N in G.
We close our extended treatment of Fitting classes “of Dark type” with a short
survey of their applications, over and above the ones already mentioned above in
Section 5 and Section 6.
1. Lockett [3] uses Dark’s original method to construct a Fitting class g and a
group G f g with the property that G/(Z(G) n G') e g. This example was designed
to answer a question of Gaschiitz about Fitting classes closed under central exten-
sions. The role of the group У satisfying Hypotheses 5.5 is played by the semidirect
product of an extraspecial group E of order 311 with a cyclic group of order 61, acting
irreducibly on £/Ф(Е). This group admits an automorphism a of order 5 which is
used to specify the set sf.
2. In [11] Hawkes uses a variation of Dark’s method to answer some questions
of Cossey, posed in [1], by constructing a <Q, E„>-closed Fischer class which is not
a formation. The noteworthy feature of this particular construction is the fact that
the specified key section has to be a direct product of copies of a certain group (as
opposed to a subdirect product found on other examples).
3. Berger and Cossey [1] use the Dark Construction to produce a counter-
example to the so-called “Lockett Conjecture” that g„ = g n for all Fitting
classes g. (This could be more accurately described as “Lockett’s question” since he
merely posed it as an “open problem” in Lockett [4].) We give full details of this
example in X, 6.16.
4. The above-mentioned example of Berger and Cossey is also exploited by Hauck
[5] to show that if I and 9) are Fitting classes, in general the two classes (I О ?))*
and I* О "D* need not be comparable. We give the details in Example X, 6.17.
5. Hauck [1] also uses a modification of the example of Berger and Cossey to
show that there exist Fitting classes g and I such that I £ g*9I and I g9I. (Thus,
in his terminology, g* can be H-normal in I while g fails to be so.)
6. In [1] Beidleman and Brewster introduce the relation of strict normality for
Fitting classes: I is said to be strictly normal in 9) if for all G e £ and for У e Inj9(G),
the radical У5 is an .E-injector of G. They show that the intersection 9)0 of all
non-trivial Fitting classes strictly normal in ф is itself strictly normal in ф and that
m с ш |n the sequel [2] they use a variation of Dark s construction to produce
a Q-closed Fitting class X which is properly contained and strictly normal in a Fitting
class ф, thereby showing that 9)0 can be strictly smaller than ?),.
bits IX. Fining classes—examples and properties related 10 injectors
7. In [3] Brison uses Variation II of this section to construct a non-nilpotent
Fitting class with a trivial Lockett section. This is described in Example X, 5.34(a).
8. In his doctoral dissertation. Hawthorn [1] (see also Hawthorn [2]) has used a
complicated variation on Dark’s construction to produce a Fitting class T with the
property that the smallest class in Locksec (T) cannot be determined by means
of the transfer Fitting pairs described below in Section 5 of Chapter X. In particular,
S' is not a Berger class (see Definition X, 5.18 and Theorem X, 5.28).
Chapter X
Fitting classes—the Lockett section
1. The definition and basic properties of the Lockett section
In this section, up to and including Corollary 1.30, we work within the universe (f;
thereafter we restrict the universe to S.
We described earlier, in IX, 2.14(b), an example of a Fitting class g and a group
G such that
(G x G)g > Gg x Gs.
It was this failure of radicals to respect direct products that led Lockett [4] to the
following construction, which not only gives a precise description of the extent of this
failure, but also associates with each Fitting class a previously unknown and often
large family of closely related Fitting classes. We follow Lockett’s original treatment
of this material with only minor modifications.
(1.1) Definition (Lockett's star operation}. For each Fitting class g of finite groups,
an associated class g* is defined as follows:
g* = (G e G: (G x G)g is subdirect in G x G).
It is obvious that g £ g*, and in due course we shall show that g* is a Fitting class,
that g is strongly contained in g*, and that the g*-radical of a direct product of finite
groups is the direct product of the g*-radicals of the direct components.
(1.2) Lemma. Let g be a Fitting class and G a finite group
(a) //(g1,92)e(G x G)s, then (gt. g[l) e (G x G)g and gtg2 e Gg.
(b) Any two of the following statements are equivalent :
(i) Geg‘;
(ii) (G x G)R contains (g, g l) for all geG:
(iii) (G x G)g = (Gg x Gg)<(g, g 1): g e G>.
Proof, (a) Let (91, g2] e (G x G)g, and let D denote the direct product G x G x G.
Then(G x G), x 1 = (G x G x l)n£>gby IX, 1.1(a), and consequently (g„ g2, I) e
Dg. Similarly Dg contains (1, gt, дЛ and since Dg char D it also contains (1, g2, g,\.
Therefore Dg contains (S1, g2. DO, 02. 0>Г‘ = '• sf^ow idenbfymg G x G
with the subgroup G x 1 x G of D, we see that (gt ,g2 ) e (G x G)g. Consequently
678
X. Fitting classes—the Lockett section
(1, S1S2) = (Si- ffz) e (G xG)sn(l x G)= 1 x Gg, and Assertion (a) is
justified.
(b) (i) => (ii): Let geG e 5*. By definition of 5* the subgroup (G x G)g contains
an element of the form (g, g2) and therefore contains (g, д'1) by Assertion (a).
(ii) => (iii): If (ii) holds, the following inclusion is obvious:
(Gg x Gg)<(g, g~'): geG) <(G x G)a.
On the other hand, if (g2, g2) e (G x G)a, by Assertion (a) we have g2 e gil Gs. Hence
(Si- S2)e (Si> sr’)(Gg x Gg), and the reverse inclusion holds.
It is obvious from the definition of g* that (iii) => (i), and the circle of implications
is complete. □
(1.3) Lemma. If A is a group of operators on a group G e R*, then [G, Л] < Gg. In
particular, G/Gg is abelian.
Proof. If G e g*, by (1.2) we have (g, g-1) e (G x G)g for all geG. Let a e A. Since
A can act on G x G via its action on the first coordinate, it follows that (g“, g~’) lies
in the characteristic subgroup (G x G)s of G x G. Then by 1.2(a) we have [g, a] =
g ‘g"e Gg, and consequently [G, A] < Gs. The final assertion of the lemma is seen
by taking for A the group G acting on itself by conjugation. □
(1.4) Theorem (Lockett [4]). The class R* defined in (1.1) is a Fitting class.
Proof. We begin by showing that R* is s„-closed. Let N < G e R*. Then (G x G)g
contains <(g, g”1): g e G) by (1.2)(b), and therefore the subgroup (N x N)g =
(N x N) n (G x G)g contains <(n, tT1): n e N>. It follows that (N x N)g is subdirect
in N x N, and so by definition the class R* contains N. Hence R* = s„R*.
To prove that R* is N0-closed, let G = Nt N2, where Nt and N2 are normal
R*-subgroups of G. The group G acts as a group of operators on each /V,- by
conjugation, and therefore by (1.3) we have [N,, G] < (NJg < Gg for i = 1,2. Hence
G'= [NjN2, G] = [N1; G][N2, G] < Gg. If geG = N2Nt, let д = n2nt with
t^eN^ and note that [n2, tqjeGg. Then (g, g-1) = (И1И2[И2> th], =
(ni,nr1)(n2>'>2I)([«2>”il OefNj x Nt)s(N2 x N2)S(GS x 1) < (G x G)g. There-
fore (G x G)g is subdirect in G x G, and so G e g*. Consequently g* is N0-closed by
П, 2.11(b). □
(1.5) Lemma. Let R be a Fitting class and G a finite group. Then (G x G)g =
(Gg x Gg)<(g, g-1): g e Gg.>; in particular, (G x G)g < Gg. x Gg..
Proof. Because of the automorphism interchanging the coordinates of the direct
product G x G, the characteristic subgroup (G x G)s has the same projection,
G* say, into each coordinate. Since (G x G)a < G* x G* < G x G, we have
(G* x G*)g = (G x G)g, which is subdirect in G* x G* by definition of G*. Con-
sequently G* e R* and G* < Gg.. Thus (G x G)s = (Gg. x Gg.)g, and the desired
conclusion now follows directly from (1.2)(b). □
1. The definition and basic properties of the Lockett section 679
ann u be Characteristic of a group G with R > S,
and let A - Aut(G). We call the section R/S characteristically hypercentral in G if for
some r e M we have
[R, A,Л] < S.
If this is true with r = I, we say that R/S is characteristically central in G.
Recall that G" denotes the direct product of n copies of a group G.
(1.7) Proposition. Let 5 be a Fitting class. Any two of the following statements are
equivalent.
(a) G e g*;
(b) G"/(G")g is a characteristically central section of G" for n = 1,2,...;
(c) G7(G")g is a characteristically hypercentral section of G" for n = 1, 2,....
Proof. (a) => (b): Let G eg*. Then G" e g* by (1.4), and Statement (b) follows at once
by (1-3). The implication: (b)=>(c) is obvious. (c)=>(a): Assume that Statement (c)
holds, let n be a prime not dividing |G/GB|, and note that n then does not divide
| G7(G")g| because G7(G")S is an epimorphic image of G7(Gg)f s (G/GB)". Let a be
the automorphism of G" obtained by permuting the components by an n-cycle thus:
a- (Sl> •••» Sn) “* (Sni Sl> •••> Sn-l)-
The automorphism <r, as an operator of coprime order acting hypercentrally, cen-
tralizes the section G7(Gn)B by A, 12.3. Therefore, if g e G, we have (g-1, g, 1...., 1) =
(g, 1,..., l)-1(g, 1,..., ire(G")g, and hence (g *, g) e (G x G)g. Consequently
(G x G)g is subdirect in G x G, and hence G e g*. □
The next theorem shows, inter alia, that Lockett’s ‘star’ operation behaves like a
closure operation when its domain is restricted to Fitting classes.
(1.8) Theorem (Lockett [4]). Let g and 6 be Fitting classes. Then
(a) g£g* = (g*)* s g9I, and
(b) if g £ 6, then g* £ 6*.
Proof, (a) From the definition of g* and (1.3) we have g S g* S g®. It remains to
show that the star operation is idempotent. Certainly g* £ (g*)* Let X e (g ) , and
put R = Xs. and S = XB. Setting A = Aut(X), we have [X, A] < R by (1.3), and
again [R, A] < S because Reg* and A acts naturally as a group of operators on R.
It follows that X/XB is characteristically hypercentral. In particular, if G is an
arbitrary group in (g*)*, we may take X = G", which belongs to (g*)* by (1.4), and
conclude that G7(G")B is characteristically hypercentral for each n - 1, 2,... Then
by (1.7) we have G e g*, and thus we have shown that (g*)* = 8*-
(b) Let G e g*. Then (G x G)g is subdirect in G x G. Therefore (G x G)e, which
obviously contains (G x G)g, is also subdirect in G x G, and consequently G e ® .
680
X. Fitting classes—the Lockett section
(1.9) Theorem (Lockett [4]). Anytwoqf the following three statements about a Fitting
class g are equivalent :
(a) 5 = 5*;
(b) (G X Я)8 = Gg X H9for all G, He ffi;
(c) (G x G) j = Gg x Gj for all G e g®L
Proof. First we prove the implication: (a)=>(b). Let (g. h)e(G x Я)в. Identifying
G x Я in turn with the appropriate subgroup of G x Я x Я, we conclude that
(G x Я x Я)8 contains (g, h, 1) and (g, 1, h) and hence contains (1, h, h'1). Thus
(h, IT1) e (H x H)j. By (1.5) and the assumption that g = g* we have (Я x Я) g <
Hs. x Я j. = Я j x Hs, and therefore h e H%. Similarly we have g e G5, and it is then
clear that Statement (b) holds. The implication: (b) => (c) is obvious. To complete the
proof we now assume that Statement (c) holds and deduce that g = g*. Let G e g*.
By definition of g* the subgroup (G x G)s is subdirect in G x G. By (1.8) we have
G e g2I, and so by our assumption Gg x Gg is subdirect in G x G; in other words,
G = Gj e g. Thus g* s g, and these two classes are therefore equal. □
We draw a conclusion from this theorem for later use.
(1.10) Corollary. Let neN, n>2, and let % be a Fitting class. Then G eg* if and
only if (G") j is subdirect in G".
Proof. For n = 2 this statement is of course just the definition of g*. First we
prove that the condition is necessary. If G e g*, then the subgroup
(G x G x 1 x ••• x l)j
projects onto the first component of the direct product. Since
(G x G x 1 x ••• x l)j<(Gn)j,
the projection of (G")j into the first component is therefore surjective. The same is
true for the other components, and hence (G") g is subdirect in G".
For the sufficiency first observe that, since g £ g* = (g*)*, by (1.9) we have
(G")j < (G")j. = (Gj.)". If Gj. < G, then (Gg.)" is certainly not subdirect in G". It
therefore follows that, if (G")g is subdirect in G", then G = Gg. e g*. □
Next we characterize the g-radical of a direct power. The result is an essential part
of Lockett’s original definition of g*.
(1.11) Theorem. Let n e N,n > 2, let g be a Fitting class, and let Gbea group. Then
(G")j = |(»i, 9t e Gs., f] gt e Gg|.
Proof Let g = (gt,..., gn) e G". Since g S g*, we have (G")8 (G")8. = (G8.)” by
(1.9). Therefore, if g e (G")g, then g, e G g. for i = 1,..., n.
1. The defminon and basic properties of the Lockett section 681
Now let g e (Gg-Г, and let i g {1, Put x, = 91... g, and
П = (1,..., l.Xj.x/1,1,..., 1),
where the entries xf and x,1 are in coordinate positions i and (i + 1) Since x, 6 Gt g iv*
it follows from (1.5) that s ’
йе(1 x x 1 x G x G x 1 x x I),(G")g.
However, we have
<9r. • -•, 9») = Сч. x-t'x2, x^xj,.... хДх,) = yty2...y„_,(l,..., 1, x„),
and therefore ge(Gn)g if and only if (1,..., 1, x„) e (G”)g. Consequently, since
(1 x ••• x 1 x G)n(G")g = lx-xlx Gg, we conclude that ge(G")g if and only
if 9i • • • 9n = x„ e Gg. □
(1.12) Definitions, (a) A Lockett class is a Fitting class ft such that ft = ft*. Thus by
(1.9) Lockett classes are precisely those Fitting classes for which the radical of a direct
product is always the product of the radicals of the direct components.
(b) (Lockett’s lower star operation) For an arbitrary Fitting class § define
ft, = Q {X: X is a Fitting class such that X* = ft*}.
Of course, the class ft, is a Fitting class, but the reason for its significance is that it
is closely related to ft. We shall see that it has the remarkable property that (ft,)* =
ft*, so that, in particular, with each Lockett class ft there is associated a smallest
Fitting class whose ‘star closure’ is ft, namely the class ft,.
(1.13) Proposition. Let {Эл}ЛеЛ be a set °f Fitting classes. Then
(La) ( 0 &Y = 0 (ftz)*-
\ЛеЛ / ЛеЛ
We shall prove (1.13) with the help of the following lemma.
(1.14) Lemma. Let ft be a Fitting class. A group G is in ft* if and only if the following
subgroup T of G x G
(1.P) T = (G' x G')<(», g“‘): g e G>
is in ft.
Proof of (1.14). Let Gcft*. By (1.3) we have G'< Gg, and therefore
tG x G а1)’ a g G> = (G x G)g by (1.2). Therefore T e s,ft - ft. Conversely,
ifTG ft thesubgroup(G x G)gcontainsTandisthereforesubdirectinG x G.Hence
G g ft* by definition of ft*.
6g2 X. Fitting classes—the Lockett section
Proof of(L 13). For a given group G let T denote the subgroup defined by Equation
If. Then Lemma 1.14 yields the following chain of equivalent statements:
G belongs to the left-hand side of Equation La-»
Te Плел
Те Sa for each 7 e Л »
G e (5Д)* for each 2 e A®
G belongs to the right-hand side of Equation La. □
By applying (1.13) to the definition of 6*. one immediately obtains: (6*)* = g*.
Using this fact together with (1.8) (a), one then readily derives the relationships of the
following theorem.
(1.15) Theorem (Lockett [4]). For any Fitting class g we have
(6,), = 6* = (6*)* S 6 S 6* = (6*)* s 6Л-
(1.16) Definition. With each Fitting class g we associate a set of Fitting classes called
the Lockett section of g and defined as follows
Locksec(g) = {®: ® is a Fitting class and ®* = g*}.
(1.17) Theorem. Any two cf the following statements about a pair of Fitting classes
g and ® are equivalent:
(a) ® e Locksec(g);
(b) g, <= ® £ g*;
(c) Locksec(S) = Locksec(g).
Proof, (a) => (b): If ® e Locksec(g), we have ® £ ®* = g*. By (1.15) we also have
5, = (6*), = (S’*)* = ®„ £ ®. Therefore Statement (b) holds.
(b) => (c): By (1.8)(b) we can apply the star operation to Statement (b) to conclude
that (5»)* £ ®* £ (6*)*- By (1.8)(a) and (1.15) we have (g*)* = 5* = (5,)*. Hence
®* = g*, and Statement (c) holds.
(c) => (a): This follows from the obvious fact that ® e Locksec(S). □
(1.18) Proposition (Bryce and Cossey [5]). Let g and ® be Fitting classes such that
g £ 6. Then g„ £ g n ®„.
Proof. By (1.13) we have (g n ®,)* = g* n (©„)* But by (1.15) and then again by
(1.13) we also have g* n (®„)* = g‘n®* = (gn ®)* = g*, and the conclusion of
the proposition now follows from the definition of g*. □
For a pair of Fitting classes g £ it follows from the preceding proposition that
(1T) X-Xng*
defines a map from Locksec(ffi) to Locksec(g). We shall see in Theorem 6.16 of this
I. The definition and basic properties of the Lockett section
683
chapter that for the pair <S £ G, this map is onto; in other words, the Lockett section
of <S is determined by the Lockett section of G. In [4] Lockett asks whether this map
is always onto when © = S, and although he only raises it in the form of a question
this has since become known as the ‘Lockett conjecture’. Whereas this conjecture is
true for certain well-behaved Fitting classes g £ S (for the primitive saturated
formations, for example—see Theorem 6.12 of this chapter), Berger and Cossey [1]
have shown that in general the Lockett conjecture is false; we describe their counter-
example in (6.16). We now extend and formalize Lockett's original question by means
of the following definition.
(1.19) Definition. For a pair of Fitting classes g £ © we say that g satisfies the
Lockett conjecture with respect to © if the map (l.y) from Locksec(©) to Locksec(g)
is surjective. We shall see in (6.1) that a necessary and sufficient condition for this is
the following equation: g„ = g* n ©t.
By Theorem 1.17 it is clear that each Fitting class belongs to one and only one
Lockett section; in other words, the Lockett sections form a partition of all Fitting
classes. Next we show that the Fitting classes of a given Lockett section all have the
same characteristic. Later in this section we consider other properties common to
the Fitting classes of a Lockett section.
(1.20) Proposition. For each Fitting class g iw have g* £ Qg, Char(g*) = Char(g),
and <r(g*) = <r(g).
Proof. It is clear from the definition of g* that each group in g* is the projection,
and hence an epimorphic image, of a group in g. If Zp e g*, then Sp £ g* by IX,
1.7. Hence Sp £ g'2l by (1.8), consequently g contains the derived group of a
non-abelian p-group, and it follows that Zp e g. Thus Char (g*) £ Char(g), and since
g £ g*, equality holds.
If G e g*. then (G x G)g is subdirect in G x G by definition of g*. Therefore
<r(G) = <r((G x G)g), and <r(g*) = <r(g) holds. □
The next two theorems give criteria for a pair of Fitting classes to belong to the same
Lockett section. The first of these exploits the property described in (1.3).
(1.21) Theorem. Let g and © be Fitting classes with g £ ©. Any two of the following
statements are equivalent:
(a) g and © belong to the same Lockett section:
(b) [G®, Aut(G)] < Gg /dr all G e G;
(c) Ge/GB<Z(G/GB)/CT-fl«GeG;
(d) Ge/GB < Z(G/GB)./br all G e ©21;
(e) [G, Aut(G)] < Gg for all Ge©.
Remarkson Terminology. H. Laue [1] calls ^centralunder © if g £^© and Statement
(c) is satisfied. If g £ © and Statement (e) holds, then Bryce and Cossey [5] say that
g is styongly normal in ©-
684
X. Fitting classes—the Lockett section
Proof. First we prove the implication: (a) =* (b). If Statement (a) holds, then g £ ®
g», and so Gp, g g* for all Ce6. Since G® admits Aut(G) as a group of operators, it
follows from (1.3) that [G®, Aut(G)] < (Со)я = Gs. Thus Statement (b) holds. The
implications: (b) => (c) => (d) are obvious. To see that Statement (d) implies (e), let
Gs®, and for aGAut(G) let H denote the semidirect product [G]<a>. Clearly
H e and since G s Я©, Statement (d) implies that [G, a] s Hs. Consequently
[G, a] < G = Gs for all a e Aut(G), and so [G, Aut(G)] < GB, as required.
Finally, we have to prove that Statement (e) implies Statement (a). Let G g © and
ne N. Then G" gd0© = (5, and therefore [G", Aut(Gn)] < (G")s by the assumption
that Statement (e) holds. Therefore by (1.7) we have Gg g*, and it follows that
(5 g Locksec(g). □
It is a consequence of the next result that for an arbitrary Fitting class g the associated
dominant Fischer class g О 9? is a determining invariant of the Lockett section of
g. Recall from II, 1.5 thatE.X = (Gg®: 3K < G such that К < Z„(G) and G/K g X).
(1.22) Theorem (Cossey [2]). Let f> he a <q, Ez>-c/osed Fitting class of finite soluble
groups satisfying the following property:
(1.5) For all peP there exists a group H e f, such that Zp H ф f>.
Then two Fitting classes g and © belong to the same Lockett section if and only if
g О S = 6 О &
Remark. It is easy to verify that any primitive saturated formation of bounded
nilpotent length and full characteristic fulfils the requirements of § in the hypotheses
of this theorem; in particular, we could take § = 91.
Proof. First observe that g О § = g* О for if G g g* О f>, then G/G5 has a
central normal subgroup Gg-/Gg with quotient in § by (1.21), and so by hypothesis
we have G g g О (Ezf>) = g О f>. Therefore g’ О fi £ g О fi. The reverse inclusion
follows from the Q-closure of § and so the two classes coincide. Hence, if g and ©
belong to the same Lockett section, it follows that gO& = g*O§ = ®* Ofi =
©Of).
Conversely, assume that g О § = © О and first suppose, by way of contradic-
tion, that g*\©* is non-empty. Let G be a group of minimal order in g*\©*, and
note that by the usual argument G®" is the unique maximal normal subgroup of G;
furthermore, since g* £ (© О £j)9I £ ©<5 = ©*<5, the index |G: Со«| is a prime, p
say. By Hypothesis 1.5 we can find a group H g § such that (Zp rljrcB H) ф & put
IF = GQjreg H. Let В = G", the base group of W, and let R = (G®»)11. Since ©* is a
Lockett class, by (1.9) we have R = Ba~. Hence R = Bo W^’, and therefore WC1-
centralizes B/R. But W/R is isomorphic with Zp Qj„8 H, whose base group, the image
of B/R, is self-centralizing, and consequently = R. However, W/R ф § by choice
of H, and so W ф ©* о & = © О f). On the other hand, В e D„g* = g*, and there-
fore W/W^- g q(H) £ f>. Thus H'g g* О § = g О § = © o §, and we have reached
the desired contradiction. Therefore g* £ ({j*. The same argument shows that ©* £
g*, and so we conclude that g and © are in the same Lockett section. □
1. The definition and basic
properties of the Lockett section
685
A Fitting class g is said to have a trivial Lockett section if Locksec(g) = {gl and
our next observation is that the classes 9l„, n s P, have this property. In view of (1.3),
any Fitting subclass of the class of finite perfect groups also has this property (for
example, the classes d03 for any 3 s 3\®)- In (5.34)(a) we construct a metanilpotent
example of such a Fitting class, which suggests that the problem of determining even
the soluble Fitting classes with a trivial Lockett section is a difficult one.
(1.23) Remarks, (a) If n s IP, then 9l„ = (9l„)* = (91„)ж.
(b) If g is a Fitting class cf characteristic n, then 91„ e gt.
Proof, (a) By (1.20) we have (9i„)* c q91„ = 91„, and hence 9l„ = (91J*. Also by
(1.20) we have Char((9l„),) = Char(9?„) = rr, and therefore 9l„ c (9i„)„ by IX, 1.9.
Consequently 'Ji„ = (9l„)„.
(b) By IX, 1.9 we have 9?„ s g, and therefore by (1.18) and Remark (a) we have
9?я = (9?„), <= g,. □
To have a trivial Lockett section is obviously a sufficient condition for a Fitting
class to be a Lockett class, and we now look for other conditions for this to be the case.
Our first is both necessary and sufficient and leads to a sharpening of the quasi-R0
lemma (IX, 1.13).
(1.24) Theorem (Hauck [6]). The following statements about a Fitting class g are
equivalent:
(a) g is a Lockett class',
(b) For all groups G with normal subgroups Nt and N2 such that Nlr>N2 = 1 and
G/Nt N2 e 91, the following condition holds:
Geg о G/Nt and G/N2 e g.
Proof, (a) => (b): Let g be a Lockett class, and let G be a group with normal sub-
groups N2 with the stated properties. If G/Nt and G/N2 e g, it follows from IX,
1.13 that G e g. Now suppose that Ge g, and let p be the embedding of G into the
group D = G/Nt x G/N2 defined by p(g) = (gNi, gN2). If Go = f(G), then as in the
proof of IX, 1.13, we have Go sn D. and therefore Go S Ds. But Go, and so Dg, is
clearly subdirect in D. Hence D e g* = g, and therefore G/N, e g for i = 1, 2.
(b)=>(a): Let G e g*. Since (G x G)g is therefore subdirect in G x G, we have
G x G = (G x G)g(G x 1). Let N, = Gg x 1 and N2 = 1 x Gg, and note that by
IX, 1.1 (b) the group (G x G)g/Nj N2 is abelian. Since (G x G)ge g, our assumption
that Statement (b) holds implies that (G x G)g/Ni e g. Therefore
G s(G x G)/(G x 1) = (G x G)g(G x l)/(G x 1)
=(G x G)g/((G x G)gn(G x 1)) = (G x G)g/Ni e 8,
□
and it follows hat g* = g.
686
X. Fitting classes—the Lockett section
The next result shows that the presence of certain additional closure operations often
yields a sufficient condition for a Fitting class to be a Lockett class. On the other
hand, in (5.34)(a) later in this chapter we describe an example which shows that
closure under none of them is necessary. Moreover, not every additional well-known
closure property provides such a sufficient condition; for certainly the operation г.ф
is ’additional’ in the sense that Еф f <s„, n0>, and it is easy to see that the class 91 О <S,
is ^-closed but not a Lockett class.
(1.25) Proposition (Lockett [4]). A Fitting class g that is closed under any one of the
operations q, r0, or sF is a Lockett class.
Proof. The sufficiency of the condition: g = Qg follows at once from (1.20).
Next suppose that g = Rog, and let G e g*. Let H denote the subgroup
(GB x Gs x Gg)<(g-1, g, g): ge G> of the direct product D = G x G x G. Then H <
Dby (1.3). By considering the projections of H onto each of the subgroups G x 1 x G
and G x G x 1 of D in turn, we see that the quotient groups Я/(1 x Ggx 1) and
Я/(1 x 1 x Gg) are both isomorphic with (Gg x Gg)<(g-1, g): geG) and therefore
belong to g by (1.2)(b). Hence HeRog = g, and we have Я < Dg. But Dg also
contains the subgroup (G x G x l)g, which by (1.2)(b)contains the element (g, g-1, 1)
for all geG. Therefore Dg contains (g, g-1, 1) (g~ ’, g, g) = (1, 1, g) for all geG. Then
evidently G = lxlxGes„g=g, and it follows that g* = g.
Finally, suppose that g is a Fischer class, and let G e g*. By (1.2)(b) the subgroup
T = (Gg x Gg)<(g, g-1): g e G> of G x G belongs to g. For geG, let Я(д) denote
the group (Gg x 1)<(д, д-1)>, which, as a cyclic extension of the normal subgroup
Gg x 1 of T, belongs to sFg and hence to g by supposition. Since <(1, g)> clearly
centralizes Я(д) and also belongs to g by IX, 1.9, we have
(Gg x 1)<(д, 1)> < H(g)<(l, g)> e Nog = g,
and then, because (G x l)/(Gg x 1) is abelian by (1.3), it follows that the group G x 1
is the join of the subnormal g-subgroups (Gg x 1)<(д, 1)>, as g runs through G.
Therefore G = G x 1 eNog= g, and again we conclude that g = g*. □
The next subject under discussion is the effect of Lockett’s star operation on a Fitting
class product go®. The obvious question as to whether it is always true that
(le) (g О ©)* = g* о ©*
has a negative answer. In Example 6.17 of this chapter we describe Fitting classes g
and © such that (g О ©)* £ g* О ®* and g* о ©* £ (g C ©)*. We therefore direct
our attention to the problem of finding sufficient conditions for (Le) to hold.
(1.26) Lemma (Hauck [5]). Let g and ® be Fitting classes of finite groups. Then
(a) (g о ©*)* = (g о ©)*;
(b) If g is a Lockett class, then g О ©* = (g О ©)*; in particular, the Fitting
product of two Lockett classes is again a Lockett class.
687
1. The definition and basic properties of the Lockett section
Proof, (a) Since ® £ ©*, then clearly g о © s g о ®. Let G e and =
G/G,. By <L2‘) have ^./Ge<Z(G/GeX and it therefore follows that
(gO®)* ° Й ~ Z °'°so ТЬеП’agalnfrom(l-2l)’weconclude that (g о ©*)* =
(b) Let g be a Lockett class, and let G e C. Since ©* is also a Lockett class by
several applications of (1.9) we have y
(G x G)S0(6./(G x G)s = ((G x G)/(G8 x 60)^
= (G/Gj)e« x (G/Gjjlu,"
= <Ggo e«/GB) x (GB0 6»/Gb)
= (Ggo x GB0 О«)/(СВ x Gg)
= (Ggo<r,’ x Ggc, О«)/(С x G)g.
It follows that (G x G)So(6. = GSo(6. x Ga<> e., and therefore g О ©* = (g о ®*)*
by (1-9). The first conclusion of Part (b) now follows from Part (a), and the second is
obvious. q
For a pair of Fitting classes g £ © consider the following property
(1.0 G’nGg«<Gg for all G e ©.
If g satisfies the Lockett conjecture with respect to © (see Definition 1.19), then
obviously ©„ n g* = gt £ g, and since by (1.3) and (1.15) we have G' < G6. for all
G e ©, it follows that Equation l.£ is satisfied.
Open Question. If Property l.£ is fulfilled for a pair of Fitting classes g £ ®, does it
follow that g satisfies the Lockett conjecture with respect to ©?
In the next theorem we show that if (1.0 holds for the pair g £ g О ©, then
g О ® £ g* О ®. From (1.26)(b) we can then deduce that (g О ®)* £ (g* О ©)* =
g* О ®*. But, as we have already mentioned, there exist Fitting classes g and ®
such that (g О ®)* £ g* о ®*. and therefore (1.0 cannot be true for all pairs of
Fitting classes.
(1.27) Theorem (Hauck [5]). Let g and © be Fitting classes of finite groups.
(a) If Property l.f is satisfied by the pair g £ g О ©, then g О © £ g* О © If it
is also satisfied by the pair g £ g* О © and if 91 £ ®, then g О © = g* О ©.
(b) If <Q, e,>© = ©,t/ieng о © = g* О ©; in particular, g О &isa Lockett class.
(c) If ©* is <Q, Effclosed or <Q, Enclosed, then (g О ©)* = g* О ©*
Proof. First assume that (1.0 is satisfied by the pair g £ g 0 ©. We suppose that
g О © £ g* О ®, choose a group of minimal order in g О ©\g* О ®, and derive
688
X. Fitting classes—the Lockett section
a contradiction. It is easy to see that, as usual in this situation, GB« 0 lf> is the unique
maximal normal subgroup of G. By assumption we have G' n GB« s Gs, and if
G' = G, then GB« = Gs, contrary to the choice of G. Consequently G' < G, and G/G'
is a cyclic p-group for some prime p. Since G £ ft, we have p e Char(ffi), and therefore
G/G'GB e ffi. Furthermore, G'GB n GB- = (G' n GB«)GB = GB by assumption, and
G/GB e © since Ge gО ffi. Applying the quasi-R0 lemma to the group G/Gs, we
obtain G/Gg« e © and therefore Geg* 6 ft, a contradiction. Hence g 6 ffi £
5* О ®.
In the case where the pair J £ J* О ® satisfies (1.0 an analogous argument shows
that g* C ® £ g О ®.
(b) The conclusion that g О © = g* О © when ® is <q, E,>-closed follows easily
from the fact that by (1.21) the normal section Gs>/Gg is a central factor of G for all
G e (£. Since g* is a Lockett class and since Proposition 1.25 implies that ® is a
Lockett class, we conclude from (1.26)(b) that g* о © is also a Lockett class.
(c) First suppose that ®* = <q, E,>ffi*. Then by Part (b) we have g О ®* =
g* О ®*, and with the help of (1.26)(a) and (b) we obtain (g О ®)* = (g О ©*)* =
(g* О ©*)* = g* о ffi*.
Now suppose that ®* = (Q, E®)©*, and observe that the Q-closure at once im-
plies that go ®* £ g* q ®»_ Therefore by (1.26)(a) and (b) we have (g О ®)* =
(g О ®*)* S (g* О ®*)* = g* О ®*. To prove the opposite inclusion, let G e
g* о 6*. We show by induction on |G| that Ge(g О ®)*. If Gg./Gg £ Ф(в/вв),
there is a proper subgroup U of G such that G/GB = (GS./GB)(G/GB), and by (1.21)
we have U <i G. By induction we can suppose that U e (g О ®)*, and since g* £
(g О ®)*, it follows that G = G-B«U e N0((g о ®)*) = (g О ©)*. On the other hand,
if GB./GB < Ф(в/вв), we have G/GBe E®ffi* = ®*, and therefore Ge g О ®* £
(g О ©)* by (1.26)(a). □
Lemma 1.26(a) implies that for arbitrary Fitting classes g and ® it is always true that
8 ° ®*s(8 O ®)*. However, if © does not have full characteristic, this inclusion
can certainly be strict (for example, take g = <5, and © = 6P).
Open question. Is it true that g О ©* = (g О ©)* for all Fitting classes g and © such
that 91 c ©?
Since g 0 ® £ g о ©♦ £ (g 0 ©)*, this question is equivalent to asking whether
g О ©* is a Lockett class when 91 s ffi; a partial answer is contained in the following
theorem.
(1.28) Theorem (Hauck [5]). Let g and © be Fitting classes with 91 £ ffi If ®* is
<Q, E,У-closed or a Fischer class, then g о ffi* = (g о ©)*.
Proof. If®* = <q, e,>®*, then g о ©* = g* о ®* by (1.27) (b), and hence by(1.27)(c)
we have (g О ©)* = g* о ©* = g о ©*. (We remark in passing that the e,-closure
of ©* automatically implies that 91 £ ffi.)
Now assume that ffi* is a Fischer class. By (1.26)(a) we have g О ffi* £(g0 ©)*.
By way of contradiction we suppose that (g О ©)* g О ffi* and choose a group
1. The definition and basic properties of the Lockett section
s " * °en(s ° 51=**• ” -»
Ga CR. < Ggi (6. < G 6 (g O ®)».
The choice of G implies that Gg0(6. is the unique maximal normal subgroup of
G, and because Ge(g о ®)*, it follows from (1.3) that G/G^ is abelian and
hence cyclic. Let G/Gg<> e = (aGs.y o>, and let D = G x G. By (1.5) we have P. =
L a 5?’ 9 9 G ЬУ (12) a,S° D*‘ e = (G«< ® x Gso a’*».
Evidently the group ’ //
L = (lx Ggo e)<(°> fl-I)>Pg/^g
is a nilpotent extension of the normal subgroup (1 x Gg<6)Pg/Pg of the group
<f>/Dn G © - ®*. Since (b* is a Fischer class, we therefore have L e tb*. Since
<(n, l)>Pg/Pg clearly centralizes L and since 91 s ©*, the group L«(a, l)>Pg/Pg)
belongs to n0S* = ®*; thus its normal subgroup (1 x Gg0 о)<(1, a)>Pg/Pg =
(1 x G)Pg/Pg belongs to s„®* = (b*. Consequently G/Gg s (1 x G)/((l x G)n Pg)s
(1 x G)Dg/Dg belongs to ®*, and therefore G e g о (b*, contrary to the choice of G.
This contradiction shows that (g о ®)* s g о ®*, and therefore equality holds.
□
The next result shows that if g is a Q-closed Fitting class, then g* inherits a measure
of Q-closure.
(1.29) Theorem (Doerk and Porta [1]). Let g and ® be Fitting classes, and assume
that g is Q-closed. If G eg,, then G/Gff, e g,.
Proof. By (1.25) we have g = g* = Qg*. Therefore g* s ® О g* S (® О g*)* =
(® О g„)* by (1.26)(a). Therefore by (1.18) we have (g*)„ S ((® О g,)*),. But
(g*)„ = g, and furthermore ((® < g,)*)« S ® О g, by (1.15). Hence g, S ® О g,
□
The argument in the proof of Theorem 1.29 seems to show more than is claimed,
namely that if g is closed under radical factor groups, then g, is also closed under
radical factor groups. But we do not know an example of a Fitting class g with
Qg g g* that is closed under radical factor groups.
(1.30) Corollary (Cossey [2]). Let % be a Fitting class. If Ge G„ then G/Gg e
This completes our discussion of the Lockett section in a general framework. In
the remaining results of this section injectors play at least an implicit role, and from
this point on we therefore confine our attention to the universe <Z. The first topic
requiring the existence of injectors concerns the behaviour of strong containment
within a given Lockett section. It turns out that there it coincides with the ordinary
partial order of inclusion between classes.
690
X. Fitting classes—the Lockett section
(1.31) Theorem (Lockett [1]). Let g and (5 be Fitting classes of finite soluble groups
in the same Lockett section. Then
g « ffi if and only if g £ ffi.
Proof. If 5 « ffi, obviously it is always true that g £ ffi. Assume that g* = ffi* and
that g £ ©. We suppose that g is not strongly contained in © and derive a contradic-
tion. Let G be a group of minimal order such that a ©-injector V of G contains no
g-injector of G. Let E be a Sylow system of G reducing into F, and V an g-injector
of G into which E reduces; set D = NGfL). Let Д' be a proper normal subgroup of G
such that G/N e 91, and put Uo = V n N, Vo = V n N. Then Uo and Vo are respec-
tively g- and ©-injectors of N into which X n N reduces. Because | <V| < j G |, the group
Vo contains an g-injector of N and therefore contains Uo by 1,6.6. Since by hypothesis
and (1.8)(a) we have ffi £ g* £ g8I, we conclude that Uo = (F0)R char f'o. Since U
normalizes U r\N = Vo and V normalizes Fo, it follows that NC(UO) contains U and
V. Therefore, if |<VC(L'O)| < |G|, by IX, 1.5(c) the choice of G implies that V < F, a
contradiction. Thus we can suppose that Uo < G. In this case by IX, 3.16 we have
U < DU0. But Uo < V, and D < Nc(f') by I, 6.8; consequently V < UV < G. By IX,
1.5(c) the subgroup U is an g-injector of UV and therefore contains Fs. Since V e g*,
we have [И L'] < FR by (1.3). Therefore [F, G] < U, and consequently U < GF. But
then UV g No© = ffi, and from the ©-maximality of V we derive the desired contra-
diction that U < V. □
With the help of IX, 1.5(c) and (1.3) we obtain the following consequence of the
preceding theorem,
(1,32) Corollary. Let g £ ffi with g* = ©*. If V is a <S-injector of a group G, then Fs
is an ^-injector of G.
For Lockett classes not only radicals but also injectors ‘respect’ direct products.
(1.33) Theorem (Lockett [4]). Let g be a Lockett class. For i = 1, 2 let Gt e S, and
let V be an ^-injector of Gt x G2. Then I’= (l’n GJ x (l'nG2); in particular,
F = Fj x V2> where V, 6 InjR(G,) for i = 1, 2, and every subgroup of this form is an
^-injector of Gt x Gz.
Procf. Let F; = V n Gj, and put D = Fj x F2. Clearly D < V e g. We aim to show
that D = F. Suppose not, and let E/D be a composition factor of V/D. Let £, denote
the projection of £ onto G, for i = 1,2, and if e,, e j are elements of £,, choose elements
e2 and ej in £2 such that (e,, e2) and (ej, ej) are in £. Since E/D is abelian, D contains
[ten e2), (ej, ej)] = ([e,, ej], *), and therefore [et, ej] e F,. Consequently £2/Gj is
abelian. Similarly £2/F2 is abelian, and hence so is (£t x £2 )/£>. Because £ 6 s„( F) £
g, it follows that £ <(£t x £2)R. But (£, x £2)R = (£j)s x (£2)R by (1.9), and
(£,-)s — Fj because Fj, as an g-injector of Gh is g-maximal in G; for i = 1,2. Therefore
£ < Fj x F2 = D. This contradiction shows that D = F. The remaining conclusions
follow from IX, 1.3(a) and the conjugacy of injectors. □
1. The definition and basic properties of the Lockett section 691
(1.34) Proposition (Hauck [1]). Let g and®be Fitting classes of finite soluble groups.
If 5 « (5, then g* « (5*.
Proof. Let V be a ®--injector of a group G, let V be an ^--injector of G, and assume
that G has a Hall system X which reduces into both U and V By (1.32) the radical
is an g-injector of G, and since g « © « ©*, we have I7g < И By (1.33) the
subgroup V x V is a ©--injector of G x G. and it follows from (132) that (U x Lj-
is an g-injector of G x G. Therefore V x V > (V x G)B =(Ug x С'й)<(и, iG1)-. и e (J>
by (1.2), and we conclude that if u e 17, then и eV. Hence U < Ц and consequently
g*«®*. 4 □
Remark. If g « ®, it does not necessarily follow that g* « 6„. For example, we have
91« G but 91* (= 91) is not strongly contained in G* (see Exercise 6).
Next we aim to show that Lockett’s star operation commutes with the Fitting class
map W ) described in IX, 1.14. The proof we offer depends on the behaviour of the
star operation with respect to classes of the form g f ® described in Construction В
of Chapter IX, Section 2. To analyse this situation, the following lemma about the
boundaries of a Lockett section will be helpful.
(1.35) Lemma (Hauck [6]). With G as the universe, let g be a Lockett class, and let
3E be an <s„, off-closed class. If 3E £ g, then
X n b(g b(g) = 3E n (Q {b(S): ® e Locksec(g))) # 0.
Proof. If G e b(g*) r> b(g), then G £ g, and G has a unique maximal normal sub-
group, which belongs to g*. Therefore, if g* G ® G g* = g, we have G e b(6), and
it is then clear that b(g*)r>b(g) = Q {b(S): ® e Locksec(g)}. It remains to show
that the class 3E r> b(g*) nb(g) is not empty. Let G be a group of minimal order in
3E\g. Then, as usual, Gg is the unique maximal normal subgroup of G, and jG : G~ j =
p for some prime p. Since Gg/Gg, < Z(G/Gg.) by (1.21), it follows that G/Gg> is a
cyclic p-group. Let
L = (Gg. x Gg.)<(p, g~l): g e Gf,
and observe that L < G x G and L e <s„ d„>3£ = X By (1.5) we have (G x G)g. < L
and so Lg, = (G x G)g.. Since by hypothesis g = g*, from (1.9) we conclude that
Lg = Ln(G x G)g = Lr>(Gg x Gg) = (Gg. x Gg.)<(p, g '): g e Gg> = (G x G)g
by (1.5), and therefore Ls = LB,. Evidently |L: Lg| = |G: Gg| = p, and by A, 14.15
it follows that L has a subnormal single-headed subgroup К such that KLg - L.
Then К e s„3E = X and clearly К ф g. Thus we have К g = К n Ls = К n LB. = Kg.
and | К : К g| = p; consequently К e 3E r> b(g) r> b(g*). □
(1.36) Lemma (Beidleman and Hauck [1]). Let g and 6 be Fitting classes such that
the class g f ® is also a Fitting class (in the universe G). Assume furtherthat the class
g f ®* is unclosed. Then (g f ®J* = 8 f ©*. in Particular, g f © is a Lockett
class.
692
X. Fitting classes—the Lockett section
Proof. We write § = 8 f © and first show that f>* £ 8 f ©*• Let G e £>*, and let R
denote the ^-radical of G. By (1.2) we have
(l.i/) (G x G)6 = (R x R)<(y, g~‘): g e Gf
Let V be an 8‘>r'jector of G, and let W be an 8-injector of G x G containing
V x V. Then clearly V x V < W, and since W'n(G x G)6 e ffi, it follows that
(V x F)n(G x G)6e ©. Let veV; then from (Li/) we see that (», »-1)e
(V x F)n(G x G)8 < (P x F)<6 < (F x F)ffi- = Fffi. x Va- by (1.9). Therefore V =
Fffi> e ffi*, and we have G e 8 f ©*, which proves the asserted inclusion.
We use an argument by contradiction to prove that reverse inclusion: 8 f ©* £
f>*. If this inclusion does not hold, then by (1.35) there exists a group G in
(8 f ®*) r' h(§) n b(S*)- Let M denote the subgroup G6 = G6., the unique maximal
normal subgroup of G, and let V e Inj S(G). Since G f § = 81 ®> it follows that V f ©,
and therefore that V f M; furthermore, V e ffi* since G e 8 f ©*• Let W be an
8-injector of G x G containing V x V. Then V x V < W, and (F x F)ffi(M x M) <
G x G. Since the subgroup (M x Л1) r> W is an 8-injector of M x M e f>, it lies in
®, and therefore the subgroup (F x V)a(M x M) r\ W = (F x F)ffi((M x M) r> FF)
lies in s„ffi = ® because (F x F)ffi and (M x M) r-, W are both normal ©-subgroups
of IF. Consequently (F x F)ffi(M x M)e 8f © = and we then obtain (F x F)e <
(G x G)6 < (G x G)8« = Gg. x G6« = M x M by (1.9). But (F x F)ffi is subdirect in
the ©*-group F x F, and so F < M. This contradiction proves that 8 f ®* — $>*
□
(1.37) Theorem (Brison [2]). Let Л he a Fitting class of finite soluble groups, and let
л P. Then
L„(8)* = MB*).
Proof. Write £ = L„(8). The first step in the proof is to show that if 8 is a Lockett
class, then £ is also a Lockett class, and this we prove by contradiction, choosing a
group G of minimal order in £*\£. Let M = Gs. Then M is the unique maximal
normal subgroup of G, and |G: M] = p for some pen because £S„. = £ by IX,
1.15(b). By (1.2) we have
(G x G)e = (M x MK(g, g~l): g e Gf
and consequently |(G x G)c: M x M| = p. Let F be an 8-injector of G. Since F r> M
is an 8-injector of M, the index ]M: F r> M\ is a n'-number, and so |G : F| is a
n'-number if F £ M; but this gives G e £, a contradiction. Therefore F < M. Because
8 is supposed to be a Lockett class, by (1.33) the subgroup F x F is an 8-injector of
G x G, and hence of (G x G)s. But then |(G x G)v: M x M\ divides the л'-n umber
|(G x G)s: F x F| and we have the desired contradiction. Therefore £ = £*.
We now prove the theorem in the general case. First note that L„(8) £ L„(8*) by
(1.32), and so by (1.8)(b) we have L„(8)* £ MB*)*- Since 8* is a Lockett class, we
deduce from the first paragraph of this proof the fact that L„(8*)* = MB*), and
1. The definition and basic properties of the Lockett section
693
therefore conclude that L„(g)* £ L.(g*). To prove the reverse inclusion, we use the
characterization of the operator L,( ) described in IX, 2.2. This gives the following
sequence of relations: e
b«(S*)=g*e„fg*
= (8SJ*tS* by (1.25) and (1.27)(c)
by (1.32)
= b«(8)*-
by (1.36) since gG„ f g* = ge, f
which is a Fitting class by IX, 2.3
□
We now return to the question of properties which are always shared by the Fitting
classes of a given Lockett section. We shall show that normal embedding and
permutability are such properties, and that dominance is not. But first we make the
elementary observation that if a Fitting class g is contained in a Lockett class 3E, then
5* £ ЗЕ* = 3E by (1.8)(b), and therefore all the members of Locksec(g) are contained
in 3E. Thus, in particular, solubility is an ‘invariant’ of Lockett sections.
(1.38) Theorem (Doerk and Porta [1]). Let Ti be a normally embedded Fitting class
cf finite soluble groups. If ® e Locksec(S), then © is normally embedded.
Proof. First we show that for a soluble Fitting class 3E and a prime p, we have
(1.0) = (L.fXje.e,,.)* = L„(.V)GpGp..
Because GpGp. is a Fischer class, it follows from (1.28) that LP(3E)GPGP- =
(Lp(3E)GpGp.)*. Then, by setting ffi = G„G„. in (1.27)(c) and noting that ® = ®* =
<Q, еф>©*, we deduce that (Lp(3E)GpGp)* = Lp(X)*GpGp. But LP(3E)* = LP(3E*) by
(1.37), and therefore Equation 1.0 holds.
Now, to prove the statement of the theorem, we repeatedly appeal to the equiva-
lence of criteria (a) and (d) of IX, 3.27. Since g is by hypothesis normally embedded,
we have g « Lp(g)GpGp. for all pe P. Then, appealing to (1.34) and substituting g
for 3E in (1.0) we^obtain g* « (L„(g) ©,«„•)* = Lp(g*)GpGp. for all pe P. Therefore
g* is normally embedded. Thus ©*( = g*) is normally embedded, and consequently
©* « L (®*)G G • for all primes p. Finally, setting Л' = 6 in (1.0), we conclude that
© « &*« for all primes p and hence that ® is normally embedded.
Next we show that the property of permutability is another invariant of Lockett
sections.
(1.39) Theorem (Doerk and Porta [1]). Let g be a permutable Fitting class of finite
soluble groups. If&e Locksecffi), then 6 is permutable.
694 X. Fitting classes—the Lockett section
Proof. In this case we use the criterion for permutability described in IX, 3.26. Let
n c p. Since g is permutable, we have g « L„(g), and by (1.34) and (1.37) also
g* « L„(g(* = L„(g*). Therefore g*( = ®*) is permutable, and consequently ® «
®* « /.„(©*). By IX, 3.26 once more, it will suffice to show that ® « L„(S).
Let IF* be an L„(®*)-injector of a group G. Then W* contains a Hall n'-subgroup,
G„. say, of G by IX, 1.16. Let W denote the L„(®)-radical of W*. By (1.37) we have
L„(ffi*) = /.„(©)*, and so IV is an L„(6)-injector of G by (1.32). Since L,(S) =
L,(ffi)S,., and because W*/W is abelian, it follows that W*/W is a л-group and that
G,- e HallK(IK). Let V be a S-injector of W*. Then U is a ©-injector of G because
© « /.„(©*). Since U W is a ©-injector of W e L„(ffi), we have W = (U r> IL)G„.,
and therefore UG„. = U(U n IK)G„. = UW, which is a subgroup of G. However, U
is a ®-injector of UG„-, and so we have UW = UG„. e L„(S); therefore, because W
is /.„(©(-maximal in W*, we conclude that U < IK Thus we have shown that an
/.„(©(-injector W of an arbitrary group G contains a ©-injector U of G, and therefore
® « L„(S). □
Finally we consider the question of whether dominance is a property of Lockett
sections. In fact, this question is easily resolved negatively. It follows from IX, 4.3
that if n с P, then S„ is the only class in Locksec(S„) which is dominant in S.
However, Locksec(S„) contains infinitely many Fitting classes when |л| > 2, as we
shall see in Section 5 of this chapter. Even for Fitting classes of full characteristic the
dominance of g* does not necessarily imply the dominance of g. For example, the
class is S-dominant by IX, 4.8 and contains in its Lockett section the Fitting
class Sp О (Sp.)., which is not dominant in S (see Exercise 7 below). In contrast there
is many a Fitting class whose Lockett section consists entirely of S-dominant Fitting
classes; for example, any Fitting class of the form g О 91 (g a Fitting class) has this
property (see Exercise 8 below). However, apart from the following theorem, we have
very little positive information about the general behaviour of dominance within a
given Lockett section.
(1.40) Theorem (Hauck—unpublished). If a Fitting class g is dominant in S, then g*
is also dominant in S.
Proof. Let G be a finite soluble group, and let GB. < W < G with W e g*. Further,
let V be an g*-injector of G. It will suffice to show that W is contained in a conjugate
of V. By (1.9) we have (G x G)B. = Gs. x Gs., and it follows that (G x G)s <
(W x W)g. Since g is dominant, there is an g-injector of G x G which contains
(W x W)~. But by (1.32) and (1.33) the subgroup (V x I')B is an g-injector of G x G,
and so there are elements g, he G such that (W x IK)5 < (V x < T® x Vh.
By (1.2) we have <(w, w"1): we W) <(W x IK)g, and it therefore follows that
W < Vе. □
Open Questions. Let g and ® be Fitting classes in the same Lockett section.
1- If g ® and g is dominant in S, does it follow that ® is dominant in S?
2. If g and ® are dominant in S, is g n ® necessarily dominant in S? This second
1. The definition and basic properties of the Lockett section 695
question would have a positive answer if the following question could be settled
affirmatively.
3. Assume that 5 is dominant in G and that GB < W <, G with W e g Let X be
a Hall system of G which reduces into W, and let V be the g-injector of G into which
X reduces. Does it follow that W < VI
Concluding Remarks. The Lockett section has a natural dual in the theory of forma-
tions. This has been investigated by Doerk and Hawkes [2] and further explored by
Schmieden [1] and Torres [1]. Although the correspondence between the two
situations turns out to be close, the dual theory is without interest in a soluble universe
because Theorem IV, 1.18 implies that the Lockett section of a formation of soluble
groups is trivial. Nevertheless, there do exist insoluble formations with a non-trivial
Lockett section, for example those described in Exercise 12(c) below, and so in the
larger universe G the dual theory merits study. The basic facts about ‘Lockett sections’
of formations are summarized in Exercises 10 and 11 below, while the developments
due to Schmieden are discussed at the end of Section 4 of this chapter.
Exercises
1. Let g be a Fitting class. If G e G and V e Inj B(G), define a map p = p(G, V) from
the lattice N(G) of normal subgroups of G to N(V) by setting p(N} = N nV.
Prove that p(G, V) is a lattice homomorphism for all G e G if and only if
g n G = G„ for some л £ p. (Compare with IV, 5.3)
2. Show that the following assertion is false: If g is a Fitting class and if V is an
g-injector of a soluble group G, then VGB. is an g*-injector of G.
3. (Hauck [5]) (a) If Condition l.< (see page 687) is satisfied by the pairs g ? g < (f>
and g £ g* О ffi*, show that (g О ©)* = g* О ©*.
(b) If (g О ®)* is Q-closed, show that g* О ©* £ (g 0 ®)*, and that if ©* is
Q-closed as well, then equality holds.
4. (Hauck [5]) Let g and ® be Fitting classes in G. Prove that
(a) (g О ©,), = (g О ©),,
(b) if 91 £ ® and g, = g n G,, then g О 6, = g, О ©*, and
(c) if g A (1) A 6, then the following statements are equivalent in pairs:
(i) g о ©, £ (g о ©),;
(ii) g, о © £ (g о ©),;
(iii) g, О ©, - (g c ®)*i
(iv) g = ffi = G„ for some prime p.
5. (Brison [1]) Let g and © be Fitting classes such that gn© = (l). Then
(g О ©)* = g* О ©*. „
6. Show that although 91« G, it is not true that 91, « G,. [Hint: Consider Exam-
7.
8.
9.
pie IX, 2.14(b).] _ , , , .
Show that G„ G„. is a dominant Fitting class, that G„ О (Sp)* e Locksec(GpGp-),
and that 5 О (G„.L is not dominant in G. . . . . _
Show that if g is a Fittingclass and if© e Locksec(g9l), then ©isdominamin G.
The following statements about a Fitting class g £ G are equivalent in pairs.
(i) a(g)\h(g) = 0 (for notation, see IX, 3.20);
696
X. Fitting classes—lhe Lockett section
(ii) o(8)\b(8) is the union of a finite set of isomorphism classes;
(iii) g* = S and G/GR is an elementary abelian p-group for some prime p and all
G e S.
10. (Doerk and Hawkes [2]) Let g be a formation of finite groups. Define a class
g° = (G: (G x G)® r> (G x 1) = 1) and show that
(a) g° is a formation,
(b) [G®, Aut(G)] < G®°,
(c) g £ 8° = (g°)° £ 81g, and
(d) any two of the following statements are equivalent:
(i) 5 = 8°;
(ii) (G x G)® = G® x G® for all G e ffi;
(iii) (G x H)® = G® x H® for all G, H e ffi.
11. (Doerk and Hawkes [2]) Let g be a formation, and let g° be the related class
defined above in Exercise 10. Then define
‘Locksec’(g) = {©: ffi is a formation and ffi0 = g0} and
go = {ffi: © e ‘Locksec’ (g)}.
Prove the following assertions:
(a) qr0(G eg: G7, Z(G) = 1) £ (g0)0 = g0 = (g°)0 £ g £ g° = (g0)° £ E„g-
(b) Any two of the following statements are equivalent:
(i) © e ‘Locksec’ (g);
(ii) g0 £ © £ g°;
(iii) ‘Locksec’ (©) = ‘Locksec’ (g).
(c) If g £ S, then ‘Locksec’ (g) = {g}.
12, (Schmieden [1]) Let pe P, and let G be a finite group such that G/Z(G) is a
non-abelian simple group and |Z(G)| = p. Assume further that Z(G) < G' and
that the Schur multiplier of G is a p'-group. Let Z(G) = <z>, and let D denote
the normal subgroup <(z, z)> of G x G. Finally, put g = qr0(G), and let g° and
go be the associated classes defined in the preceding exercises. Prove that:
(a) g° = g and g0 = QR„((G x G)/D);
(b) g0 = g° if and only if G has an automorphism which acts non-trivially on
Z(G);
(c) If n = 5 or n > 8, there exists an extension
1 - Z2 - G - Alt(n) -> 1
with Z(G) < G'. (This group G is called the ‘representation group’, or sometimes
the ‘double cover’, of Alt(n). For more information on these representation
groups the reader is referred to the original article on the subject by Schur [1]
or, perhaps more accessibly, to the book of Karpilovsky [1], Theorem IV, 8.5
on page 183.) Show that for this group G, we have g0 # g° and ‘Locksec’
(8) = {8o> 8°}; in particular (G x G)®° # G®° x G®°.
2. Fitting classes and wreath products 697
2 . Fitting classes and wreath products
For Fitting classes the wreath product is not merely a technical device for the
construction of examples but has a significant part to play in the general theory In
particular, groups of the form G rLreg Zp for p e P may be regarded as the counterparts
for Fitting classes of primitive groups in the Schunck class setting We know, for
example, that a Schunck class § coincides with Gp$j if and only if § contains all
primitive groups G such that G/Socp(G) e the corresponding result for Fitting
classes is Theorem 2.15, which is one of the main objectives of this section. Another
way of expressing this analogy would be to say that regular wreath products by
p-groups are to Fitting classes as modules over Fp are to Schunck classes and
saturated formations, and in this connection the relevant result is Theorem 2.12,
which gives a detailed picture of what can happen to the wreath product G"4i P
when P is a p-group and G is a group in some Fitting class 8- Most of the results in
this section prepare the way for the two theorems just cited, and applications follow
later in Section 3 of this chapter and in Sections 4 and 5 of Chapter XI. Our treatment
of this material is based on the work of Hauck [1] and [6]. Except where otherwise
stated, the universe in this section is (£.
Because the base group of a wreath product is a direct product, it is hardly
surprising in view of (1.9) that Lockett classes play an important part in this inves-
tigation. This is the case in the first result of this section, which depends only on
elementary properties of radicals and wreath products.
(2.1) Proposition (Cossey [2]). Let 8 be a Lockett class, and let G be a finite group.
(a) If G ф g, then (G rLreg H)s = (Gg)" for all НеЧ.
(b) If pis a prime, and if P and G/Gs are non-trivial p-groups, then GgrL„g P £ 3-
Proof, (a) Let W = GrLreg H, and suppose that JPg/lGg)* * 1 Then =
(Cjg = (Ggf by (1.9). Since the wreath product W = (G/GR)Qjreg H is isomorphic
with' IV/tG-)*, it follows that IF has a non-trivial normal subgroup which has trivial
intersection with the base group (G/Gg)11 of W. But by A, 18.8(b) this can only happen
if G/Gg = 1, which is contrary to the hypothesis that G ф 8- Therefore Ws = (Gg)".
(b) Let IV = G rL g P, and suppose that 8 contains the group Gg Qj„g P, which is
obviously isomorphic with (G^P. Since W/W is a p-group by hypothesis, it
follows that (Gg)1,P is a subnormal 8-subgroup of W and, as such, is contained
in But = (Gg)" by Part (a) and so P < G'nP = 1 by definition of a
wreath product. Since this contradicts the hypothesis that P # 1, we conclude that
From Cossey’s results we now deduce two simple consequences. A characteristically
central section is defined in (1.6).
(2 2) Corollary Let 8 be a Fitting class, and let G be a finite group.
(a) If G/G^is not characteristically central in G (in particular, ifG/G* is not abelian),
then (G %, H)R < G" for all НеЧ.
698
X. Fitting classes—the Lockett section
(b) If P and G/G?. are non-trivial p-groups, and if G/G* is not characteristically
central in G, then G~, Qj„g P ф g.
Proof, (a) By the implication: (a)=>(b) of (1.7) we have G ф g*, and so from (2.1)(a)
we conclude that (GQjreg H)s < (Grljrel! H)R. = (GR.)b < G".
(b) Again by (1.7) our hypotheses imply that G ф g*, and so by Part (a) we have
(Gli„g Р)я < G". Since (GQj„g P)/(G~,f is a p-group, the desired conclusion now
follows as in the proof of Proposition 2.1 (b). □
(2.3) Lemma. Let g he a Fitting class, G an arbitrary finite group, and H a nilpotent
group. If G Qjreg H e g, then G" Qjreg H e g for all n e N.
Proof. By A, 18.6 the group G" can be embedded in the direct product
(Gli„g Я)” in such a way that (G")" maps onto (G")". Since H e 91, we conclude that
G" Qjreg H is isomorphic with a subnormal subgroup of (G Qjreg Н)" e Dog = g. Hence
G"Qjrcg H e s„g = g. □
(2.4) Lemma. Let ft be a Fitting class, G a finite group, and H a nilpotent group. If
there exists a natural number m such that Gm Qjreg H e g, then G" rljreg H e g* for all
n e N.
Proof. By (2.3) it will be enough to show that Grljrcg H e g*. By A, 18.6 the group
Gm Qj„g H can be embedded as a subdirect subgroup, S say, of (G Qjreg Hf", and
since H is nilpotent, S is a subnormal g-subgroup of (G Qjreg Hf”. Consequently
((GQjreg H)m)s contains S and is therefore also subdirect in (GQj,cgH)m. Hence
GQj„b H eg* by (1.10). □
(2.5) Lemma. Let % be a Fitting class, let G and H be finite groups, and assume that
H is the product of two normal nilpotent subgroups and M2. Assume further that
there exist natural numbers nt and n2 such that G"1 Qj„g e g for i = 1, 2, and let
n = IcmJnj, n2}. Then Gnfli„g H e g.
Proof. Let m, = |H: M,| for i = 1, 2, and let В denote the base group of the wreath
product G" Qjreg H = BH. Let i e {1, 2}; then by A, 18.8(a) the normal subgroup BMt
of BH is isomorphic with G"m‘ Qjreg Mit which belongs to g by (2.3). Hence BH —
(BM1)(BM2)eNog = g. □
(2.6) Lemma. Let peP and ne N. Further, let g be a Fitting class, let G e g, and
assume that G Qjreg Zp, e g for all natural numbers i < n. If P is a p-group of exponent
at most p", then G Qj„g Peg.
Proof. We proceed by induction on | P|, taking as our starting point the case |P| = p,
when the desired conclusion is a part of the hypotheses. Therefore assume that the
conclusion has already been proved for all groups of order less than |P|. If P is not
cyclic, then P is the product of two proper normal subgroups and M2, each of
exponent at most p". By induction we have G Qjreg Mi e g for i — 1, 2, and therefore
1. Fitting classes and wreath products
699
G Qjreg P e 5 by (2.5). On the other hand, if P is cyclic, then P г Z , for some i < n
and the desired conclusion holds by hypothesis. This completes the induction
(2.7) Theorem (Cossey [2], Hauck [6]). Let ре P, let g be a Fitting class, and let
Ge %. If there exists a non-trivial p-group Po such that G Qj, Po e g, then G Qj P e
g* for all p-group P. "8
Proof By (2.6) it will suffice to show that G %, Zpl e g* for all i > 1, and we achieve
this by an induction argument on i.
The case i = 1: Let Z < Po with |Z| = p, and let G” be the base group of the regular
wreath product G Qj Po. Then by A, 18.8(a) we have
G^IQj^Z = G'ZsnG'P» = GQj„gPoe g.
Consequently, G|Pl,:Z| Qjreg Z e g, and by (2.4) it follows that G Qjteg Z e g*.
The case i > 1: Suppose inductively that we already know that GQjreg Zp, e g* for
1 < j <i — 1. By A, 18.11(c) the wreath product Zpi i is a product of two
normal subgroups of exponent p1-1. Therefore by (2.6) and (2.5), together with the
inductive supposition, we have
Grb„g(Zp..,fb„,Zp)eg*.
But by A, 18.11(a) the group (Z„.-i Qjreg Z„) contains a subnormal subgroup Zpl, of
index m say, and therefore
G" Qj„g Z„. sn G ^e(Zpl- Qj„g Z„).
Thus Gm Qj„g Zp, e g*. and consequently by (2.4) we have G1 ureg Z„, e (g*)* = g*.
(2.8) Lemma. Let G he a group in a Fitting class g, and let nelLIfG Qjreg Z„ e g*,
then G2 rL„g Z„ e g.
Proof. Let Z„ = <2>, and put Я = G%, 4 By hypothesis and (1.2)(b) we have
(H x H)~ = (Яя x H,)<(h, h e H>, and since G" e Dog = g, we conclude that
(H X H)% = (Яд X Нд)<(г, 2-1)>. It is straightforward to verify that the mapping ty,
defined as follows, is a monomorphism from G ’breg Zn to (Я x Я)д.
((Xj, У1),..-, (x„, ув))г' - «Х1.•x”)’• • ’nz
Since evidently ф((С2)'} = (G*)2, it follows that the group G2%g Z„ is normally
embedded by ф into (Я x Н)я and therefore belongs to s.g - g. U
700
X. Fitting classes—the Lockett section
The next result and the remark after it show that the prime 2 has an anomalous
position in this theory.
(2.9) Theorem (Hauck [6]). Let % be a Fitting class, and let Geg. Further, let Po be
a non-trivial p-group for some p e P, and assume that G Qjrcg Po e g*. Then for all
p-groups P we have
(a) G2 Qjreg Peg, and
(b) if p # 2, then even G Qjreg Peg.
Proof, (a) By (2.7) we have G Qj„g Peg* for all p-groups P, and so, in particular,
by (2.8) we have G2 Qj„g Zp„ e g for all n e N. It then follows from (2.6) that G2 Qjreg P e
g for all p-groups P.
(b) Now suppose that p # 2. We proceed to show by induction on i that G Qj„g Zpl e
g, for if this is true. Assertion (b) follows at once from (2.6).
The case i = 1: Let Q be a non-abelian p-group with |Q| = pm. By (2.7) we have
GQjreg Q e g*, and since G e g, it follows from (1.3) that G’ < (GQjreg Q)$. Hence
there exists a subgroup Z of Q such that | Z| = pandG^Z e g.Letq = p"1-1 — IQ:Z[.
Then G’rli„g Zp = GbZ, and so G’Qjreg Zp e g.
Furthermore, since 2 |(q — 1), from (2.3) and Part (a) we also have G’ -lfb„BZpeg,
and because of the decomposition G" = G’ 1 x G, an obvious application of IX, 1.13,
the quasi-R0 lemma, yields G Qjrcg Zp e g.
The case i > 1: Suppose inductively that GQjreg ZpJ e g for all 1 < j < i. Since
.^eBZp is the product of two normal subgroups of exponent at most p‘ ’, it
follows from (2.6) and (2.5) that G Qjreg(Zpl , Qjreg Zp) e g, and because Zp, , Qj„g Zp
contains Zp, as a subnormal subgroup, of index q say, we conclude that G’ Qjreg Zpi e
g. Since q is odd and G2 Qj„g Zpl e g by Part (a), the argument used for the case i = 1
again applies, and we have G Qjreg Zpl e g, as required. □
Remark. If g denotes the Fitting class T>({3}) described in IX, 2.14(b), it is easy to
see that the group Z31|гев Z2, which is isomorphic with Z3 x Sym(3), belongs to the
class g*\gt. Thus, if p = 2 and GQjreg Po e g*, it does not in general follow that
G 1|гев Po e g. However, by strengthening the hypotheses of (2.9), we can obtain
Conclusion (b) even in the case p = 2, as the next result shows.
(2.10) Theorem (Hauck [6]). Let ^bea Fitting class, and let Geg. If G Qj„g Z2i e g
for some j e PJ, then G Qjreg Peg for all 2-groups P.
Proof. By (2.6) it will be enough to show that G Qjreg Z2„ e g for all n e FJ, and this
we do by induction on n.
The case n = 1: If the integer j in the hypotheses is 1, there is nothing to prove.
Therefore suppose that j > 1, and write
D, = Dih(2j+1) = <a, b: a21 = b2 = l,ab = a~'>.
Fitting classes and wreath products
701
Let a denote the permutation representation of Dj on the cosets of <h> and let G"
denote the base group of the wreath product W^G'b.D,. By 2.9(a) we have
G breg Z2 e g, and since j > 1, it follows from (2.3) that G2'"1 Ii Z, e S From A
18.13(d) we know that G\ay G%cgZ2J and that G'faby ~ G2'4 z° and
because IF is the normal product of these two groups, we conclude that W e'Lg = g
PU‘9Xn ~ 1B5'A’,813(b)wehaveG2 x (G,rlireg Z2) G1,<b>sn IVe g, and
hence G breg Z2 e sng - g. But we have shown above that G,+1 Qj„ Z2 e g, and so
by IX, 1.13, the quasi-R0 Iemma, we have G rlireg Z2 e g. "
The case n > 1: By induction assume that GQjreg Z2, e g for all i with 1 < i < n, and
let — D*h(2 ) — (a, b: a2 = b2 = 1, ab = a 1 \ Let r denote the permutation
representation of Dn on the cosets of and let Gb denote the base group of
W = GlitD„. By A, 18.13(a) the subgroup G'°fa2,aby of IF is isomorphic with
G rijreR Dih(2"), and since Dih(2") has exponent 2""1, it follows from (2.6) that
G rUeE Dih(2") e g. Let r = 2" 1 — 1. Since G Qjreg Z2 e g, it follows from (2.3) that
G'rbreg Z2 e g, and hence that G"<b> s G2 x (G'rli,eg Z2) e g. Since IF is the normal
product of G\a2, aby and G *</>>, we therefore have IFei^g = g. Consequently
the group G rlireg Z2„, which by A, 18.13(d) is isomorphic with the normal subgroup
G"<a> of IF, also belongs to g, and the induction step is complete. □
(2.11) Lemma. Let g be a Fitting class, let H, Hu ..., H„e9l, and assume that
a(H) c (J?=1 olHf Let G e g.
(a) If for each ie {1,2,.... nJ there exists a natural number m, such that
Gm‘ rlireg Ht e g*, then G2 Hj„g
(b) G2 rli„g Heft if and only if G Qjrcg H e g*.
Proof, (a) Leta(H) = {р2,..., p,}, and by renumbering the groups Ht and allowing
repetitions if necessary, suppose that each p, divides |H,|. Let I < i < r. Since
Gm‘ rlireg H, e g*, then G’"1'1 Qire, ZP1 e g* with tf = IHJ/Pi- By (2.4) and (2.7) we
then have G Qjreg Zpr e g* for all n e M, and therefore by (2.8) we conclude that
G2 rbrcg Zpj, e g for all n e M. It then follows from (2.6) that G2 Op.(H) e g, and
consequently G2 rbreg H e g by (2.5).
(b) If G2 Qjreg H e g, then G 4jreg H e g* by (2.4). Conversely, if G H e g*, then
G2 rlircg H e g by Part (a) of this Iemma. □
The following theorem describes the location of the wreath product Gnrb„gH in
relation to a Fitting class g, when G belongs to g and H is a nilpotent group.
(2.12) Theorem (Hauck [6]). Let g be a Fitting class, let Ge ft, and let He 9b Then
exactly one of the following cases obtains*.
(i) G"%„gH^g* for oIlneN;
(ii) G2" rli„8 H e g and G2" 1 Htftfor all n e N,
(iii) Gn He % for alined. . .
Furthermore, Case (ii) cannot arise unless O2(H} is a non-trivial cyclic group.
Remark. Case (ii) of this theorem can actually arise, for example when G = Z3,
H = Z2 and g = S*.
702
X. Fitting classes—the Lockett section
Proof. Suppose that 6’ 1i„B H e g* for some m e PJ. Then by (2.4) we have
G" Qj„b H e g* for all n e PJ, and consequently G2" Qjreg H e g for all n e PJ by (2.11 )(b).
For any n e PJ the assertion that G2"-11ireB He is therefore equivalent to the
assertion that G 1ircB H e g by IX, 1.13, the quasi-R0 Iemma. Therefore if Case (i) does
not hold, it follows that either G rbreg H $ g and Case (ii) holds, or G 1ireB H e g and
Case (iii) holds.
Now suppose that O2(H) = 1. If H = 1, then obviously Case (iii) holds. Therefore
letpbeaprime divisor of|H|. Suppose that Case(i) does not hold. Then G rlircB H e g*
by (2.4). If t denotes the index |H: OP(H)|, then G‘ 1i„B Op(H) e s„g* = g*, and con-
sequently G 1ireg Op(H) e g* by (2.4). Because p is odd, we can then conclude from
(2.9) that G 1ireg Op(H) e g, and hence from (2.5) that G 1ireB H e g. Thus Case (iii)
obtains.
Finally suppose that O2(H) is not cyclic, and write O2(H) = MtM2, where and
M2 are normal subgroups of index 2 in O2(H). Set t = |О2(Я)| and t' = |O2.(H)|. If
Case (i) does not hold, then G rb„B H e g*, and therefore G' rlirce O2.(H) e g*. By (2.4),
however, this implies that the group G 1i„B O2 (H) belongs to g*, and hence to g by
the preceding paragraph of this proof. Furthermore, G' 4eg O2(H) e g*, and con-
sequently Glireg O2(H) e g* by (2.4). From (2.7) we have G 1ireB M, e g*, and there-
fore G21ireg Mj e g by (2.11)(b) for i= 1, 2. Let G" denote the base group of
Grb„g O2(H). Since |О2(Я): MJ = 2, we have CM, = G21|гев Mf e g for i = 1, 2,
and therefore G Qjrei O2(H) e Nog = g. As already shown, G ^lircB O2 (H) e g, and so
finally from (2.5) we conclude that G 1ireB H e g. □
The next theorem shows, in particular, that if g is a Lockett class with the property
that G Qjreg Zp e g for all Geg, then gSp = g. This is not true for a general Fitting
class, as the example g = S* and p # 2 shows; for by (2.16) below we have G rlireB Zp e
<5„ for all G e <5*, whereas if q is a prime congruent to 1 modulo p, then the ©-normal
Fitting class T>({qJ) described in Example IX, 2.14(a) shows that E(p/q) e <5*Sp\£t.
(2.13) Theorem (Hauck [6]). Let g be a Fitting class contained in a Lockett class f),
and let pbea prime. Assume that for each Geg, there exists a natural number n and
a non-trivial p-group P such that G" Qjreg P e Sy. Then g* <Zp c Sy.
Proof. Suppose, by way of contradiction, that the conclusion of the theorem is false,
and let G be a group of minimal order in g*<5p\J>. By a familiar argument G is
single-headed, and since g* £ §’ = Sy, we have G ф g*. Hence G/G~. is a non-trivial
p-group. Because the section GB./G5 is central in G by (1.21), (a)=>(c), the quotient
group G/GB is therefore nilpotent and consequently, as a single-headed group, cyclic
of p-power order. By hypothesis, there exists a p-group P 1 such that (GB)n rlircB P e
Sy- In the wreath product G" rbrcg P, the subgroup ((G^)")11 P is subnormal and belongs
to § because ((G^P =? (GB)nQjrrg P. Therefore P < (GnrbreB P)6. On the other
hand, by (2.1)(a) we have (G" rb„B P)6 = (Gg)11 = (G#)6 because G$ S> and S> is a
Lockett class. Thus P < G", which contradicts the fact that P # 1. Hence our initial
supposition is false and g* Sp E S>. □
The following consequence of this theorem will be applied in the proof of Theorem
3.13 in the next section.
2. Fitting classes and wreath products
703
5 and !et П be a set of primes that
3 SP 8 for all pen. Then for all finite subsets {p„...,p ) с л there exists a
group G e g such that G %ct Pi ф g* for all i = I,...% and
Proof. By (2.13) there exist groups G; e g with Gf rbre|! P, ф g* for all p-groups P, * I
and for i - I, .... tn. Then by induction on m it follows from (1.24), the sharpened
version of the quasi-R0 lemma, that
(G, x - xGJ^P^g*
for all p-groups Pf 1 and i = 1,..., m. q
The following theorem is a generalization of a criterion for a Fitting class to be normal
which was first proved by Blessenohl and Gaschiitz [1] and Makan [1].
(2.15) Theorem (Hauck [6]). Let g be a Fitting class and p a prime. Any two of the
following statements are equivalent.
(a) g* = g*e„;
(b) G2rlireg Peg for all G eg and all p-groups P;
(c) For each group Geg there exists a non-trivial p-group P and a natural number
n = n(G, P) such that G"rbteg P e g*.
Furthermore, if p A 2, the group G21ireg P in Statement (b) can be replaced by G 1i„g P.
Proof. The implication: (b)=>(c) is obvious, and the implication: (c)=>(a) follows
directly from (2.13). It therefore remains to prove that (a)=>(b). Let Geg and let P
be a p-group. Since g* = g*®p, we have G rlireg P e g*. If P = 1, then there is nothing
to prove. If P 1, then it follows from (2.9) that G2 rlireg Peg, and, if p 2, even
that СЪ,.,Peg. □
(2.16) Corollary (Blessenohl and Gaschiitz [1], Makan [1]). Let % be a Fitting class
of soluble groups. Then g is normal in S if and only if whenever g contains a group
G, it also contains the wreath products G2 rlircg Z2 and G rbrtg Zp for all odd primes p.
Exercises
1. (Hauck [6]) Let g be a Fitting class, let G„ G2 e g, and let Я e 91. For i = 1, 2
let Wj denote the wreath product G; rbrcg H. and for m, n e PJ put lVm.n =
(Gf x G;)rb„ H. Prove the following statements:
(a) If Wi e g for i=l,2, then Wm,„ e g for all m, n e N.
(b) If e g and W2 e g*\g, then lVm.„ e g if and only if n is even.
(c) If Wj e g*\g for i — 1, 2, then one of the following occurs:
(i) Wm „ e g if and only if m - n is even;
(ii) IFm „ e g if and only if m and n are both even.
(d) If WФ g* for i = 1, 2, then IF„.n Ф g for all m, n e PJ.
2. Show that in Part (c) of the previous exercise Cases (i) and (n) can both actually
occur.
704
X. Fitting classes—the Lockett section
3. (Hauck [6]) Let 5 be a Fitting class, and let Ht,H„ e 91. Further, let Gt,...,
G„ be groups in 5 such that G, rlircg ф 8 for i = 1,..., n.
(a) Show the existence of non-negative integers kt, k„ such that the group
L = Gi' x • x G*" satisfies L rbrcg ф 8 for all i = 1,.... и and if О2(Н7) is either
trivial or non-cyclic for some j e {1..n}, then even LH2 ф 8*-
(b) If 8 = 8* or the Sylow 2-subgroups of all Ht are either trivial or non-cyclic,
then the statement in Part (a) is fulfilled for all natural numbers kt, ...,k„.
4. (Hauck [1]) Call a Fitting class 8 repellent (originally ‘abstofiend’) if whenever
G e 8. P e P, and G Qj„g Zp e 8, then G e Qp.
(a) If Ge <5p.<5p\9l with Op.(G) abelian, show that G is not contained in any
repellent Fitting class.
(b) Let 8 (£®) be a Fitting class of characteristic л with the property that the
class 8 n 9l2 is Q-closed. Show that the following statements are equivalent in
pairs:
(i) 8 is repellent;
(ii) E(p/q) ф 8 for all primes p, q such that p + q;
(iii) В =
(c) Let 8 = <R0, s„, n0>8 S 9l2. Prove the equivalence of each pair of the following
statements:
(i) 8 is repellent;
(ii) E(p/q) ф 8 for all primes p, q such that p q\
(iii) в e 3 = (G: Soc(G) < Z(G)).
Remark. A complete determination of repellent Fitting classes is not known.
5. (Hauck [1]) Suppose that for each л s P a Fitting class 8„ is given. Let p £ P,
and put 8 = Sp n p 8„®,®я-- Show that for each p e P and for each G e 8
such that Op (G) = G 1, there exists an ne N such that G”rbreg Zp e 8-
6. (Hauck [1]) Show that any two of the following statements about a Fitting class
8 of soluble groups are equivalent:
(a) If G e 8 and p is a prime such that G" rlircg Zp e 8 for all n e N, then for each
prime q p there exists an m e M such that Gm rbreg Z, e 8;
(b) There exists a prime p such that for each prime q p and for each G e 8> there
exists an m e PJ such that Gm rlireg Z, e 8-
(c) There exists a prime p such that 8*®p- = 8*
7. (Hauck [1]) Let л с P, and let 8 be a Fitting class in 6. Prove that <5„8* = ® if
and only if for each p e P and for each G e 8 such that O„(G) — 1 the group
G2 rbreg Zp also belongs to 8-
8. (Hauck—unpublished) Let 8 and © be Fitting classes, and assume that 8* 8*®P
for all primes p. Prove that the dominance of the product 8 О ® implies the
dominance of ©. (See Chapter IX, Theorem 4.6.)
3. Normal Fitting classes
We recall from Chapter IX, Definition 2.13(b) that a Fitting class 9) is said to be
normal in a class X if (1) ^4 ')) £ X and Gv is '((-maximal in G for all G in JE. We
remarked that X-normal Fitting classes can be obtained as kernels of X-Fitting
3. Normal Fitting classes
705
pairs when X is a Fitting class, and in IX, 2.14 we described some non-trivial
examples of this phenomenon. (More examples of normal Fitting classes derived
from Fitting pairs will be presented in Section 5 later in this chapter.)
The first investigation of this concept, by Blessenohl and Gaschiitz [1] was
devoted entirely to ©-normal Fitting classes, and the theme was taken up by other
authors in the soluble context. H. Laue [1] was the first to extend the study to the
universe of all finite groups. We shall begin our treatment of this material with
some general remarks about X-normality when X is an arbitrary Fitting class satisfy-
ing X2 = X ± (1). We shall then specialize X to © and to 6 in turn and in these cases
aim to give a detailed account of the present state of knowledge. Except where
otherwise indicated, the universe throughout this section will be 6.
Our requirement that X2 = X in our study of X-normality is somewhat restric-
tive and, in the soluble case at least, confines attention to the classes X = £„,
л e P, as our first lemma shows. Less restrictive frameworks for the study of normal-
ity have been considered by Beidleman and other authors (see the Concluding
Remarks of this section).
(3.1) Lemma. Let X be a Fitting class of characteristic n such that X2 = X. Then
X is a Lockett class and X n £ = £„.
Proof. Let G e X*. Since G/Gj e 91 by (1.3), it follows from IX, 1.8 and (1.20) that
G/Gj e 9l„ £ X. Hence G e X2 = X, and so X is a Lockett class. Furthermore, we have
9l„ £ X n £ £ £„, and because X2 = X, evidently (9l„)г £ X for r = 1,2,.... But then
£„ = U“=1 (9V S X, and therefore X n £ = ©„. □
Remark. If 3 is an arbitrary class of finite simple groups, it is easy to see that the
class X = X(3) comprising all finite groups whose composition factors belong to 3
is a Fitting formation satisfying X2 = X. Moreover, it is also not difficult to verify
that any <q, s„>-closed class 9) satisfying 9)2 = 9) must have the form 9) = X(3) for
a suitable class 3 of finite simple groups. But not all Fitting classes g satisfying
g2 = g have this form; for example, it is easy to see that the class
g = (G e (E: G has no subnormal subgroups isomorphic with SL(2, 5) or Alt(5))
is a Fitting class with g2 = g which does not have this form. Characterizations of
Fitting classes g satisfying g2 = g, and also of those satisfying g 0 g = g, can be
found in VI, 3.11(d).
(3.2) Lemma. Let X be a class of finite groups satisfying X - X #(1),and let g be
an X-normal Fitting class. Then g contains all simple groups in X, and if Char(X) - n,
then 9l„ £ g; in particular, 91 is contained in both £» and
О Г Т7- . u - »r - n X because X2 = X. By definition of X-normality, the
Proof. First observe that X = D„* Because л j H/_rn, r
class ft contains a group G I. Let J be a simple group in X, and set W J U G.
Thin И' e to X) X = X and Ice G 6 g, it follows that 1. By A 18.8(b) we have
I # И4, n J’e g and since any minimal subnormal subgroup of J is isomorphic
706
X. Fitting classes—the Lockett section
with J by the Jordan-Holder theorem, it follows that J e s„8 = 8- If p e n, we then
conclude that Zp e 8, and hence that 91, e 8 by Iх. 1-8. □
The next result shows that the set of X-normal Fitting classes is a union of Lockett
sections.
(3.3) Theorem (H. Laue [1]). Let X be a non-trivial Lockett class. A Fitting class 8
is X-normal if and only if 8* is X-normal.
Proof. Let 8 be a Fitting class, and first suppose that 8* is normal in X. Let G e X,
and let Gg < H < G with Я e 8- By (1.21) we have
(3.a) Gs./Gg < Z(G/Gg),
and consequently H < HG%.. Hence HG^ e n„8* = 8*> and from the X-normality
of 8* we conclude that Я < Gg«.Thenby(3.a)wehaveW < G, and therefore Я < Gg,
which proves that 8 is X-normal.
Now suppose that 8 is normal in X, let Gel, and let Gg« < Я < G with Я e 8*.
Since by hypothesis X is a Lockett class, by (1.8)(b) we have 8* — £• Since Gg < Яй,
Lemma 1.5 yields the following:
(3./J) (Я X Я)д = (Яд X Яд)<(р, g-'f. дену
> (Gg х Gg)<(p, д~1): д е Gg.>
= (G х G)g.
But 8 is normal in X by supposition and G x Ge D0X = X; hence (Я x Я)д =
(G x G)g. Then, by considering the projection of (Я x Я)д onto the first component
of G x G, we conclude from (3./J) that Я < Gg«. □
(3.4) Definitions. Let X be an arbitrary class of finite groups.
(a) (The class Xй). For an R0-closed class 8 we define an associated class Xй as
follows:
Xs = (G8: G e X).
(b) (The class X/8). For an No-closed class 8 we define an associated class X/8
thus:
X/8 = (G/Gg: GeX).
Observe that if 8 e Locksec(X), then X/8 S 91 by (1.3).
(3.5) Theorem (H. Laue [1]). Let Xbea Fitting class satisfying X2 — X (1), and let
8 be an X-normal Fitting class. Then 8 e Locksec(X) if and only if X f X/8-
3. Normal Fitting classes
707
Proof. If g 6 Locksec(X), then X/g e Щ by (1.3), and then obviously X X/g
Now suppose that g ф Locksec(X), and, by way of contradiction, sup^se further
that there exists a group X e X\(X/g). Since g* c X* = X by (3.1), by (1.7) there exists
a group G in D„X - X such that G/GB is not characteristically central in G. Let W =
G rbrt8 X and note that W e (d0X)X = X. Then by (2.2)(a) we have < G\ and
because X ф X/g, it follows that ITn < (IV5X)B. Thus W is an X-group whose g-
radical is not g-maximal, which contradicts the X-normality of g. □
We now turn our attention to G-normal Fitting classes. These were the first to be
studied and are often referred to in the literature simply as ‘normal’ Fitting classes.
One of our first goals is to show (in Theorem 3.7, (a) -=>(c)) that there exists only one
Lockett section of G-normal Fitting classes, namely Locksec(G) itself. This is in
striking contrast to the situation for G-normal Fitting classes, as we shall see later in
the section.
(3.6) Lemma (Blessenohl and Gaschiitz [1]). Let nbea non-empty set of primes, let
g be an <Z„-normal Fitting class, and let pen. Then for each G in g there exists a
natural number m = m(G, p) such that for all n e M,
(3.T) G”mrb„g Zp 6 g.
Proof. Suppose, by way of contradiction, that the lemma is false, and choose a pair
(G, p) for which it fails with | G| as small as possible. Since Zp e S g by (3.2), clearly
G / 1. Let M be a maximal normal subgroup of G, of index q say, and first suppose
that p = q. Then by the minimality of |G|, there exists anm, eM such that
eg
for all n e N. The wreath product W, = IVJn) = G"'”rljtISZf is generated by the
normal subgroup (G’"1")1’ e Dog = g together with the subgroup (JWmi")l,Zps
Л1’"1" 1ire Z 6 g, the latter subgroup being subnormal in Щ because is
a p-group. Hence Wt (n) e Nog = g for all n e M, and so G satisfies Condition 3.y with
m = mr.
Since G is not a counterexample in this case, it follows that p ^q. Let E denote
the primitive group E(q/p) (see B, 12.5), and suppose that | E | = p q. Let Q e Syl,(E),
and let n be an arbitrary natural number. Since Q = Z, and q e a(G) £ n, the choice
of G ensures the existence of a natural number m2, independent of n, such that
(3.3) M^-^Ceg.
Now consider the wreath product И
base group, and put R — (AT'F12'1)t:- B]
belongs to g by (3.<5). Because BQ/R
В e D„g = g, we then conclude th;
maximal in W2, and it follows that В
If N denotes Op(£), the group BN/В ii
'2 = G"2" E e G„ let В = (G'”2")1' denote its
i A, 18.8(a) we have RQ =? rbrce Q, which
is a q-group, RQ is subnormal in BQ, and since
Jt BQ = B(K0 e Nog = g. Thus В is not g-
< (IV2)B because by hypothesis g is G„-normal.
; the unique minimal normal subgroup of W2/B,
708
X. Fitting classes—the Lockett section
and consequently BN < (^z)x- Let P < N with P = Zp. Then BP e s„g = g. How-
ever, with m = m2p'~Iq, it then follows again from A, 18.8(a) that BP s Gmn ^.Z,,
and therefore Equation З.у is satisfied for the group G and the prime p. This contradic-
tion proves that no counterexample exists. □
We are now in a position to prove three important characterizations of ©„-normal
Fitting classes. The first (Statement (b) in the following theorem) is due to Makan
[1]; the second (Statement (c)) is due to Lockett [4]; and the third (Statement (d)),
which historically precedes the other two, comes from Blessenohl and Gaschiitz [1].
(3.7) Theorem. Let n be anon-empty set of primes. Any two of the following statements
about a Fitting class g of finite soluble n-groups are equivalent :
(a) g is normal in ©„;
(b) For each pen and G e g, there exists a natural number n such that G" rbre8 Zp e g;
(c) g* = ©„;
(d) G/Gg is abelian for all G e ©„.
Proof. That Statement (a) implies (b) is an immediate consequence of (3.6). Next we
justify the implication: (b) => (c). If Statement (b) holds, it follows from (2.15), (c) = (a),
that g* = g*£pfor all pen. Since g S £„, we have g* c (£„)* = £„, and therefore
g* = £„. The implication: (c) => (d) follows directly from (1.3), and the final implica-
tion: (d) => (a) to complete the circle is obvious. □
Remarks, (a) The equivalence of Conditions (a) and (c) in the preceding theorem
shows that Locksec(S) is the only Lockett section of ©-normal Fitting classes and,
in particular, that £„ is the smallest ©-normal Fitting class.
(b) With the help of the Lausch-Laue-Pain maps described in Section 5 below, we
shall show in (5.32) that (£„)* £„ if and only if | zr| > 2.
(c) If T denotes the class (G' :G e £), the equivalence of Conditions (c) and (d) of
(3.7) implies that 35 £ £„ and further that <s„, N0>T = <Zt. An example to show that
D is suggested in Exercise 8 below.
The following theorem was first proved in a somewhat weaker form by Blessenohl
and Gaschiitz [1]. It shows in particular that the implication: (a) => (b) of (3.7) holds
with the natural number n equal to 2.
(3.8) Theorem (Hauck [6]). Let g be an <Z-normal Fitting class, and let G e g.
(a) One of the following two statements holds:
(1) G^ He % for all He 91:
(2) If H e 91, then G Qj,^ H eg if and only if either O2(H) = 1 or O2(H) is
non-cyclic.
(b) If G rbreg Z2 e g, then G rlireg H e g for all H e 91, and in any case G2 rliieB H e g
for all H e 91.
Proof, (a) We suppose that Statement (1) of this part of the theorem fails to hold and
deduce that Statement (2) holds.
3. Normal Fitting classes
709
By this supposition there exists anilpotent group Ho such that G^„g Ho * g. Since
G Ho e © 5 , it follows from (2.12) and, in particular, from the final sentence
of its statement that GOi,., O2.(H0) e g, and hence by (2.5) that GO2(Ho) ф g
Now let H denote a nilpotent group with a non-trivial cyclic Sylow 2-subgroup. Since
G b„g O2(Ho) ф g, it follows from (2.10) that G %g O2(H) ф g. If the group G Qj H
were in g, and if t denotes the odd natural number |H:02(H)|, we should have
G' “teg 6 S„g = g, and could then conclude from (2.12) that G Qj O2(H) e g,
which is not the case. Thus we have shown that if G 01„, H e g, then 62(H) is either
trivial or non-cyclic. Conversely, if O2(H) is either trivial or non-cyclic, Case (ii) of
Theorem 2.12 is ruled out, and it then follows from that theorem that G Qj„g H e g.
This completes the proof of Part (а). ""
(b) If GrbrcB Z2 e g, Case (2) of Part (a) fails to hold, and therefore Case (1) obtains.
Since g* = ©, it follows directly from(2.11 )(b) that G2%,,H eg. □
By (1-25) an ©-normal Fitting class which is closed under one of the operations q,
Ro, sf must coincide with ©. As a consequence of (3.8)(b) we obtain the following
stronger result for the closure operation s.
(3.9) Proposition. If g is an S-normal Fitting class, then sg = £.
Proof. Let 1 A G e S, let M <• G, and, proceeding by induction, assume that there
exists a group H e g which contains a subgroupM* s M. Let | G: M\ = p e IP. From
A, 18.9 we know that G is isomorphic with a subgroup of M Tjreg Zp. But the group
M Qj„g Zp is isomorphic with the subgroup (M* x l)’Zp of the group (H x H)riircllZp,
which belongs to g by (3.8)(b). Thus G e sg, and the induction step is complete. □
Proposition 3.9 implies in particular that s£„ = ©, and since we already know from
Examples IX, 2.14(b) that ©„ is a proper subclass of ©, it follows that ©„ is not
s-closed. Nevertheless, the smallest ©-normal Fitting class does display some measure
of subgroup-closure, for we shall show in X, 6.7 below that if g is a normally
embedded Fitting class, and if V is an g-injector of a group G in ©„ then V e ©„
The next theorem contains two further criteria for a Fitting class of soluble groups
to be ©-normal. Criterion (a) is due to Blessenohl and Gaschiitz [1] and Criterion
(b) to Lockett [1].
(3.10) Theorem. Let g be a non-trivial Fitting class cf finite soluble groups. Then each
of the following conditions is both necessary and sufficient for g to be normal in S:
(a) s(©/g) # ©;
(b) Q(©/g) + ©.
Proof. If g is ©-normal, it follows from (3.7), (a) => (d), that <s, Q>(©/g) S 91, and so
both conditions are necessary. .
Now suppose that g is not normal in S, and let H e S. By (3.p, (d)^ (a) there
exists a Zup G in © such that G/GB ф 91. Let W = G%g H, and observe that by
(2.2)(a) we hfve < G". Thus lP/fPB has both a subgroup and a quottent group
710
X. Fitting classes—the Lockett section
isomorphic with H, and it follows that s(©/ft) = © = o(S/ft). Consequently both
conditions are also sufficient. □
Next we touch on the normality of a Fitting class product.
(3.11) Proposition (Cossey [2]). Let ft and & be Fitting classes contained in S. If
either ft or ® is <5-normal, then ft 0 ® is <5-normal.
Proof. We appeal to the equivalence of Conditions (a) and (d) in (3.7). If ft is
©-normal, then for all Ge © we have G' < Gs < Gs ce; therefore ft 0 ® is ©-
normal.
If ® is ©-normal, then G'GR/GR = (G/Gg)' < GR 0 e/GR for all G e S. Therefore
G' < Gso e for all Ge S, and again ft 0 ® is ©-normal. □
It can easily happen that the Fitting product of two non-S-normal Fitting classes is
©-normal. Take, for instance, the Fitting classes Sp and 3P (soluble groups with
p-socle central), neither of which is normal in S; their product Sp 0 3P however, is
clearly S itself. Nevertheless, there is certainly some positive information to be
derived about two Fitting classes whose product is normal in S, the most significant
being Statement (e) of Theorem 3.13 below, and to prove it we need the following
lemma.
(3.12) Lemma (Beidleman and Hauck [1]). Let ft be a Fitting class, and let л be a
set of primes such that ft©, = ft. Then ftr, 2, ft* ©„..
Proof Since (ft,£„f = ft*S„- by (1.27)(c), it follows from IX, 2.3 that ft f (ft*©„)
and ft f (ft*©„)* are both Fitting classes. Hence the hypotheses of (1.36) are satisfied
with ® = ft*S„., and therefore we have
(8t(8.Sn))* = 8t(8*s.-) = S-
Consequently ©* c ftf (ft*S„-), and if GeS, nft, it follows that G e Injg(G)
□
(3.13) Theorem (Hauck [5]). Let ft and ® be Fitting classes of finite soluble groups.
Any two of the following statements are equivalent:
(a) ft 0 ® is normal in ©;
(b) ft 0 ®* is normal in ©;
(c) ft* О ® is normal in S;
(d) ft* 0 ®* = S;
(e) There exists a set л of primes such that ft*©„ = ft* and S„ 0 ®* = S.
Proof. Lemma 1.26(a) states that go® and ft о ®* belong to the same Lockett
section, and since there is only one Lockett section of ©-normal Fitting classes, the
equivalence of Statements (a) and (b) is clear. Lemma 1.26(b) states that ft* о ®* =
(ft* 0 ®)*, and so the equivalence of Statements (c) and (d) is also clear.
3. Normal Fitting classes
711
of^XentX‘h И ‘O.Tplete the P™fit wi" be enough to show the equivalence
of Sta ements (b)and (e). For suppose we know that (b)o(e). If Statement (d) holds,
then it follows that Statement (b) is true with g* in place of g. Since Statement (e)
remains unchanged when g is replaced by g* we can therefore deduce that (d) => (e)
But (e) obviously implies (d), and so our claim is justified. Thus we prove first the
implication
(b) => (e). Let л - {p e P: g G„ = g*}. Then obviously g* G„ = g* We suppose
that G„ 0 ® ± G and derive a contradiction; note that л # P in this case Let L be
a group of minimal order in G\G„ о ®*. Then clearly 0„(L) = 1, and L $ ®*. Let
1 ± H e G, put W = LQjteg H, and observe that O„(lVj = 1. By (2.1 )(a) we have
H7®- = (L®-); and so W/W^ is not abelian. Let {Pi,denote the set of
primes dividing |Soc(H7)|. Since 0„(H7) = 1, we have {p,,..., Pm} c and therefore
8* SPi 8* for i = 1,..., m. By (2.14) there exists a group G e g such that G Qj„g P ф
g* whenever P is a non-trivial p-group for some p e {Pi, • •., pm}. Consider the wreath
product G Qj„g H7, and suppose that (Grljteg H7)g > G“. Then there exists a minimal
normal subgroup N of Wsuch that G < (G IV)g. It follows that G,H':N| N =
G^N e s„g = 8, and consequently by (2.4) we have GQjreg N e g*. But this contra-
dicts the choice of G since N is a p-group for some pe {pj,..., pm}, and therefore
(G Qjreg H7)g = G11. However, in this case we have
(GQj„g ИОДСЧр, И7)-, ® s W/W*,
which is not abelian, and by (3.7), (d) => (a), the Fitting class g о ®* is not normal
in G. This contradiction shows that G„ 0 ®* = G.
It remains to prove the implication
(e) => (b): If л' = 0, then g* = G, and g 0 ®* is G-normal by (3.11). We therefore
suppose that G„ G and aim to show that
(3.e)
(g 0 6*)G„ = G,
for if this is the case, then s(G/(g О ®*)) £ sG„ = G„ G, and it follows from (3.10)
that g 0 ®* is G-normal, thereby completing the proof. Therefore suppose, by way
of contradiction, that the class G\(g О ®*)G„ is non-empty. Then it contains a group
G of minimal order, and by the usual argument G has a unique maximal normal
subgroup and G/G' S Z^ for some pe л” and n e N. By (3.7), (a)=>(d), we have
G < Gs , and hence Gg./(Gg- n Ge.) e G„.. Since g*G„ = g* it follows from (3.12)
that (G,‘. n Ge )/Gg. e G„.. Hence Gg./Gg e G„-. Again because g*G, = g , we
have 0 (G/G^-) = 1 and can therefore conclude that 0„(G/Gg) = 1. Since by assump-
tion g” 0 ®* = S, it follows that G e g 0 ®* £ (g 0 ®*)S,- This is a contradic-
tion. and therefore (3.e) holds. □
We now present two interesting applications ofthe theory of G-normal Fitting dasses
to the structure of a general finite group. They are due to Gaschutz [13], and we give
his original proofs. The first of the two results shows that certain abe ian quotients
of the soluble radical Gs of a group G can be lifted to abelian quotients of G.
712
X. Fitting classes—the Lockett section
(3.14) Theorem (Gaschiitz [13]). Let g be an S-normal Fitting class, and let G be an
arbitrary finite group. Then the section (GSG')/(GSG') of G/G' is isomorphic with
g5/g„.
Proof. Put R = G%, and observe that Gs/R < Z(G/R) by (1.21), (a)=>(c), since g e
Locksec(S). We assert that
(3.0 (Ge/R)n(G/R)' = l.
Suppose that (3.0 is false, and let x be an element of (Gs/R) n (G/R)' of prime
order, p say. Let H/R be a Sylow p-subgroup of G/R. Since R < H n Gs, then
R < NC(H' n Gs). Therefore H'nGe < (H‘ n GS)R < Gs, the latter because G/R is
abelian. Since H is obviously soluble and g is S-normal, by (3.7), (a) => (d), we have
H' < H7„ and so H' n Gs is subnormal in Thus H' n Gs is a subnormal g-
subgroup of Gs, and so H' n Gs < (Gs)g = R. Consequently (H'R/R) n (Ge/R) = 1.
Let v denote the transfer homomorphism с(С/д-.н/Д). and let n = |G: H|. Since x e
Z(G/R), it follows from A, 17.2(c) that r(x) = x"(H/R)'. Since x e (G/R)' < Ker(v), we
have that x" e (H/R)' n (Gs/R) = 1, and hence that x = 1 because (n, |x|) = 1. This
contradiction proves that (3.0 holds and hence that Gs n RG' = R. Thus we obtain
GSG'/RG' = GSRG'/RG' Gs/R. □
(3.15) Corollary. If G is a finite perfect group, then G, e S*.
The second application gives a sufficient condition for the central normal section
Gs/Gs to be complemented.
(3.16) Theorem (Gaschiitz [13]). Let g be an S-normal Fitting class with the property
that S/Ss is elementary abelian for all S e S, and let G be an arbitrary finite group.
Then Gs/Gs is a direct factor of G/G^.
Proof. Let R = Gg, let H/R e Sylp(G/R), and let P/R e Sylp(Gs/R) for some prime p.
Since Ge and H are soluble groups, by hypothesis P/R and H/H^ are elementary
abelian p-groups. Because R is g-maximal in Gs and because R < n Gses„g =
g, we have R = n Gs = Hs n P. Let L/H^ be a complementary subgroup to
the subgroup РНЪ/НЪ of the elementary abelian group H/H%. Then L n P <
H% r. P = R, furthermore PL = PHSL = H, and therefore L/R complements P/R in
H/R. Thus the extension G/R of the abelian normal subgroup Gs/R splits over its
Sylow subgroups. Then by A, 11.1 the central subgroup G^/R is complemented in
G/R and is therefore a direct factor. □
We now move on to G-normal Fitting classes and first describe some examples which
do not lie in Locksec(G). These examples are formulated in terms of Construction D
(Chapter IX, Section 2) and the following hypotheses.
(3.17) Hypotheses. Let £ and '!) be Fitting classes of finite groups such that £ s ']).
For each GeG, let
713
3. Normal Fitting classes
S(G) = (G9/Gx) r> Soc(G/Gx).
Let 3 c J, and define a Baer function / = /3 as follows:
if Je3n9I,
f(J)= d0(J) ifJe3\9I, and
if Je3\3.
By IX, 2.8 the following class:
HS(f3, S) — (G e G: S(G) is /-hypercentral in G)
is a Fitting class. We shall now prove that this class is G-normal (and hence, in
particular, dominant in G). and for this purpose we introduce the following special
notation.
Notation. Let H/K be a chief factor of a finite group G. Then define
Q(H/K) =
CG(H/K) if H/K e ill, and
HCG(H/K) if H/K $ 91.
Thus CG(H/K) is a normal subgroup of G and is the largest subgroup of G which
induces inner automorphisms on H/K (see Chapter IX, Lemma 2.6).
(3.18) Lemma. Let g denote the Fitting class HS(f3, S) of Hypotheses 3.17, and let
G e G. Then
(3-r?)
Gg = {Cg(H/Gx): H/Gx is a minimal normal subgroup of
G/Gx of composition type 3 contained in G9/Gj}.
Proof. Let H/Gx be a minimal normal subgroup of G/Gj contained in G9/Gj and of
composition type 3. Let N < G. Then Nx = N n Gx. and (H n N)/Nx is a direct
product of a (possibly empty) set of minimal normal subgroups of N/Nx of composi-
tion type 3 contained in N$/Nx. If N e g. it follows easily from Proposition A, 4.13(c)
describing the structure of (H/GX}K that N < C'g(H/Gx), and hence that Gs is con-
tained in the right-hand expression of Equation 3.»;.
To prove the reverse inclusion, let N denote the right-hand side of Equation 3.»;,
and let R/Nx be a minimal normal subgroup of N/Nx of composition type 3 contained
in Ni}/Nx. Let T = <KG> < Nv, and let H/Gx be a minimal normal subgroup of G/Gx
contained in TGX/GX T/Nx. Then H/Gx has composition type 3 and is contained
in Gv/Gx. and so by hypothesis N < Clc(H/Gx). Since H/Gx % (H n T)GX/GX 2
(H n T)/Nx, it follows that if RB/NX is a minimal normal subgroup of N/Nx contained
m (H n T)/Nx, then N = C^Ro/NJ- Next we point to the obvious fact that T/Nx is
completely reducible as an N-module and hence that
T/Nx = (R/Nx) x (Rs‘/Nx) x • • • x (Rs‘/Nx)
714
X. Fitting classes—the Lockett section
for suitable g,, g2, , gs e G. By A, 4.9 the minimal normal subgroup R0/Nx of N/Nx
is A'-isomorphic with R9/Nx for some 9 6 {9i,.... 9,}. Thus A = Q(R0//Vj) =
C‘n(R9/Nx), and it follows that Cj,(R/Nj) = № ‘ = N- Hence N eg and N < Gg.
□
(3.19) Lemma. Let g denote the Fitting class HS(fz, S) described in Hypothesis 3.17.
Let H be a subgroup of a group G such that Gg < H < G. Then Hx = Gx, and each
minimal normal subgroup of H/Gx lies in G^/Gx.
Proof. By definition of g we have Gx < Gg. Let R/Gx be a minimal normal subgroup
of G/Gx. Since R centralizes every minimal normal subgroup of G/Gx that is distinct
from R/Gx, and since it is clear from its definition that C'g(R/Gx) contains R, it follows
from (3.18) that R < Gg.
Let S/Gj be a minimal normal subgroup of H/Gx. With R as above we have either
S < Rot RoS = Gx. Since R < Gg < H by hypothesis, R normalizes S, and in the
latter case we have [R, S] < Gx. Therefore S < Cg(R/Gx) in any case, and con-
sequently S < Gg by (3.18). Were Gx properly contained in Hx, the subgroup S could
be chosen in Hx, and we could then derive the following contradiction
Gj = (Gg)j = Gg П Hx > S > Gj.
Therefore Gx = Hx, and the final statement is clear. □
(3.20) Theorem (Blessenohl and Laue [2]). Let g denote the Fitting class HS(f3, S)
of Hypotheses 3.17, and assume that 3 n 91 = 0. Then g is normal in G.
Proof. Let R/Gx be a minimal normal subgroup of G/Gx of composition type 3
contained in G9/Gj. Then by (3.19) we have R < Gg. Suppose that Gs < H < G with
Я e g; we aim to show that H = Gs. First note that R < Gs n Gr — (Gg)» <
and also that Gx = Hx by (3.19). Since the composition factors of R/Gx belong to 3
and are non-abelian by hypothesis, it follows that R/Gx = R/Hx is completely reduc-
ible as an Я-group by A, 4.14. Therefore R/Gx < S(H). Since Я is an g-group, it
induces only inner automorphisms on the direct factors of R/Gx, and consequently
Я < Cg(R/Gj). Hence Я < Gg by (3.18), and equality holds. Therefore Gg is g-
maximal in G. □
As an example of the family of G-normal Fitting classes g described by Theorem
3.20, we mention the one obtained by taking for 3 the class of all non-abelian finite
simple groups and setting S(G) = Soc(G) for all GeG. Then G belongs to this
particular g if and only if the non-abelian component of Soc(G) is a direct factor of
G. Thus, if A = Alt(5) and Я e G, then the g-radical of the group W = A Qjreg Я is
obviously the base group 4". Therefore W/Wg = H, and it follows that g is an
G-normal Fitting class which is not contained in Locksec(G) by (1.3).
Our next objective is to show that the intersection of an G-normal Fitting class of
the type described in Theorem 3.20 with an arbitrary G-normal Fitting class is again
G-normal. The following lemma contains information about a minimal counter-
example for this situation.
3. Normal Fitting classes
715
(3.21) Lemma. Let g and ® be G-normal Fitting classes. Let Gbea group of minimal
^der^bject to the requirement that is not (g n &)-maXimal in G. Then for every
(g n G>)-subgroup X of G properly containing Ggrffi we have
GgTf — GaX = GgG(6 = G.
Proof. Let
(3.0)
Ggn® < X < G
with X e g n ®. Suppose that, if possible, the subgroup H = GSX is a proper
subgroup of G. By the choice of G the radical Hgr(5 is (g n ®)-maximal in II, and
consequently Gs,iffi < If Hgr(6 < Gg, then H^rl!, < Gg, and so Hgr(5 < Ggr l6,
which is a contradiction. Hence Gg < (Hg„ffi)Gg e Nog = g. However, this con-
tradicts the hypothesis that g is G-normal, and it therefore follows that GSX = G.
Similarly we have G&X = G.
It remains to show that GgG® = G. By hypothesis G has an (g n ®)-subgroup
X satisfying (3.0). If G® < Gg, then G® = Ggn® < X e ®, which contradicts
the hypothesis that ® is G-normal. Hence Gg < GgGlr„ and since GgG® n G^X =
Gg(GgG® n X), it follows that GgG® n X Gg. Thus we obtain
(GgG®)gn® — Ggr,® < GgG® n X e s„(g n ®) — g n ®,
and therefore by the minimal choice of G we must have GgG® = G. □
(3.22) Proposition (Blessenohl and Laue [2]). Let g be the Fitting class HS(f3, S)
described in Hypothesis 3.17, and assume that 3 n 91 = 0. IfG> is an arbitrary G-normal
Fitting class, then g n ® is also G-normal.
Proof. Suppose, by way of contradiction, that g n ® is not normal in G. Let G be a
group of minimal order subject to the condition that Ggni is properly contained in
some (g n ®)-subgroup X of G. Then the conclusions of(3.21) hold for G. Recall from
(3.17) that in the background we have Fitting classes X and 9) such that S(G) =
Gs/Gx n Soc(G/Gj) for all G e G. Let R/Gx be a minimal normal subgroup of
G/Gx, of composition type 3, contained in G9/Gj. We aim to show that, in the
notation of (3.17),
(3.1) G = CgIR/GJ.
Put D = G^ = G^n G«, and first consider the possibility that R n D = Gx n D
(= Dj). With H = G in Lemma 3.19 we obtain R < Gg and therefore
[R, G®] <RnGgnG® = RnD = Dj <Gj.
Thus G® < C‘g(R/Gx), and since Gg < C‘c(R/Gx) by (3.18) and G = GgG® by (3.21),
we conclude that G = C*g(R/G$).
716
X. Fitting classes—the Lockett section
Thus, in proving (3.t), we can suppose that R n D Dx. In this case we have
(R n D)GX = R, and hence R/Gx = (Rr' D)/(R n Dn Gx) = (Rn D)/Dx. Since X >
D, we also have Xx n D = Dx < R, and therefore (R n D)X x/Xx = (R n D)/Dx. Con-
sequently R/Gx is X-isomorphic with (R n D)Xx/Xx. But R n D < GvnD — Dv <
Xs, and so the normal section (R n D)Xx/Xx, as a direct product of copies of a
non-abelian simple group in 3, is by A, 4.14 a direct product of minimal normal
subgroups of X/Xx, of composition type 3 and contained in X^/Xx. Since X e g, it
follows that X < C^IR/Gj), and therefore again by (3.21) we have G = GgX <
Cg(R/Gx). This proves that Equation 3.t is true. But then by (3.18) we have Geg,
and therefore GSr,e = G® < X e ®, contrary to the hypothesis that ® is G-normal.
This contradiction proves that g n ® is indeed normal in G. □
This proposition, and, incidentally, the fact that the intersection of S-normal Fitting
classes is again S-normal, offer some evidence for a positive answer to the following
question.
Open Question. Let ЗЕ = JE2 = <s„, n0>3E. Is the intersection of two X-normal Fitting
classes always 3E-normal? In particular, is this the case for JE = G?
We now direct our attention towards Theorem 3.26, which gives H. Laue’s descrip-
tion of the uniquely determined smallest JE-normal Fitting class for certain classes X
containing insoluble groups.
(3.23) Hypotheses, (a) Let X be a Q-closed Fitting class satisfying JE2 = JE S. Let
3 denote the class of non-abelian finite simple groups in X and set D = d03. Let л
be the set of primes defined by the equation
JEns = s„,
whose validity was established in (3.1).
(b) Assume that S„ satisfies the Lockett conjecture with respect to S if л / 0.
Remarks, (a) As we pointed out in the Remark following (3.1), the requirement that
3E2 = £ = <Q, s„>JE implies that X consists of all finite groups whose composition
factors belong to (Zp: ре л)иЗ. Furthermore, since JE S, it follows that 3^0,
and so T> is a non-trivial Fitting formation by II, 2.13; thus, in particular, g 0 D = g35
for any Fitting class g.
(b) We shall show independently in X, 6.12 that if it 0, then S„ always satisfies
the Lockett conjecture with respect to S.
(3.24) Lemma. Assume that Hypotheses 3.23 hold, and let 91 be an G„-normal Fitting
class. If Ge JE\S„, then С,л < С,Л1>.
Proof. Let I = <91, S,\ the Fitting class generated by 91 and S*. Clearly I is
S-normal. By Hypothesis 3.23(b) there exists an g in Locksec(S) such that
3. Normal Fitting classes
717
5 n = 9?. Since ©, s g, evidently 2 E g, and therefore
9? <= I n ©„ <= g n G„ = 9?.
Consequently 9? = I n ©,. Since GeX, we have Ge e sJE n S = X n G = G„, and
% к e"’,He™e G1 - Glr'e- = G”- Since G t S« by hypothesis, we have G' g
Gx and therefore Gz is a proper subgroup of GZG'. Let N/Gz be a minimal normal
subgroup of GZG /Gz, and note that N sn G. It follows from (3.14) that Ge n (GjG') =
Gz, and consequently Gz = (GtG')e. We can therefore conclude that N., = Gz =
G« = N*. Hence N/Nx( = N/Gz) is a direct product of non-abelian simple groups
belonging to qs.X = X, and so N/N* e d03 = £. Therefore N e 9i£\9i, and the
conclusion of the lemma is now clear. q
(3.25) Proposition. Assume that Hypotheses 3.23 hold, and let 9? be an ^„-normal
Fitting class. Then 9?T> is normal in X.
Proof. We suppose that 913? is not X-normal and derive a contradiction. Let G be
an X-group of minimal order such that G4tz is properly contained in some 9?T>-
subgroup H of G. If G were soluble, Hypothesis 3.23(a) would imply that G e S„, and
it would follow from the ©„-normality of 9? that Gj, = G„T. is a maximal 9?T>-
subgroup of G. Since this is not the case, we have G e X\S„ and hence
(З.к) Gj, < Gj,^
by (3.24). If Gj, = Яя, then the group H/Gx belongs to T>, and therefore its normal
subgroup Gj, к/Gj, has a normal complement in H/Gx, call it N/G&, which centralizes
G«t>/G«- On the other hand, if Gy, < Н,я, then [Я,л, G„T] < GST = (Gsr,)s =
Gj,, and in this case we set A' = Н,я. Thus we have shown the existence of a normal
subgroup N of H which properly contains Gj, and centralizes СЯ1>/СЯ. Let C =
Сс(Сят/Ся). Since C < G and 2(СЯ11/СЯ) = 1, we have = C n Gxz = Gj, <
N < C es,X = X. But N < H e 9i3>, and therefore СЯ1> is not 913\maximal in C.
From the choice of G we conclude that C = G and therefore that Gj, = Gj,r. But this
contradicts (З.к); therefore 913? is normal in X. □
(3.26) Theorem (H. Laue [ 1]). Assume that Hypotheses 3.23 hold and additionally that
3 c (£*. Then (S„)tT> is the uniquely determined smallest X-normal Fitting class.
Proof. Let Я be an X-normal Fitting class, and let G e (©„)*3> We show that G e Я.
Let R denote the (S„)t-radicalofG. Since G/R = GJR x x GJR, a direct product
of non-abelian finite simple groups, it will be sufficient to show that G, e Я. Therefore
without loss of generality suppose that either R = G or G/R e 3. Since Sir © is
obviously normal in X n S = ©„ when л 0, it follows from (3.7) 'ha'either
(©„). £ Я or X n S = (1); in any case R < GB, and if R = G we have Ge ft Other-
wise R is a maximal normal subgroup of G, and therefore G„ is either R or G Iret
H/R be a minimal non-abelian subgroup of the non-abehan simple group G/R. By
A, 10.7 we have H/R 6 9I2, and by hypothesis H/R is a л-group. Hence H 6 S„. But
718
X. Fitting classes—the Lockett section
by (3.7), (a) =>(<!), the (G„)t-radical of H contains H'R, which properly contains R,
and so R is not Я-maximal in G. The assumption that К is £-normal and the fact
that G e (G,),!1 £ £ therefore force the conclusion that G = Glt e Я.
Thus (G„)tT> £ Я, and since (G„)tT> is itself £-normal by (3.25), the desired
conclusion now follows. □
If £ = G in Hypotheses 3.23, then T> = d0(3\9I) and G„ = G. As a special case of
Theorem 3.26 we therefore obtain the following.
(3.27) Theorem. Let T> denote the class of all direct products of non-abelian simple
groups. Then G*T is the smallest G-normal Fitting class.
Finally, we prove a theorem which suggests the existence of a wide variety of
G-normal Fitting classes.
(3.28) Theorem (H. Laue [1]). Let G> be an G-normal Fitting class. Then for all Fitting
classes g the Fitting class g О ® is normal in G.
Proof. Suppose by way of contradiction that g 0 ® is not normal in G, and choose
a group G of minimal order such that G~ , (r, is properly contained in an g 0 ®-
subgroup H of G. If Hg = Gg, then H/Hs is a ©-subgroup of G/Gs properly contain-
ing G.R,, (s/Gjj = (G/Gg)®, contrary to the hypothesis that ® is G-normal. Hence
Hg > Gg. Since Hg n G8< 6 = (Gg,-. ffi)g = Gg, it follows that [Hg, Gg0 < Gg. Let
C = Ce(Ggc e/Gg). Then Hg < C < G, and we have
Cgo ® = Gg<) (r, n C < (Gg, lr, n QHg < H e g 0 ®.
Thus the subgroup (Gg c e n C)H8 of C belongs to s„(g 0 ®) = g 0 ®, and it there-
fore follows from the choice of G that C = G; in other words, we have Gg c e/Gg <
Z(G/GS).
Let N/Gg,. be a minimal normal subgroup ofG/Gg<, e. lfN/G~(l is abelian, then
N/G% is nilpotent, and by (3.2) we have N e g 0 ®, which is a contradiction. On the
other hand, if N/Gg (6 is non-abelian, on setting T> = d0(3\9I) we then have H/Gg e
£ GtT> by (3.2); hence N/Gg e ® by (3.27), and so N < Gg 0 lr„ a further and final
contradiction. Therefore g 0 ® is normal in G. □
Concluding Remarks, (a) In generalizing the concept of an G-normal Fitting class
from G to a general universe £ = <s„, n0>£, we might have chosen as the defining
property any one of the Conditions (b), (c), or (d) of Theorem 3.7 with £ in place of
G„. A full investigation of the relations between these possible definitions could prove
fruitful. In the case where £ = £G at least, it is possible that they are all equivalent
to the condition that g belongs to Locksec(£), which, we have seen, yields a more
restrictive concept of £-normality than the one actually chosen.
(b) Within the universe G the relation of strong containment between Fitting
classes can be strengthened by the addition of a "normality’ property. For example,
3. Normal Fitting classes
719
concept °f PaperS Beidleman and Brewster ([1], [2], [3]) have studied the following
Let S “d ® be Fitting classes contained in S. Then g is strictly normal in ©
provided that, for each G e G and for each ©-injector U of G, the subgroup U, is an
g-injector of G. It is not hard to see that g is strictly normal in © if and only if
(i) g « ©. and
(ii) g is normal in ©,
and further that, within a given Lockett section, strict normality is simply the relation
of inclusion. There is clearly scope for ringing the changes on this concept by sub-
stituting for Condition (ii) other generalizations of normality mentioned in Remark
(a) above.
Exercises
1. Let 3 £ 3\ЭД, and let T = d03. Verify that T> is a formation and a Fitting class.
Show further that if M and N are normal subgroups of a finite group G = MN,
then (a) (Lockett [1]) GT = and (b) (Fitting [1]) Gj, = MZNT.
2. (H. Laue [1]) Let T> = d0(3\9I). Show that the map X -»X о defines a bijec-
tion from Locksec(G) to Locksec(G 0 D).
3. (Dark [1]) Let G be a finite group which has a subnormal subgroup isomorphic
with Sym(3). Show that G is not perfect. (Hint: Apply Corollary 3.15.)
4. (Hauck [5]) Let g and © be Fitting classes such that g о © = G. Show that if
tt = {p e P: g*Sp = g*}, then G„ О © = G (compare with Theorem 3.13).
5. (Hauck [5]) Let л £ P, and let g be an Enclosed Fitting class. Prove that G„ 0 g =
G if and only if g contains the Fitting class (G e G: 0„(G) < ZK(G)). (More
difficult) Show that this is false without the assumption that g is closed under
central extensions.
6. (Beidleman [3], Hauck [5]) A Fitting class g £ S is said to satisfy Property
a if the following holds: Whenever X is a Fitting class such that I 0 g is
G-normal, then X is G-normal. Prove that any two of the following statements
are equivalent:
(a) g satisfies Property a;
(b) g* satisfies Property a:
(c) Sp. О g* ± S for all pe P;
(d) G . 0 g is not normal for all p e P.
7. (Hauck [5]) A Fitting class g £ S is said to satisfy Property у if the following
holds: Whenever X is a Fitting class such that g О X is G-normal, then I is
G-normal. Prove that any two of the following statements are equivalent.
(a) g satisfies Property y;
(b) g* satisfies Property y;
(c) g* 0 Sp * g* for all primes p;
M) If g О X = S for some Fitting class X, then X — S.
8. Show that although Dih(8) belongs to S„ there exists no finite group G such
that G' = Dih(8). ,.
9 If gtU = S for a Fitting class g, then g is normal in S (see (3.1 )).
Iff (Pense [1]) Let g be a Fitting class of finite groups, and let
720
X. Fitting classes—the Lockett section
= (G e g: If § £ g = <s„ n0>(G, §), then g = <s„, No>§).
(a) If there exist maximal Fitting classes in g, then Ф<5_. No>(g) is the intersection
of all such maximal Fitting classes of g. Otherwise <h<s„.N(1>(g) = g-
(b) If gS = g*, then <J><Sn.No>(g) e Locksec(g).
(c) If g = (gS)*, then there exist no maximal Fitting classes in g.
4. The Lausch group
Let g be a Fitting class of finite groups. We recall from Definition IX, 2.10 that Set(G)
denotes a fixed set containing one representative of each isomorphism class of finite
groups, and that
Set(g) = g n Set(®).
We also recall that (A, d) is called an g-Fitting pair if A is a (possibly infinite) abelian
group and if for each G in Set(g) there is a homomorphism dG: G -> A such that
FP1: dG = a ° dH for all 6, He Set(g) and all normal embeddings a: G -» H, and
FP2: A = {gdG: g 6 G 6 Set(g)}.
As we pointed out in IX, 2.12(d), the definition of an g-Fitting pair remains unchanged
when the word “normal” is replaced by “subnormal” in FPL Moreover, the require-
ment that A is abelian is a consequence of FP1 and FP2 by IX, 2.12(c). By IX, 2.11 (ii)
and (1.7) the kernel
Я(Л, d) = (G 6 Set(g): Gdc = 1)
of the g-Fitting pair (A, d) belongs to Locksec(g).
The starting point for the discussion in this section is the question of the existence
of an g-Fitting pair whose kernel is g*; it will turn out to be a universal g-Fitting
pair in the following obvious sense.
(4.1) Definition. An g-Fitting pair (A, 5) is called universal if for any g-Fitting pair
(A, d) there exists a homomorphism ф: A -» A such that
dG — bG ° Ф
for all G e Set(g). In this case we write d = д о ф.
г л
G -----—► Л
Ф
A
4. The Lausch group
721
Remark. From this definition it is not hard to establish that if (Д', Й') is a second
universal Д-Fitting pair, then there exists an isomorphism 0: A -> A' such that £ ° 0 =
d . Thus to within isomorphism a universal Д-Fitting pair is unique.
Lausch [1] was the first to describe a universal Д-Fitting pair. He carried out the
construction for the case Д = G in the context of G-normal Fitting classes, but as
Bryce and Cossey [5] subsequently pointed out. his method applies equally well to
an arbitrary Fitting class Д. We start this section by describing Lausch's construction
of the universal pair, for which we reserve the notation (A, <5). The group Л = Л(Д)
is a certain quotient of the restricted direct product of the groups in Set® and is
called the Lausch group of Д. After deriving the basic properties of this universal pair
(Л, b), we go on to show that there is a lattice isomorphism between the subgroups
of A on the one hand and the Fitting classes of Locksec® contained in Д on the
other. In the light of this result the Lausch group becomes an important tool for the
study of Lockett sections, especially since it is possible to determine its structure
completely for a wide variety of Fitting classes (although, as yet, by no means for all
of them). The question of finding the structure of Л(Д) for a given Fitting class Д will
be a theme of Section 5, where we shall use the Lausch group as a framework for our
proof of a theorem of Berger’s on the determination of the class Д„. Although sections
of possibly infinite direct products frequently come into play, all classes of groups
considered in this section are subclasses of the universe (E.
In order to frame the definition of the Lausch group of a Fitting class we need
some suitable notation.
Notation. First we recall that if is a set of groups, their restricted direct
product
= X
цсМ
consists of all functions f:M~* IJpeM G,, of finite support such that /(/i) 6 G» for
all p 6 M; the group operation is defined ‘pointwise’ thus: (fg)(fi) = f(/i)s(/i)- Let
N c Af, and let DK = Gv.. By the natural embedding eN: DN -> Du we mean the
map which sends an element j0 6 DN to the following element f of Du:
f(» =
'Ш
1
if/eA,
if / £ N.
In the special case when N = {v},
to denote the natural embedding
Obviously sN is a normal embedding of DN into DM.
a singleton, we set G = Gv and use the symbol cG
°f Fiially,' if GandH are groups, we denote the set of all subnormal embeddings of
G into H by Snemb(G - H) and the subset of normal embeddings by Nemb(G H)
(see IX, 2.10(b) for the appropriate definitions).
(4.2) <> 1» 1Г S i. а of » ,™,ps. let A®
denote the following restricted direct product.
722 X. Fitting classes—the Lockett section
A(g) = X{G:G6Set(g)}.
Let T(g) denote the following subgroup of A(g):
(4.a) T(g) = <(0_1sG)(sasH)-' G, H 6 Set(g), g e G, a e Nemb(G -► H)>.
By Remark 4.3(a) below F(g) is in fact normal in A(g). The Lausch group A(g) of g
is then defined as the following quotient group:
A(g) = A(g)/T(g).
(b) (The associated map 6). If G 6 Set(g), we define a map dG: G -* A(g) as follows:
(4./i) g6c = (0®с)Г(5) for all geG.
Since the map 6G is the composition of the monomorphism eG with the natural
homomorphism from A(g) to A(g)/T(g), it is certainly a homomorphism.
(4.3) Remark. Let g be a Fitting class, and let G e Set(g). Then
[G, Aut(G)]ec < T(g).
In particular, we have
(a) A(g)' < T(g), and
(b) if G is perfect, then Gec < T(g).
Proof. Let a e Aut(G). Then a e Nemb(G -» G), and if g e G, then by (4,a) we have
[g, a]eG = (0-1eG)((ga)sG)6 Г(<5)-
Since [G, Aut(G)] is generated by such elements [g, a], the stated inclusion is now
clear.
It follows that (Gcg)' = G'eg = [G, Inn(G)]ec < [G, Aut(G)]ec < r(g), and since
A(g)' is generated by the subgroups (Gec)' as G runs through Set(g), we conclude
that A(g)' < T(g). Finally, if G is perfect, we have Gcc = G'eg < T(g), which is
Assertion (b). □
(4.4) Proposition (Lausch [1]). If ft is a Fitting class of finite groups, then the pair
(A (S)> ^1 defined in (4.2) is an ^-Fitting pair.
Proof. Let G,H e Set(g) and a e Nemb(G -> H). If g e G, we deduce from the defini-
tion of the subgroup Г = T(g) in (4.a) that ((да)еи)Г = (g-1eG)-1r = (gsG)F, and
from (4./J) it therefore follows that
g&G = ((ga}BH)f = (д<№н = g(a O <5„)
4. The Lausch group
723
for all g e C Hence Requirement FP1 of an g-Htting pair is fulfilled. Although, as
we indicated earlier, the fact that A(g) is abelian will follow from FP1 and FP2 we
note that it is also a consequence of Remark 4.3(a).
To complete the proof it remains to show that Requirement FP2 is fulfilled, and
this is clearly a consequence of the following more general result.
(4.5) Lemma. Let Lbea o„-closed subclass of a Fitting class g, and let E(3£) denote
the following subset of the Lausch group A(g):
(4.y)
- {gdc- G e Set(I), geG}.
Then:
(i) E(I) is a subgroup of A(g);
(ii) If e denotes the natural embedding cf A(X) into A(g), then E(£) = (А(ЗЕ)е)Г(Я)/Г(®-
(iii) E(g) = A(g).
Proof, (i) Let g e G e Set (I) and he H e Set(X). Then gSG and hdB represent two
typical elements of E(I). Let L be the element of Set(g) isomorphic with G x H. Then
L e = I. Let a and ft denote the natural embeddings of G and H respectively
into G x H, and without loss of generality identify G x H with L; then clearly
a e Nemb(G -> L) and P e Nemb(H -» L). Using Property FP1, already proved in
(4.4), and the fact that is a homomorphism, we have (g, h)bL = ((g, 1)(1, h))bL =
((ga)(hp))6L = (g(a □ bL))(h(P о bL)) = (gdG)(hbH), and therefore (gbG)(hdH) e L6l s
E(3£). Consequently E(I) is closed under products. But because 6G is a homo-
morphism, we also have (g<5c)-1 = (g-1)^G, and it follows that E(3E) is indeed a
subgroup of A(g).
(ii) By definition of 6G we have
S(I) = {(0®g)П&): g e G 6 Set(X)},
and this set clearly generates (A(3E)e)F(g)/F(g) because the direct product A(I) s
A(I)e is restricted. Since E(3E) is a subgroup of A(&) by Part (i), we therefore conclude
that E(X) = (А(Х)£)Г(&)/Г(&). _
(iii) Taking 3£ = g in Part (ii) and f. to be the identity map, we obtain s(g) =
A(g)/T(g) = A(g). This completes the proof of the lemma and also the proof of
Proposition 4.4. I-'
The folio wing concept is useful when passing from a Fitting pair to the Lausch group.
(4.6) Definition. The roof map d.
Let g be a Fitting class and (A, d) an g-Fitting pair. Let M be an index set for the
groups in Set(g) (since Set(g) is countable, we could take M <= I for example). T
we can write
A(g) = X G, = (/:Af-
ты
X. Fitting classes—the Lockett section
Define the map d: Д(&) -> A as follows:
(4.<5) fd = fl
Since/has finite support and A is abelian, the product on the right-hand side of this
equation is well-defined and independent of the order in which it is taken.
The following elementary observations about this definition will be useful.
(4.7) Remarks. Let (A, d) be an ^-Fitting pair.
(a) If g e G eSetffi), then (geG)d = gdG.
(b) The map d is an epimorphism with T(g) < Ker(d).
(c) Ker(dG)r.G = Gr.G n Ker(J).
Proof, (a) If G e Set(^), then G = GA for some 2 e M. If g e G, then the element gi:G
of Д is the function / such that /(>.) = g and f(p) = 1 for all p e M\{2}. Therefore
from (4.<5) we have (geG)d = f(X)dGi = gdG.
(b) That d is a homomorphism follows at once from the definition of a direct
product and the fact that each dGu, p e M, is a homomorphism; and that d is surjective
is a consequence of Part (a) and Property FP2 for the pair (A, d). If G, H e Set(g),
a e Nemb(G -» H), and geG, then
((0_1eo)(0aeH))d = (<; '</;)( = 1
because d satisfies Property FPI. Since the elements of the form (g 'ес)(даен)
generate F({V), it follows that F({V) < Ker(d).
(c) From Part (a) we have gi:G e Ker(d) if and only if g e Ker(dG), and Part (c) is
now clear. □
(4.8) Theorem (Lausch [1]). If ft is a Fitting class, the Fitting pair (A(&), 5) defined
in (4.2) is universal.
Proof. A typical element of A(g) (= Д(&)/Г(5)) has the form xT(g) with x 6 Д(&). If
(A, d) is an ^-Fitting pair, define a map A(g) -» A as follows:
(хГ(&)№ = xd.
By (4.7)(b) we have T(g) < Ker(d), and therefore ф is well defined. If g 6 G e Set(Jtj-),
we then have
9(&g °Ф) = ((дгс)Г)ф = = gdo
by (4.7) (a). Therefore dG° ф = dG, and the Fitting pair (A(8), 6) has the desired
universal property of Definition 4.1. □
4. The Lausch group
725
°rUInT ,mP°rtant obJectlve will be the isomorphism between the subgroup lattice
of the Lausch group of a Fitting class g and the sublattice of Locksec(g) below g.
But first we present some more elementary facts about the newly-defined concepts of
this section.
(4.9) Lemma. If g is a Fitting class, the definition of T(g) remains unchanged when
the set Nemb(G —* H) is replaced by Snemb(G —* H) in Equation 4.a.
Proof. Set Г = r(g), and let Г* denote the subgroup of A(g) defined when Nemb
is replaced by Snemb in Equation 4.a. Then obviously Г < Г*. Now suppose
that G, He Set(g) and let a e Snemb(G -»H). Then there exist groups Go ( = G),
Gi> G„ ( = H) in Set(g) and a;e Nemb(G,_, -»G() such that a = ala2...a„. Let
g e G, set g0 = g, and for i = 1,..., n define д,- = af. Then a typical generator of
Г* has the form
(в 1eG)(gaefl)= П (ft-iCc,
and since the right-hand side is a product of generators of Г, we have shown that
Г* < Г. □
(4.10) Lemma. Let g be a Fitting class, let G, Kt,..., K„ be groups in Set(g), and
assume that there exist a, 6 Snemb(Kj -» G) such that
G = <K1a1,...,KA>.
For i = 1,..., n let <5,- and c, denote the homomorphisms <5K, and cK. respectively, and set
Г = T(g). Then
(a) GSC = <KA:i = l,...,n),and
(b) (Gec)r = (XJe j К,е,)Г for some subset J of {1,.... n}.
Proof (a) Let 1 < i < n, and let L, e K,. Since b satisfies Property FP1 by (4.4), we
have
(4.e)
= kpAa-
Consequently KA < Gdc. On the other hand, by hypothesis each g in 6 is the
product of elements of the form к{а( with fc, e Ks and । e {1,.. •, n}, an so rom
(4.8) we conclude that gbc is the product of elements of the form kA- Hence
Gbc < (Kfi:1=1.......n>’ and Assertion (a) holds.
(b) From Part (a) and the definition of the map 6 we have
(Geg)F =<K.8.r:i = 1, ,">
But the right-hand side of this equation is equal * <*£ " = b - -
if {Ky.jeJ} represents the set of distinct groups in the list .... K„
726
X. Fitting classes—the Lockett section
<Klel:i = 1, ...,n> is obviously the direct product of the component subgroups
{KjEj: j e J} of A(g). □
(4.11) Lemma. Let ffbe a Fitting class, and let G.He Set(g). Let К be a subnormal
subgroup of G, and let a e Snemb(K —» H). If x e K, then
xsc = (xa)e„ (mod F(g)).
In particular, if G = H, then (х~'(ха))ес e F(g) for all x e K.
Proof. Since Kes„g=g, there exists a group L e Set(g) and a map
P e Snemb(L -» G) such that LP = K. If x e K, then x = yP for some у e L, and so
by (4.9) the group F(g) contains the element (y~'eL)(yPec) = (уЕеГ'(хео)- Thus
xec = yeL (mod F(g)).
Since p о a e Snemb(L -> H) and y(P о a) = xa, we likewise conclude that
yeL = (xa)eH (mod F(g)),
and the conclusions of the lemma are now clear. □
Since we frequently need to refer to the Fitting classes lying between g* and g, we
introduce the following terminoloy.
(4.12) Definition. Let § be a Fitting class of finite groups. The Lockett subsection of
g is defined as follows:
Locksub(g) = {(5: (5 e Locksec(g), (5 £ g}.
Obviously Locksub(g) is a sublattice of the lattice of Fitting classes in Locksec(g).
(4.13) Lemma. Let ‘ft be a Fitting class, and let ffi e Locksec(g). Let G 6 g, let H be
an arbitrary finite group, and let a e Nemb(G -> H). Then (g~', ga) e (G x H)a for all
geG.
Proof Because Ga = G e g £ ffi*, it follows from (1.2)(b) that (g~\ ga) e(G x Ga)ffl.
Since G x Ga < G x H, we have (G x Ga)([, < (G x H)K„ and the result follows. □
We now come to a theorem of fundamental importance, describing the intimate
connection between the Lockett subsection of a Fitting class and its Lausch group.
(4.14) Theorem (Lausch [1], Bryce and Cossey [5]). Let g be a Fitting class of
finite groups, and let A(g) denote its Lausch group. For I e Locksub(g), let E(X) =
{xdx: x 6 X e Set(X)}, clearly a subset of A(g). Then the map
4. The Lausch group
727
H:X^E(3E)
is a lattice isomorphism from the lattice of Fitting classes
g to the subgroup lattice of A(g).
in the Lockett subsection of
Lock[ub(&)- ‘J™ certainly I = o0X, and E(I) is a subgroup of A(g)
by (4A)(i). Hence the map a has the stated target. Our initial aim will be to prove
that c. is surjective. If S is a subgroup of A(g), define an associated subclass Xs of g
as follows:
<4-f) = (G 6 Set(g): Gdc < S).
First we show that 3£s is a Fitting class. Let K„ e s,3£s. Then there exist
(i) groups К and G in Set(g), and
(ii) a subnormal embedding a: K->G
such that № Ko and Gdc < S. Since (A(g), <5) satisfies FPI, it follows that
= (Ka)dG < S, and therefore Ko e Is. To see that 3ES >s N„-closed, suppose that
Ho, Ko < H0K0 = Go with Ho, Ko e Xs. Then Go e N„g = g, and there exist
(i) groups H, K, G in Set(g), isomorphic with H0,-K0, and Go respectively, and
(ii) normal embeddings а:Я-»6 and [I: К — G
such that Ндн, Кёк < S and G = From (4.10)(a) we then conclude that
G6g = (HbH, Кдк~) < S. Hence Go els, and by II, 2.11 the class Is is N0-closed. Next
we show that 3£s e Locksub(g). Let G 6 Set(g), and note that [G. Aut(G)]5c = 1
by (4.3). Since [G, Aut(G)] < G e g, there exists a group H in Set(g) and a map
a 6 Nemb(H -> G) such that Ha = [G, Aut(G)]. Therefore Нён = (Ha)bG = 1 e S,
and it follows that [G, Aut(G)] is contained in the Is-radical of G. Consequently, by
(1.21), (e) => (a), we have Is e Locksub(g).
To complete the proof that the map E is surjective, we now assert that E(IS) = S.
If s e S, Condition FP2 for the pair (A(g), 6) ensures that s = gs6G for some g.eGe
Set(g). Set
KG = {geG-.gbce S}.
Then s e Ko, and since Ko is obviously the 3£s-radical of G, we can find a group К
in Set(Is) and a map a in Nemb(K -» G) such that Ka = Ko. If ka = gs, then
kbK = 9S6G = s, and so s e E(3£s). It follows that E(3£s) = S, and hence that E is
surjective.
Our next goal is to prove that the map E is injective, and we begin by showing
that the Fitting class 3^ (obtained by setting S = 1 in (4.0) coincides with g,. We
have shown above that 3Ej e Locksub(g), and therefore g„ To prove the
reverse inclusion, let G 6 Set^,) and geG. Since gbc = 1, we have geG e T(g) by
definition of 6G (see Equation 4.(3), and so from Equation 4.a we have
gec = П (д7'^,)((дл)ен)
i=l
for suitable G„ H|6Set(g), д>еС(, and a,-6Nemb(G,--»HJ, i=l, r. Ut
728
X. Fitting classes—the Lockett section
D = <Gec. G,ec , H,r.H : i = 1,..., r>, a direct product of finitely many components
of A(S) and hence a finite subgroup of A(&), and let denote the element
(0i”1£g )((0i“i)eH,) of D, i = 1, r. Let 1 < i < r. If Gt = If, then cq e Aut(Gf),
and л,' = [g„ а,]еС( 6 [G,-, Aut(G,)]eGi, which by (1.21), (a)=>(e), is a subgroup of
((G,)B )eG = (G,eG.)B>. the other hand, if G, H,, by (4.13) the element л, belongs
to the subgroup ((G(eG_) x (Н^н ))Яг of D. Thus we have shown that gi:G = x, x2... x,
belongs to the following subgroup R of D:
R = П ((^-Сс.НН,-^.))^.
it
Since the terms of this product are evidently normal g^-subgroups of D, it follows
that R is a normal g*-subgroup of D; therefore gt:G e Gec r> R < (Ge0)B< because
Gr.c < D. Since g is an arbitrary element of G, we conclude that G S Gbg = (Gec)B< e
5*. Hence 3Ej c and equality holds.
To complete the proof that the map E is injective, let S < A(8) and let 9) be an
element in Locksub(g) such that E(9)) = S. We have already proved that E(IS) = S,
and so it will suffice to show that 9) = Is. By definition of we certainly have
9) о 3£s„ To prove the reverse inclusion, let G e Set(Is) and g e G. Since g6G e S,
by definition of E(9)) there exists some he H e Set (9)) such that gdG = hdH. Set
L = G x H, let a: G L and (3 . H L denote the natural embeddings into the direct
product, and for notational simplicity identify L with the element of Set(g) which is
isomorphic with G x H. Then 1 = (g ^c)(h5H) = ((g~1a)(hp))6L, and from the fact
that Ij = we conclude that (g-1a)(h/J) e LB>. Since 9) 6 Locksub(j^), we have
Ls><L9. Moreover, since H e 9), we have Hf < L^, and it follows that
ga e Ly n(G x 1)= G4) x 1. Since g is an arbitrary element of G, we conclude that
G s Ga = Gs e 9). Thus 9) = 3ES, and we have now shown that E is a bijection.
Finally, the fact that E preserves the lattice operations follows directly from the
following obvious remarks:
(a) Both lattices are induced by the partial order of inclusion;
(b) The map E preserves inclusions and is therefore an order isomorphism. □
We now draw a series of conclusions from this theorem. The first of these is the
following observation, which is implicit in its proof.
(4.15) Corollary. Let % be a Fitting class, let Sbea subgroup of its Lausch group A(g),
and let & denote the uniquely determined Fitting class in Locksub(5) such that
E(G>) = S. If G e Set(§), then Ga = {g e G : gbG e S}. In particular, we have
(a) GB< = Ker(dG), and
(b) G^eG = GeG n Г(§).
Since A(&) is the restricted direct product of finite groups, it is a torsion group,
and therefore the Lausch group of a Fitting class is an abelian torsion group. The
subgroup lattice of an arbitrary group is complete in the sense that an arbitrary
collection of subgroups has a least upper bound, namely their join, and if the group
is abelian, then its lattice is modular by the Dedekind law. Furthermore, every
4. The Lausch group
729
subgroup of a torsion group contains an element of prime order, and so its subgroup
lattice is atomic. (An atomic lattice is one in which every element contains a minimal
element of the lattice.) Thus we obtain the following corollary to Theorem 4.14.
(4.16) Corollary. If g is a Fitting class, the lattice of Fitting classes lying between g
and § is complete, modular, and atomic. *
We show at the end of this section that Locksec(G), for example, is not dually atomic.
Another interesting consequence of Theorem 4.14 is the fact that every subgroup
lying between the g+-radical and g*-radical of a group is itself a radical.
(4.17) Corollary. Let g be a Fitting class, let Gbea finite group, and let Gg < R <Gg.
Then there exists a Fitting class 91 e Locksub(g) such that R = G*.
Proof. Without loss of generality we identify Gg with the isomorphic subgroup К in
Set(g). Thus R < K, and the homomorphism ёк: К -»A(g) sends R to a subgroup
S = R6K of A(g). By (4.14) there is a unique Fitting class 91 e Locksub(g) such that
E(91) = S, and by (4.15) the 91-radical of К is the subgroup {ke К : kbK e S}, which
coincides with R because R > Ker(<5K) = GB>; in other words, Kv = R. Since 91 e g,
we have Gs < Gg = K, and therefore Gs = K9 = R. □
(4.18) Corollary. Let be a Fitting class, and let G e Set(g) n 91. Then Gec < T(g).
Proof. By (1.23)(b) we have G 6 gt, and it follows from the final statement of (4.15)
that Gcg = Gcg m T(g). □
The next result also relies heavily on Theorem 4.14. It provides a comparison of
the Lockett section of a Fitting class t5 with that of a Fitting subclass g of <5.
(4.19) Theorem. Let g and ® be Fitting classes with g S ©. Let ®0 = <©,, g>, the
smallest Fitting class containing <5* and g. For 9) 6 Locksub(G>0), the map
9)-gn9)
is a lattice isomorphism from Locksub(®0) to the lattice <£ of Fitting classes which lie
between g n and g.
Proof. Since Set(g) <= Set((6), we can regard A(g) as a subgroup of A(®) by means
of the natural embedding described at the beginning of this section. It is then clear
from the definition of Г( ) in (4.a) that T(g) < Г(<5). . , .
Let S = A(g)F((5)/r(C5). Then E(©0) Э 3(5) = s b* On the other hand’
in the proof of (4.14) we saw that the class
3ES = (G e Set(<5): G6C < S) (where 6G maps G into A(©))
is a Fitting class containing g and and also that s(3Es) - S. Hence ®0 - *s and
730
X. Fitting classes—the Lockett section
S(®0) £ E(JES) - S- H therefore follows that E(ffi0) = S and hence by (4.14) that a
induces (by restriction) a lattice isomorphism from Locksub(©0) to the subgroup
lattice of Д(8)Г(®)/Г(®) < Д((5)/Г(®).
Now let ?) e Locksub(®0), and let
So: Locksub(g) -> the subgroup lattice of A(g)
denote the lattice isomorphism described in (4.14). Since by (1.18) we have
g* s § n ffi* £ g n 9), it follows that g n 9) e Locksub(g), and from (4.5)(ii) we can
conclude that
(4.^) Eo(gn9)) = A(gn9))r(ff)/r(g)
= (Д(8)пД®г(8))/г(8)<г>
where T denotes the subgroup (Д(fy) n Д(?))Г(©))/Г(g) of A(g). Let
Ir = (G e Sct(g): GSC < T) (where Sc maps G into A(g)),
and let G 6 3tr. Then Gec < Д(9))Г(®), where cc is the natural embedding of G into
Д(©). Thus G<5ft < Д(9))Г(®)/Г(®) = E(9)), and it follows from the proof of (4.14)
that G e 9). Consequently XT <, g n ?). Therefore T = E0(JET) < E0(g n 9)), and
equality holds in (4.,/). In particular
(4.0) A(g) n A(9))F(g) = A(g) n Д(ф)Г(®).
Setting 9) = ffi*, we deduce that
So(S ffi*) = (A(g) n F(ffi))/r(g), = К say,
and from (4.14) we then observe that Eo induces a lattice isomorphism, E, say, from
the lattice S’ of Fitting classes lying between g n ffi* and g onto the subgroup lattice
ofA(g)/K.
By the standard isomorphism theorems there exists a group isomorphism 0 from
A(g)F(ffi)/r(ffi) onto (A(g)/T(g))/((A(g)о T(ffi))/T(g)) (= A(g)/K), which therefore
4. The Lausch group
731
Z™-TT- V‘he tOrresPondinS sub^P bttices. The compos.tion
' of the maps 0, and (a,) is therefore a lattice isomorphism from Locksub((S0)
G Lw*sub((S0), then 0(^(9))) = ((A(g)mA(9))r((5))/r(g))/K, and
-У J4'6™ ((A(S) П Д('?,’Г^))/Г(Й))/К = (A(g m ?))Г(5)/Г(8))/К =
c-i(B r> 1)). thus p('JJ) = g ci4).
Next we describe a criterion due to Lausch [1] for a group to belong to g*. (Lausch
presents the criterion for <5*,but, as Bryce and Cossey [5] point out, his proof applies
equally well to a general Fitting class.) Unfortunately the criterion is of little practical
use for deciding whether a given group belongs to g*. But it is of theoretical
interest and applies to all Fitting classes, whereas Berger’s more practical algorithm,
described in the next section, appears to work only for certain well-behaved Fitting
classes.
(4.20) Theorem (Lausch [1]). Let 5 be a Fitting class. A necessary and sufficient
condition for a group G to belong to g* is the existence of a group R, together with
normal ^-subgroups No, ...,Nmof R and a normal embeddingsuch that
(О « = П"о^- and
(ii) G£<n"0[^Aut(N;)].
In the proof of the necessity, the group No may be taken to be G.
Proof. To prove the sufficiency of the condition, assume that a group R with the
stated properties exists. Since e g e Locksec(g*). by (1.21), (a)=»(e), we have
[N;, Aut(Nj)] < (N;)g> < Kg> for i = 0, ..., m. Therefore Ge < Rg> 6 g*, and con-
sequently G = Ge e s„g*, as required.
To demonstrate the necessity, suppose that G e g*. Without loss of generality
suppose that G 6 Sct(g*) and enumerate its elements thus: G = {<j,,..., <?,}. By (4.15)
we have Gec < T(g), and so from (4.a) we conclude that for each i 6 {1,.... r}
(4.1) gfiG = П (SA.)
AeAj
for suitable Gx, Ял e Set(g), e Gt, e Nemb(Gx - Ял) and suitable index sets Af.
We now define R to be the subgroup
R = <Gsg, СлеОл, (G^Ch, : A e A,, i = 1.r>.
of A(g). Let Мл = <GA„ (GAaA)f.„,> for A 6 A;, i = 1. r Tben set No = G^ and
let N, N denote the distinct groups among the {Мл}. Since is a normal
subgroup of the group <G^. Яде„,>, which is a direct^^^«foUowstat
<i A(g). Therefore for j = 0, 1,.... m we certainly have « - 1 L=o Nt. Next
let e = ec; since ec 6 Nemb(G-» A(t5))> we have e 6 Nemb(G-»K). .
Let A e A, for some ie {1,.... r}. If GA = Ял the map geG ->
clearly an automorphism of G^ = Мя. On the other hand, if G, ^Ял t he imp
which sends a typical element (geG,)(g*^m) of G^ x в eGJ
to the element (g'e^ga^) is again an automorphism of Мл. Thus each term
732
X. Fitting classes—the Lockett section
the product on the right-hand side of Equation 4./ belongs to some [M2, Aut(Af,)].
Thus we conclude that Ge < f7!"=i ПЧ, Aut(Aj)], and the necessity of the condition
is now obvious. □
We bring this section to a close by touching briefly on the question of how one
Fitting class can be contained maximally in another. In Corollary 4.16 we pointed
out that within the lattice of Fitting classes of a given Lockett section every element
contains a minimal element. Now we describe an example to show that the dual
statement need not hold; it yields a Fitting class which is not contained in any
maximal element of its Lockett section.
(4.21) Example (Bryce and Cossey [4]). Let pbe a prime, and let g be a Fitting class
containing 912. Then the Lausch group A(g) has a direct factor isomorphic with the
restricted direct product X„" i particular, A(g) has a quotient group isomorphic
with the Priifer p-group Zp,, and so Locksec(g) is not dually atomic.
We shall justify this assertion in stages.
(4.22) There exists a sequence of primesq1,q2,...suchthatp"\\(q„ — l)foreachn e hl.
Proof. Let n e hl. By Dirichlet’s theorem on primes in arithmetic progression (see
Apostol [1], Theorem 7.9) there exists a prime q„ such that
q„=l+p" (modp"+1).
Since p"|(q„ — 1) and (q„ — l)/p" = 1 (mod p), it follows that p”||(q„ — 1). □
(4.23) For each neN there exists an ^-Fitting pair (Zp„, d{n>} such that
GdM = !iZp" when G = £(P7gJ. and
° (1 when G = E(pmlqm) and m e hi\{n}.
Proof. Let n e hJ, and consider the ‘determinant’ Fitting pair, described in Example
2.14 of Chapter IX, for the set л = {q„}. The abelian group A of this pair is
and we denote the map (called simply d in Example 2.14) by A<n). Let Q denote the
Sylow p-complement of F^, and let v: F“n ->• kfJQ be the natural homomorphism.
Now define
4"> = Ag> о v, for all G e Set(g).
By (4.22) the Sylow p-subgroup of is cyclic of order p", and therefore we can
identify Im(v) with Z^. Since the homomorphisms Ag1 satisfy Axioms FP1 and FP2
for an g-Fitting pair, obviously so also do the homomorphisms d1,"’, and it follows
that (Zp„, d1"1) is an g-Fitting pair.
Now set G = Elpm/qm\ If m = n, then Soc(G) = Z4„ and G/Soc(G) S Zp„, and it
follows from the definition of the ‘determinant’ Fitting pair that GA^’eSylp(F^).
4. The Lausch group
on the other hand, if m * n, lhen G is a (v.groupi and consequently сд(, = ,
From these observations rt is now clear that the homomorphisms dg”1 have the stated
I Illei^CS.
□
Now put A X“ j Zp„ (the restricted direct product! and let a(n) denote the n'h
component of an element a e A. With this notation we define a map dG G A for
each G 6 Set(g), as follows:
(^C)(«) = Sdg"
for each geG and n e M. Since | G | involves only finitely many primes, G is almost
always a (qj-group, and so Gd<£> + 1 for only finitely many values of n. Thus the
target of the map dG really is the restricted direct product A.
Since for each n e N the homomorphisms dg1’ satisfy Axiom FPI for g-Fitting
pairs, the maps dG are also homomorphisms satisfying this axiom. To verify the
second axiom, let z„ denote a generator of Z^, and let a„ be the element of A defined
thus:
u„(m) =
z„ for m = n, and
1 for m n.
Thus a„ generates the nth component of A. Let n e LJ, and let £ be the element of
Set(5) isomorphic with £(p"/q„)-here we recall that 'JI2 £ § by hypothesis. By (4.23)
we have EdE = <a„>, and since A = <о„: и 6 N), we deduce from (4.5)(i) that Axiom
FP2 is also satisfied by the pair (A, d). Thus we have shown that
(4.24) (A, d) is an ^-Fitting pair.
By (4.8) there is an epimorphism ф: A(§) -» A such that dG — dG ° ф for all
G 6 Set(g), and our next task is to find a complement to Ker(^) in A(g). Let n e N,
and let £ be the element of Sct(Jy) isomorphic with E(p"lq„). Then E(bE ° ф) = Z^,
and so |££c| > p". On the other hand, £' < Ker(<5E), and therefore |£<5E| =
|E: Ker(<5E)| < |£: £'| = p". Consequently |E<5e| = p", and we deduce that E6e s Z^.
It follows that the restriction of ф to E6e is an isomorphism onto <an>. We can
therefore define a homomorphism 0: A -»A(5) by setting
= а„Ф
on the set of independent generators {«„: n e M of .4 and then extending the: domain
of 0 linearly to A. It is then clear that Оф is the identity automorphism of Л; thus 6
is a monomorphism, and we can conclude that
A(g) = Л0®Кег(#
Since AO A = У’-, Zp„, the first part of the statement of (4.21) is proved.
To justify the finaf assertion of (4.21) we need the following observation.
734
X. Fitting classes— the Lockett section
(4.25) There exists an epimorphism p: A -» Zp«.
Proof. Write the components of A = X”=t additively, so that a(n) is a residue
class (mod p") for each a e A and n e 14 Then define
p(a) = fj exp(2ma(n)/p").
n=l
Since only finitely many a(n)’s are non-zero, it is clear that p is a well-defined
homomorphism from A to Zp< = <exp(2ni/p"): n e Finally, we remark that each
of the generators exp(2rti/p") of Zp., lies in Im(p), and therefore p is an epimorphism.
□
When the epimorphism^: A(g) -» A is composed with the epimorphism p of (4.25),
we obtain an epimorphism ф° p: A(g) -» Zp,. Let К = Kcr(^ ° p), and let Я denote
the Fitting class in Locksub(g) such that Е(Я) = К. By (4.14) the lattice of Fitting
classes lying between Я and § is isomorphic with the lattice of subgroups of
A(g)/K = Zp«. Since Zp, has no maximal subgroups, we conclude that Я is contained
in no maximal Fitting subclass of 5; in other words, the lattice Locksub(g) is not
dually atomic. Thus the assertions made in (4.21) have all been justified.
Remark. In the next section (Corollary 5.30) we shall prove that, for a large family
of Fitting classes g (including all Fischer classes), the Lausch group of g is a restricted
direct product of finite cyclic groups. In this case therefore, it follows from (4.14) that
Locksec(g) always contains maximal elements. It is an open question whether there
exists a Fitting class whose Lausch group is isomorphic with a Priifer group Zp,.
Next we state and prove a useful criterion for a Fitting class to be maximal in 6.
(4.26) Proposition (Bryce and Cossey [4]). The following statements about a Fitting
class g are equivalent:
(a) § is maximal in 6 (among Fitting subclasses of S partially ordered by inclusion);
(b) There exists a prime p such that | G: G a| e {1, p} for all G e <S.
Proof. Throughout the proof E(g) will denote the subgroup of A(6) defined by
Equation 4.y on page 723 and v the natural homomorphism from A(6) to A(6)/E(g).
(a) => (b): Consider the Fitting class §91, which certainly contains g. If g'Ji = 8,
then g = (Ji=i 0®1‘ = ©> contrary to the assumption that g is maximal in S.
Consequently g91 = S, and so s(g91/g) e 91. Therefore by (3.10)(a) we have
g 6 Locksec(6), and it follows from (4.14) that S(g) is a maximal subgroup of A(6),
whence A(6)/E(g) Zp for some prime p. But Corollary 4.15 implies that G/Ga is
isomorphic with a subgroup of A(6)/E(g) for all Ge £ and Statement (b) now
follows.
(b)=>(a): Assume that Statement (b) holds. It certainly implies that G/Ga e '21 for
all G e S and hence that g is normal in S. Suppose for a contradiction that g is not
maximal in ©.Then by (4.14) the torsion group A(S)/E(g)is not simple and therefore
4. The Lausch group
735
contains a subgroup, B/S(R) say, of order qr, where q and r are (not necessarilv
distinct) primes. If the group B/E(g) is cyclic, let x be a generator and set у = 1- if it
,S n°‘ CyC‘,C’ hfen q - r^d “ has two generators, x and у say. Let v denote the natural
ep-morphism from A«5) onto A(S)/S(&). Since (6 о v, A(S)/S(&)) is evidently an
®.’f ‘ g p r’there exlsl soluble groups X and Y such that Х(ёх о v) contains x and
Y(6r о v contains у. Let В; = X x Y. Then B/H(g) = <x, y> < D6d by (4.10), and И
follows from (4.15) that qr||D: £>s|, in contradiction to Statement (b). □
The more general situation, where X is a Fitting class maximally contained in a
Fitting class 9) (£ S), has been considered by Bryce and Cossey [4]. They give two
conditions for this, one necessary and the other sufficient, which are described in
Exercise 1 below. Neither condition is both necessary and sufficient, and the problem
of closing the gap is still unresolved.
The partial order of strong containment offers another framework in which to
study the maximally of one Fitting class in another. But whereas for Schunck classes
this problem has been satisfactorily settled, at least for those strongly contained in
6 (see VI, 1.12), a characterization of Fitting classes maximally strongly contained
in £ is still lacking.
Conjecture. Let S be a Fitting class. If g «mM 6, then § is 6-normal.
In view of (1.31) the truth of this conjecture would mean that the Fitting classes
maximal in 6 are the same for each of the partial orders « and £ and are thus
characterized by Condition (b) of (4.26). At the time of writing the truth of this
conjecture has been verified only for normally embedded Fitting classes.
(4.27) Proposition (Lockett [1]). Let g be a normally embedded Fitting class and
assume that § is maximal with respect to « in S. Then § is normal in 6.
Proof. First suppose, by way of contradiction, that there exists a prime p such that
Lp(g) is not normal in £. Then by (3.10) (either part) we have Lp(jy)£pSp^ £.
Furthermore, since ft is by hypothesis normally embedded, we have ft « Lp(ft)£pGp,
by IX, 3.27, and so ft = Lp(5)6pSp. by maximality. Since ft £ Lp(ft), we deduce that
ft = £ . = (J"= I g(SpSp)" = e. a contradiction. Therefore Lp(ft)* = S for all
primes p₽ But thenby (1.37)₽we have Lp(ft*) = <5 for all primes p, and from the
definition of the operator Lp( ) in IX, 1.14 we conclude that ft* = 6. □
Concluding Remarks. In Exercises 10 and 11 of Section 1 of this chapter we indicated
how the concept of a Lockett section could be dualized in the theory of formations
(for a detailed account, see Doerk and Hawkes [2]). Building on this Schmieden has
taken the process a stage further. In Schmieden [1] (see also Torres [1]lhe proposes
definitions for the dual concepts of a -formation pair and of the Lausch group of a
formation and explores their natural consequences^ The essential ideas res^U
from Schmieden’s work are presented in Exercises 4-7 below. Here we simply note
the significant difference between the two situations. ,
(1) gJn contrast to the case of Fitting classes, both the Lockett section and
736
X. Fitting classes—the Lockett section
‘Lausch group’ of a formation are trivial in a soluble universe. This is a direct
consequence of IV, 1.18.
(2) Whereas the Lausch group of a Fitting class 8 is a quotient of a restricted direct
product of the groups in Set(g), the ‘Lausch group’ of a formation g, is a subgroup
of an unrestricted direct product of the groups in SetJgJ; in fact, no non-identity
element of the subgroup has finite support.
(3) The dual of the fundamental Theorem 4.14 holds with full force only for finitely
generated formations. In general there is no bijection between ‘Locksub’(g) and the
subgroups of the ‘Lausch group’ of g.
(4) In further contrast to the situation for Fitting classes, the only known ‘Lausch
groups’ of formations are elementary abelian 2-groups.
Open Question. Is there a formation whose ‘Lausch group’ has exponent greater than
two?
Exercises
1. (Bryce and Cossey [4]) Let 3E and 9) be Fitting classes with X s 9) c (5, and
consider the following 3 statements:
(A): 3E is maximal among the Fitting subclasses of 9);
(Cl): There exists a prime p such that, for all Ge?), the J£-injectors of G have
index 1 or p;
(C2): There exists a prime p such that, for all G 6 9), the group G/Gj is a p-group.
Prove that for Assertion A to hold. Condition Cl is sufficient but not necessary,
whereas Condition C2 is necessary but not sufficient.
2. Let g = Fit<G>. Prove that A(g) = GT(g)/F(g), and deduce that |Locksub(g)|
is the number of subgroups of G/GB>. (Hint: Use (4.14) and (4.15).)
3. (Gaschiitz [12], Simoneit [1]). Let g be a Fitting class. On the set of pairs
{(g, G): Ge Set(g), geG} define a relation ~ by (g, G) ~ (h, H) if and only if
(g, h '} e (G x Я)8. Show that ~ is an equivalence relation. Let [(g, G)] denote
the equivalence class containing (g, G), and let L denote the set of equivalence
classes. Show that a well-defined binary operation on L is obtained by setting
[(g, G)] [(/i, Я)] = [((g, /i), G x Я)], where without loss of generality we suppose
that G x Я 6 Set(g). Prove that with respect to this operation L is an abelian
group isomorphic with the Lausch group A(g).
The next four exercises summarize work of Schmieden [1] dualizing the Lausch
group, etc. to formations.
4. Let g be a formation of finite groups. A pair (Л, ё) is called a universal ‘formation
pair' for g if A is a (possibly infinite) group and
d: Set(g) -»(J {Hom(A, G): G e Set(g)}
is a map with the property that the image 6G of each G e Set(g) is a homomorphism
from A into G such that:
5. Examples of Fitting pairs and Berger’s theorem
737
m П aeG’ H 6 Se,(R) and for a11 ^morphisms a: H - G,
(2) Q {Ker(8c): G e Set(g)} = 1, and
Wllh properties «"responding to (1) and (2), then there exists
a normal embedding ф-. A -> A such that dG = ф о dc for all G e Set(g).
Let D be the unrestricted direct product of the groups in Set(g), and denote the
Gth component of an element g e D by Sc. Let A denote the subset consisting of
all geD with the property that gG = g„a for all G, HeSet(g) and all epi-
morphisms a. H -» G. For H e Set(g) let Л -» H be the map defined by setting
96н = 0И- Prove that A is an abelian subgroup of D and that (Л, 8) is a universal
formation pair’ for g. Verify the obvious uniqueness properties of (A, 8), in
particular that A is unique up to isomorphism. (The group A is called the ‘Lausch
group of g.)
5. Let g be a formation and (A, 8) a universal pair for g. Let g0 denote the associated
formation defined in Exercise 11 of Section 1 of this chapter. Prove that
(') So = (G e Set(g): A8C = 1);
(ii) If G 6 Set(g), then A8C = Gs°;
(iii) A is finite if and only if there exists a Ge Set(g) such that g = QR0(g0> G);
(iv) If g = QR0(g0, G), then 8G: A -» GSo is an isomorphism.
6. Let (A, 8) be a universal pair for a formation g, and let ‘Locksub’(g) = {f>: is
a formation and g0 S S S g}. For each § in ‘Locksub’(g) let (A(§), 8(§)) be a
universal pair for f>. Then there exists an epimorphism >Я5)- A ->• A(§) such that
8С — ° <?(S)C for all G 6 Set(g). Prove that the mapping E: ‘Locksub’(g) -»A
defined by §E = Ker(V<(§)) is injective. Show further that if A is finite, then E
is also suijective, and that in this case E is a lattice anti-isomorphism from
'Locksub’(g) onto the subgroup lattice of A. In other words, show that
(Si^g2)E = (g1S)(g2,E), and
(QRo(Si, S2))s = (Si2)^^)
for all gn g2 e 'Locksub’(g).
7. Let n > 8, and let G„ denote the representation group or “double cover” of Alt(n)
(see Exercise 12 of Section 1 of this chapter.). Let g„ = qr0(G8, ..., G8+m). Prove
that if m 6 M, then the ‘Lausch’ group Am of gm is elementary abelian of order 2m,
whereas if m = co (with the obvious meaning), then Am is an abelian group of
exponent 2. Show further that in the latter case the map E of Exercise 6 above is
not surjective.
5. Examples of Fitting pairs and Berger’s theorem
In Laue Lausch and Pain [1] the authors describe an ingenious way of constructing
Fitting ntirs by using the transfer. We shall call them ‘transfer Fitting pairs and wtll
ittngpa У g nart of this section to their construction and properties. Their
devote а со-“^т ot^e ris wh.ch states that for certain Fitting classes
importance lies m a theorem oi ovig
5. Examples Ы Fining pairs and Berger's theorem 7Э9
(5.2) Propowlion. Hsswner/wr Hypor/ieses 5.1 hold. ForeochGc Sel(ff)de/uieamap
do: G — {1. — I j as follows:
tSJJ) g^c - Sgn<kc><9). (9 e G)
the sign of the permutation induced by gon O(G). Let A — {g^c: Я 6 G e Set(ff)}. Then
(A. d)isan Я-Fitting pair.
Proof It is dear that dc is a group homomorphism, and hence that И is a subgroup
of the multiplicative group {1. - 1 J. Thus the requirement FP2 of IX. 2.10(c) in the
definition of a Fining pair is satisfied We now vcnfy that FPI is also fulfilled. A
normal embedding is an isomorphism onto a normal subgroup, and so we must bnng
into play botli Part (a) of Hypotheses 5.1. which deals with isomorphisms, and Part
tb), which handles normal subgroups. Let G. H e Set(ff) and a e NernblG -» Hy It
follows easily from Equation 5л that (G.,)* = (G*)^ for each u> e 0(G) The G-orbil
of 0(G) containing w is isomorphic with the coset space G/G„, while the G‘-orbit of
O(G’) containing aa is isomorphic with the coset space G'HG*}^. and is therefore
isomorphic with (G7Gwf. It follows that
(5-7) Sgn<ac><9> - Sgnttc.,(9*).
Next we apply Hypothesis 5.1(b) with N « G* S H. Condition (i) ensures that g‘
induces a permutation of the same sign on 0(H) and lm(P). and since by Condition
(ii) an odd number of isomorphic ^'-orbits of 0(G*) are mapped by 0 to a single
9*-orbit of lm(pL it follows from Condition (iii) that д' has the same sign on lm(p)
and O(G'). Thus the permutations induced by g' on 0(H) and O(G') have the same
sign, and by combining this fact with (5.',). we obtain
Scnnd?) - 5япЛМ|(9'>
for all g e G. This is obviously equivalent to the statement that do = a о <fN. which is
the desired requirement FPI of an ff-Fitting pair. О
(5.3) Examples. Let ff continue to denote a fixed Fitting dass.
(a) Let S denote a set of odd natural numbers including 1. and let JE be another
Fitting class. For each G e ff define
O(G) — {9 • G>: o(g) e $}.
and regard 0(G) as a G-set under the action of conjugation. To avoid possible
confusion, we distinguish the G-action on О in this example with a dot: thus <u • g -
g~'a>g forшe 0(G) and geG.
We now verify that Hypotheses 5.1 are satisfied. If a: G — G* is an isomorphism,
we define a map d:O(G) — O(G") by setting wa - u>' for oitO(G) Then (w-p)d
denotes the element Ig'wgf of G. and this equals (9‘)“‘u>'9* because a is an
X Fitting duces—lhe Lockett section
rt (including 3 and <f) it is possible to compute the Я.-radical of any group in Я in
a finite number of steps by means of suitable transfer Fitting pairs. This theorem is
proved at the end of the section, but instead of following the original approach of
Berger (31 we give a presentation due to Bnson [1 J. [4]. This involves computation
within the Lausch group, and requires a version of the transfer pairs which lies in
complexity somewhere between lhe pairs of Laue. Lausch. and Pain cited above and
lhe more elaborate pairs constructed by Berger [3]. This version, due to Bnson. has
the advantage of being easily comprehensible and at the same time sufficiently
comprehensive to cope with a formulation of Berger's theorem which captures its
salient features.
Although Berger's theorem shows that the transfer Fitting pairs contain all the
information needed to identify the ft(-radical. explicit calculations with them call for
a detailed knowledge of the subgroup lattice and can be laborious, even for quite
small groups. Thus, for everyday purposes, such as testing a conjecture or illustrating
a theorem, it is useful to have at hand a selection of Fitting pairs for which the
calculations are relatively easy and the identification of the associated kernel straight-
forward. In IX. 2.14(a) we have already described one family of such Fitting pairs,
involving the determinants of linear maps induced on chief factors by conjugation.
Before we discuss the transfer pairs, we therefore describe some other useful families,
which are based on the sign of a permutation. Except where otherwise stated, the
unnerse throughout this section u C.
Fining pairs from permutation repmeatatiom
For the purposes of this discussion, let Я be a fixed, but arbitrary. Fitting class of
finite groups. w
($.1) Hypothecs. Assume that to each G in Я is assigned a non-empty G-set 0(G)
satisfying the following two conditions:
(a) For each isomorphism a: G — G*. there exists a bijection a: 0(G) — O(G') such
that
(5.x) (uj)j •
for all g e G^.d all cu e 0(G).
(b) Whenever N 3 H t ft there exists an W-set homomorphism
4-d(fV):O(W)-.O(H)w
such that
(i) each element of N induces an even permutation on lhe set 0(H)\Im(d),
(ii) ld'*(«>)l is odd for each a> e Im(d). and
(iii) for each ne N and each <n>-orbit Г of O(N). the map dr is injective.
Notaitonal remark. In this discussion we will sometimes write maps exponentially to
avoid dumsy bracketing in certain expressions.
740
X. Fitting classes—the Lockett section
automorphism of G. But by definition of a and the prescribed G“-action on 12(G“),
this is precisely the element (coa)ga, and therefore Equation 5.a holds.
Now let N <! H e g. Then Nj = N n Hz, and so Q(N) £ 12(H). Therefore, taking
ф to be the inclusion map, which is obviously a monomorphism of N-sets, we have
|^-1(m)| = 1 for each m e Im(^). and therefore Requirements (b), (ii) and (iii), of
Hypotheses 5.1 is satisfied. To show that Condition (b) (i) holds, let В be an N-orbit
contained in the A'-set Q(H)n\Q(N), and let В 1 = {m-1: m e B}. Then obviously
B-1 £ 12(H)\12(A'), and because CN(m) = CN(co1), the sets В and B1 are isomorphic
as A'-sets. We assert that В # В-1. For, if not, then co = (co-1)" for some сое В and
ne N, and it follows that
co2 = con-1co-1n = [co-1, nJ.
Therefore co2 e N because N <1 G, and since co has odd order, it follows that
coe N n 12(H) = Q(N), contrary to the choice of B. Thus В B-1, and it follows
that the A'-orbits of 12(H)\12(A') fall into isomorphic pairs. Hence Requirement
(b)(i) also holds, and we have shown that Hypotheses 5.1 are fully satisfied by this
example.
Let denote the kernel of this Fitting pair (see IX, 2.13(a) for the definition). By
IX, 2.1 l(ii), (1.7), and (5.2) we have e Locksub(g), and, in particular, is an
g-normal Fitting class. This example is due to Camina [1], who used it to prove that
(5.<*>) Fit<Sym(3)> * S3S2.
To see this, let g = let £ = S3, and take {L 9} for the set S of odd integers.
Since 12(Sym(3)) = {1}, we have Sym(3) e й(, 9), whereas Dih(18) ф йц,9) because
O3(Dih(18)) contains 6 elements of order 9, on which an involution induces an odd
permutation of cycle type (2, 2, 2). Thus Fit<Sym(3)> £ Яр.в) G3G2-
(b) The next example is due to Laue [1]. Let 3 be a class of non-abelian simple
groups. For each Geg put
12(G) = (KsnG: Ke 3u(l)}.
By II, 2.13 the class D = d03 is a Fitting class, and it is easy to see that 12(G) consists
of the identity subgroup together with the simple direct factors of Gr. Then G, acting
by conjugation, induces an obvious G-set structure on 12(G), for which we shall now
show that Hypotheses 5.1 are fulfilled.
If a: G -» G“ is an isomorphism, and if К e 12(G), then clearly № e 12(G“). Therefore
a must be defined as the map which sends К to K°, and one easily verifies, as in the
preceding example, that Equation 5.a holds.
If N < H eg, then NT. = Nr. Hr. Therefore 12(H) £ 12(H), and we can take ф to
be the inclusion map. By A, 4.14 there exists a normal subgroup R of H which
complements Hr in Hr, and 12(H)\Q(H) is evidently the set of simple direct compo-
nents of R. If n e N, we have
[R, n] < R n N = R n Hr n N = R n Nr = 1.
5. Examples of Fnting pairs and Berger's theorem 741
There<ore the elements of N induce the identity permutation on Q(H)\Q(N), and it
follows that (5.1) is satisfied.
By Proposition 5.2 we obtain an g-Fitting pair from this construction, and if we
denote its kernel by R(3) when g = G, it follows that R(3) is an G-normal Fitting
class containing 6. Furthermore, if J e 3, it is clear that the R(3)-radical of the wreath
product J 4.,^ Z2 is its base group, and therefore R(3) / G when 3/0.
There is another prescription for deriving a Fitting pair from the sign map, which
can be regarded as the dual of the one described in Hypotheses 5.1.
(5.4) Hypotheses. Assume that to each G in g is assigned a non-empty G-set U(G)
satisfying the following two conditions.
(a) For each isomorphism a: G -> G" there exists a bijection a: U(G)-> U(Ga)
satisfying Equation 5.a (on page 738) for all g e G and m e U(G).
(b) Whenever N < H e g, there exists an N-set homomorphism
^ = ^(N): U(H)„-U(N)
such that
(i) each element of N induces an even permutation on the set ?5(7V)\Im(^), and
(ii ) |^”1(m)| is odd for each co e Im(^).
We can now repeat the arguments of Proposition 5.2. Hypothesis 5.4(a) ensures
that the equation
(5.e) gdG = Sgnn,C)(0)
is invariant under isomorphisms of G and so defines a homomorphism dG. G —»
{1, -1} satisfying dG = aodc.. Hypothesis 5.4(b) then ensures that, with N =
G“ < H, the map d^. coincides with the restriction of dH to G“. Thus we obtain the
following proposition.
(5.5) Proposition. If Hypotheses 5.4 are fulfilled, the map dG defined by Equation 5.e
gives rise to an ^-Fitting pair (A, d), where A = (gdG: geGe Set(g)} < {1, -1}.
(5.6) Example (Blessenohl and Laue [1]). Let g be a Fitting class, and let © be a
second Fitting class such that for all G in g the following conditions are satisfied:
(i) G has a unique conjugacy class of ©-injectors:
(ii) If F e Injff,(G), then |G: F| is odd.
For example, when g = G, we could take for ©any Fitting class satisfying G2 « © £
G; and if g = G, we could take © = G2.
In any case, for each G in g we then define
U(G) = Injffi(G),
regI,d <xo ,ta' H”wh““54 ”
1X1
X. Fitting classes—the Lockett section
satisfied. Let a: G -> Ga be an isomorphism. The fact that injectors are preserved
under isomorphisms ensures that (5.a) holds when a is defined as the map which sends
V e Injffi(G) to L'“ e Injffi(G“). To verify (5.4)(b), suppose that N < H e g, and for each
V e Inj <f, (//) define
Ф(У) = VnNeInjK(N).
Since V" n A = (F n N)n for n e N and since Injffi(/V) is a conjugacy class of sub-
groups of N, the map thus defined is an A-set epimorphism from U(H) onto O(A);
in particular, Condition (i) of (5.4)(b) holds. If W e Injffi(A), we have
^-1(H/) = {FeInjffi(H): VnN = W}.
Evidently Inj([,(H) is partitioned by the subsets ф 1 (IT), and these are permuted
transitively under conjugation by N. Therefore |^-1(H/)| is independent of W, and
consequently = IInj<f,(/7)|/|Inj<r,(A')|. Because |Inj([,(H)| = |H: AH(F)| for
V e Injt6(H), and because |// : lz| is odd by hypothesis, it follows that |^-1(И/)| is odd.
Thus (5.4)(b)(ii) is also satisfied, and our assertion is justified.
Thus, by taking g = ® and ® = G2, we can deduce from Theorem 5.5 that the
class of groups in which every element permutes the Sylow 2-subgroups in an even
permutation is an (£-normal Fitting class. It is distinct from ® because it does not
contain Sym(3) for example.
A variety of examples of Fitting pairs have been published which exploit the above
procedures involving the sign of a permutation, and a selection of them is included
in the exercises at the end of this section.
In preparation for our discussion of the transfer Fitting pairs we prove the follow-
ing elementary consequence of the quasi-R0 lemma.
(5.7) Lemma. Let g be a Fitting class, and let X < XY, where Y is a nilpotent
^-subgroup of XY.
(a) Let C be a normal subgroup o] Y such that [X, C] = 1, and assume that
X n C = 1. Then XYe^if and only if XY/C e g.
(b) Let Y* denote the group of inner automorphisms of XY induced by Y, and let L
denote the semidirect product L = [XY] Y*. If XY eg, then Leg and XY* e g.
(c) If XY e g and if Y denotes the group of automorphisms of X induced by Y by
conjugation, then the semidirect product [X] Y is in g.
Proof, (a) The hypotheses ensure that С < X У, and so, by taking N, = X and
N2 = C in IX, 1.13, we obtain the desired conclusion.
(b) Since Y e 91, then YY* e 91 by A, 8.8(c), and consequently L is the join of
subnormal subgroups X Y and X Y*. Thus if we can show that X Y* e g, it will follow
that L e Nog = g. Consider the direct product G = X Y x Y, and let D denote the
subgroup {(у, у). у e Y} of G. Since Y e 91, we have (X x l)Des„g = g; furthermore
(X x 1) n D = 1. Let Co = Cr(X Y) and put C = {(c, c): re Co}. Then from Part (a)
of this Lemma (with D in place of Y) we have ((X x 1)D)/C e g. But clearly
5. Examples of Fitting pairs and Berger’s theorem 743
ЦХ X 1)D)/C^[A- X 1](D/C)s [X] F*, whence the subgroup XY* of L belongs to
(c) Take G and D as in Part (b), let CB = Cr(X), and let С = {(с, с): c e Co}. Then,
by the same argument, we have [X] Y ((X x 1)D)/C eg. □
The transfer Fitting pairs of Laue, Lausch, and Pain
Let p be a prime and g a Fitting class. Our first task in this discussion is to identify
certain favourable properties of a fixed group U with respect to p and g. When these
are fulfilled, we shall be able to construct from U an g-Fitting pair (P/Po, dv), where
P/Po is a certain canonical section of a Sylow p-subgroup of Aut(G). The map dv,
applied to a given group G in g, involves the transfer of G into subnormal subgroups
of G which are isomorphic with V. When it is required to emphasize the role of p,
and possibly g, in the construction of dv, we will denote it by dV p or dv-p'R- Berger’s
theorem then states that for certain Fitting classes g the g,-radical of any group G
in g can be described as the intersection of the kernels of these Fitting pairs (P/Po, dv);
here U must range over the isomorphism types of subnormal subgroups of G and p
over those prime divisors of |G| for which U has the favourable properties described
below.
(5.8) Definitions. Let g be a Fitting class, let p be a prime, and let Geg.
(a) Let P* be a Sylow p-subgroup of Aut(U), and put
P = P* n([G]P*)8.
Then the semidirect product [G]P is called an (g, pycompletion of G. By the
usual abuse of notation we identify G and Aut(G) with their canonical images in
[G]Aut(G); thus GP < GP* < GAut(G).
(b) If H < Aut(G), then define
r(//) = <a e H: [G, a] < G>.
(c) If GP is an (g, p)-completion of G, we shall consistently use Po to denote the
following subgroup of P
(5n Po = HP){P: Aut(G)}.
We recall from A, 17.4 that {P; Aut(G)} denotes the normal subgroup
^(dfAfinhegX'F^
if
(i) Peg, and
(ii) P/Po / 1 for some (g, p)-completion FPo
(fGlP*),. it follows that [G]P = ([G]P*)s and
two (g, pfcompletions of G are isomorphic.
(5.9) Remarks, (a) Since G <
hence by Sylow’s theorem that any
744
X. Fitting classes—the Lockett section
(b) If U is p-active, then clearly U = [L/, Р]. Therefore U = OP(UP), and, in partic-
ular, U is a characteristic subgroup of UP.
(c) If U is p-active, then p e Char(g) by IX. 1.7.
Throughout the subsequent discussion leading up to Theorem 5.16 the following
hypotheses will be in force.
(5.10) Hypotheses. Let g be a Fitting class and p a prime. Assume that g contains a
p-active group U and choose a fixed P* e Sylp(Aut(L/)). Let UP denote the associated
(g, pfcompletion of U defined in (5.8)(a), and let Po be the subgroup of P described in
(5.8)(c).
We have now identified the abelian p-group which appears as the first term of the
putative g-Fitting pair (P/Po, dv). Our next task is therefore to define the map dv,
which means defining a homomorphism dg: G -♦ P/Po for each group G in g. This
we do in stages, proving invariant properties of the various constituent homo-
morphisms defined along the way. In what follows, maps will be written on the right,
albeit sometimes exponentially; this will be consistent with our earlier practice for
Fitting pairs. We finally arrive at our definition of dg in (5.15), and until then G will
denote an arbitrary, but fixed, group in g.
Stage I
Let X and S be subgroups of a group XS such that (i)U = X < XS e g and (ii) S is
a p-group. Note that Hypotheses 5.10 imply that p e Char(g) by (5.9)(c), and therefore
Segby IX, 1.9.
Step (a): Let t/z: X -> U be an isomorphism. Then the action of XS on X by conjuga-
tion induces a map ф: XS -> Aut(U) defined as follows: if g 6 XS, let
u —(p“1(uV'“1)0)V'
for all ue U. It is straightforward to verify that дф e Aut(l/) and also that ф is a
group homomorphism whose kernel is Cxs(X).
Step (b): First we assert that because XS e g, the semidirect product [L/J (Si//) also
belongs to g. If s e S, let s denote the automorphism of X induced by s by conjugation,
and put S = js:se S}. Then it is easy to check that the map which sends a typical
element xs of [X]S to the element (хф)($ф) is a group isomorphism, and since
[X]S e g by (5.7)(c), our assertion is therefore justified.
Since 5ф is a p-subgroup of Aut(U), there exists an element p e Aut(U) such that
(Sipy is contained in the Sylow p-subgroup P* of Aut(U), fixed according to Hypoth-
eses 5.10. Since [l/](S^y = [l/](Si^)e g and [L/](Si^y sn [l/]P*, it follows that
(Si/z)(i < P* n ([U]P*)g = P. Thus, if pp denotes conjugation by /? in Aut(U), the map
Pf is a monomorphism sending 5ф to a subgroup of P.
Step (c): Let So denote the subgroup {S; XS} of S. Since ф ° pe: XS -» Aut(l7) is a
homomorphism mapping S into P, we have ($0)ф ° pe< {(S)i// о pf-, (Х5)ф ° pf] <
{P; Aut(l/)} <P0.
5. Examples of Fitting pairs and Berger’s theorem 745
Now let v denote the natural homomorphism from P to P/Po, and define
4x.s = &s°Pn°v.
Evidently the map tjx s is a homomorphism from S to P/Po with Ker(iJx s) > So.
Ostensibly tix.s depends on the choice of the isomorphism ф: X -> V and the conju-
gating element p e Aut(C'). We now show that:
(5.11) The map t]x s: S Р/Рэ defined above is independent of both our choice of ф
and our choice of p.
Proof. Suppose that ф is another isomorphism from X to U and that у is an
automorphism of 17 such that (S<j>)y < P. Let tjx s denote the corresponding map
defined in terms of ф and y; further, put r = ф~1ф e Aut(Cj, and let s e S. By direct
calculation one easily verifies that the automorphisms sf and (si//)' of U are equal.
Let x denote the element (stfif of P, and put a = /T‘ry. Then x‘ = (si)/)" = (sj>)y e P,
and consequently x *x“ e {P; Aut(l7)} < Po. Hence x = x” (mod Po), and therefore
Wx.s = S1A ° Pf о V = (/?-1(s^)/?)v = XV = x“v = G>_1(s<?)y)v = sf о py о V = stjx.s for
all s e S. □
As a consequence of (5.11) we have the following observation:
(5.12) Let p: XS -> YT be an isomorphism such that Xp = Y and Sp = T. Then
4x.s — Ps 0 4y.t-
Proof. Suppose that the map ijiT is defined in terms of the isomorphism ф: Y-> V
and the conjugating element y. By means of direct computation it is easy to verify
that (р~°ф) = p о ф, and from this it follows that the map p о tjr T sends an element
s e S to the following element in P/Po:
(y“’(s(p о ^))y)v = (s)(p ° ф) ° py ° v.
But this element coincides with the image of s under t]x.s when the map tix.s is defined
with respect to the isomorphism p о ф-. X - U and the conjugating element y, and
so (5.12) is true.
Now let X be a subgroup of the given R-group G such that X = U, andHet
S e Syl (Ne(X)\ Further assume that XS e R. (This assumption is an essentia pre-
requisite fo! the construction of for, without .tHhe homomo^sm ?^(5T1)
cannot be defined.) Since the image of t)x,s.S-P/P0 is aoeuan,
S' < Ker(t)x.s), and therefore that the map
(5.6)
4x.s: sS srlx.s
746
X. Fitting classes—the Lockett section
is a well-defined homomorphism from S/S' to P/Po. If t>c_s denotes the transfer
map from G into S/S' (see A, 17.1), we can therefore define a homomorphism
Pg.x.s- G P/Pq thus:
(5-0 @g.x.s = vg-s ° 'lx.s-
On the face of it, the map 0G.x.s depends on the choice of X and S. We now show
that in fact it depends only on the conjugacy class of X.
(5.13) Let geG, and let T e Sylp(Nc(X9)). Then 0c,.x«.t = 0C,X,S-
Proof. Since T9 1 e NC(X9)9 1 = NC(X), by Sylow’s theorem we have T9 1 = S" for
some n e NG(X). Therefore Xе = X"9 and T = S"9, and so without loss of generality
we can suppose that T = S9. Let xe G, and suppose that .wc_s = sS' with s 6 S.
Applying (5.12) with /i as the isomorphism from XS to (XS)9 induced by conjugation
by g, we therefore have sgxs = s9rjxe r. But by A, 17.3(a) we have xnG_T = s9T’, and
so the definition of the map 0 in (5.i) now yields the desired conclusion. □
Notation. In view of (5.13), instead of 6G.x,s wc shall now write 0G,tx]> or simply 0m
when G is understood. Here [X] denotes the conjugacy class of X.
Stage III
We recall that G denotes a fixed group in g. Let ifnfU, G) denote the set of subnormal
subgroups of G which are isomorphic with U. (We remark in passing that the
subsequent discussion is also valid, with appropriate modifications, when .9v(U, G)
is replaced by the set Sf(U, G) = {X < G : X = I/}.)
Suppose that £/n(U, G) is non-empty, and let [Xj],..., [X*] denote the distinct
conjugacy classes which make up Уп(1), G). For i = 1, ..., к choose an St in
Sylp(AG(XJ). Let p‘ denote the exponent of P/Po, and for i = 1,..., к define integers
tf such that
ti|Nc(Xi): SJ = 1 (mod p‘).
Since pt|Nc(Xj): S;|, such an integer tf exists, and its residue class modulo pe is
uniquely determined. In order to be able at last to define the map dG, we need each
subgroup X.Sj of G to be in g, and to ensure this we make the following general
hypothesis about g.
(5.14) Hypothesis. If U is an (g, p)-active group, assume that for each G e g,
X e S^(U, G), and S e Sylp(NG(X)) we have XS e g.
With this hypothesis in operation we can now make the following definition.
(5.15) Definition (The map dv,F’s). For obvious notational reasons we usually
suppress p and g and write the map simply as dv. Let g e G e g, and define
5. Examples of Fitting pairs and Bergers theorem
747
1 if Уп(и, G) = 0, and
П if Sb(U, G) = (J [xj 0.
* * /=1
STZThomomo”“OrPhrm in'° the abeHan gr°UP P/P°’ « is obv*°us that
dc is also a homomorphism from G into P/Po; furthermore, since each t, is uniquely
determined modulo the exponent of P/Po, it is clear from (5.13) that depends only
on G and the isomorphism class of U. We shall now prove that (P/Po, d1') is an
Fitting pair.
(5.16) Theorem (Laue H„ Lausch, and Pain [1]; Berger [3]; Brison [4]). Assume
that Hypotheses 5.10 and 5.14 hold. Then the map d% defined in (5.15) and the abelian
group P/Po satisfy Requirements FP1 and FP2 in the Definition IX, 2.10(c) of an
^-Fitting pair.
Proof. If a: G — G° is an isomorphism, it follows from (5.12) and (5.13) that
for all g e G and i = 1,..., k. Since |NG(Xf): SJ = |NG(X?): S?|, and since obviously
Gf = .e/n(U, G“), we deduce from the definition of dG that for all g in G
In other words, we have dG = “ ° ^c1- In order to verify FP1 it will therefore be
sufficient to prove that if N < G e g, then d% is the restriction of the homomorphism
dG to N.
First we show that the conjugacy classes [XJ which lie in 5^(U, G)\£Fx(U, N)
make a trivial contribution to the right-hand side of the definition of ndG when neN.
To this end let X be a subnormal subgroup of G such that X N and such that there
exists an isomorphism if : X -> U. Let n e N, let nt>G_s = sS’ with se Se Sylp(NG(X)),
and recall from A, 17.3(b) that s may be chosen from SrN. Then [X, s]<
X n N < X, and evidently the corresponding automorphism sif induced on V satisfies
[t/, si/i] < U; hence sif e r(P) < Po, and so srjx s = 1. Consequently nPc.m = 1, and
in calculating the restriction of d% to N, we need only consider the G-conjugacy classes
[X ] which are contained in N, namely those in N).
Thus let X e ^«(G, A'), and let if: X -> U be an isomorphism. As usual we choose
an S in SyL(Nc(X)), and we put T = NnS, L = NS, and M = NNG(X). Observe
that Те SylANn(X)) because Nn(X)<Ng(X). Let {u„ ..., u } {р,,..., 4»}. and
(w w 1 be right transversals to T in N, to NL(X) in NC(X), and to M in G
respectively (see Figure 5.1). Let t|Nc(X): S| 1 (mod p‘), where p‘ is the exponent
of P/Po, and put r = tb. Then
r|N„(X): T| = rblNcW: S| = t|Nc(*): S| = 1 (mod p‘).
748
X. Fitting classes—the Lockett section
Let ne N. We aim ultimately to show that
(5-^) = П
j
where the G-conjugacy class [X] is expressed as the union of N-conjugacy classes
[Y£J. It will then follow from the preceding paragraph and Definition 5.14 that the
maps dg and coincide on N. We shall prove Equation 5.7 in stages, proceeding
from N, via L and M, up to G.
Step 1: Let Тщп = Tut- for i = 1, a; thus the map i-»i' is the permutation of
[1,o} induced by multiplying the cosets of T in N on the right by n e N. Let
s0 = f]?=i uin(ui ) 1 e T.Then by A, 17.1 we have nvK~T = s0T' and also nvL_s = s0S'
because {tq,..., u„} also serves as a transversal to S in L and obviously Su:n = Su(..
If ф'. X -» U is the isomorphism and p the conjugating element used for calculating
4x.s, then clearly ф and p can also be used to compute rjx T, and from the defining
Equation S.t] we obtain soi/x s = soi/x T. Hence from Equations 5.0 and 5.1 it follows
that
n@N,txi ~ (soT )r/x.r — (so$ )*lx.s —
Step 2: Let i e {1,..., b} and put = Lv‘. If we replace L by f,; = NS"1 in Step 1, we
obtain
5. Examples of Fitting pairs and Berger’s theorem 749
because v, e Kr(X\ and beca..«> n
by (5.13). Since n e A < G and N ' we^”^ * replaced Ьу Т'"
П^т’Г‘UAandby A, 17.3we hfve ’ " L”‘” = Lt,‘ therefore nvM^
п(1м.т - nvM~s ° >lx.s
= Il(^l)vL^soijx
1=1
b
b
= П И4...1Л-)) by (5.k) with a = pC(
— (п®л.[%])1> by (5.;<).
Step 3: As before we have Mw,n = Mwt for i = {1,..., c}_ Using A, 17.3 as in Step 2
we have ’
= f] («’."«’Г'вм.от)
1=1
= П Hm-jx-.j) by (5.Л-) with a = p„t
= JJ (n0N,(x».|)6
i=l
by Step 2 with MWl in place of M. Since r = bt and the terms of the above products
belong to an abelian group, we conclude that
(n®G,tx])' — fl
t=i
Since M = NNG(X), we have [X] = {Xе} = (Jf=1 {X""'} = |Jf=1 {(X-f}. More-
over, if {(X",,)iv} = {(Х“’)л}, then w.nuf1 e AC(X) < M for some ne N; consequently
Mwj = Mwtn = Mn'w, = Mwh and so i = j. Thus the G-conjugacy class [X] is the
union ofc distinct A-conjugacy classes fX“'],..., [X^J. It follows that Equation
5.2 holds and hence that Requirement FPI is satisfied.
To complete the proof of this theorem, it remains to show that Requirement FP2
is also fulfilled, and this is clearly implied by the following lemma.
(5.17) Lemma. Let G = VP be the (8, pfcompletion of a p-active group V. Then the
restriction of the map d% to P is the natural homomorphism v from P onto P/Po. The
kernel of dG: G-* P/Po is VPQ.
750
X. Fitting classes—the Lockett section
Proof. Since U is by hypothesis p-active, we have U = OP(G) = Op( U], and it follows
easily that №(G, G) = {G}. Now recall that P* denotes the fixed Sylow p-subgroup
of Aut(G) such that P = P* n (GP*)B and that we view G = UP as a subgroup of
UP* in the semidirect product [G] Aut(G). Choose S* e Sy\p(UP*) with S* > P*,
and put 5 = S’ r, UP and T = S*r\U = Sr>U. Clearly T is normalized by P* and
S = TP e Sylp(G).
In order to define the homomorphism i/t, s of (5.i/), we take for ф the identity
automorphism i of G, and note that by definition of P we have
(5.v) xT = x
for all x e P. Since T is normalized by P*, it follows easily that TT is normalized by
P* and hence that TT < P* since P* e Sylp(Aut(G)). Thus ST — (TP}T < P*, and from
(5.7)(c) we conclude that ST < P* n (GP*)g = P. Thus in Equation 5.1/ we may take
= i and define
4v.s = C °
where v is the natural homomorphism from P to P/Po.
Let x 6 P. Let n = |G: S|, and let tn = 1 (mod pe), where pe is the exponent of P/Po.
According to (5.15) and Equation 5.1 we have
xdG = (xvG^s)'rjvs.
But by A, 17.5(a) we have xvG-s = x" (mod {S; G}); furthermore, as pointed out in
Step (c) of Stage I, we have {S; G}/S' < Ker(i/[/ s)/S' = Ker(^[, s). With the help of
(5.v) it therefore follows that
xdG = (x^Sfiju s = (xn')T о v = x"'P0 = xP0 = xv.
Consequently (dG)P = v, as desired.
Since P/Po is a p-group and OP(U) = G, we conclude at once that G < Ker(<fg),
and as Po = Ker(v), the final assertion of the lemma is clear. □
Thus we have finally established the fact that for each (g, p)-active group G we
obtain an g-Fitting pair (P/Po, d[j, provided that Hypothesis 5.14 is fulfilled. In our
present state of knowledge we are therefore justified in distinguishing the following
family of Fitting classes for special attention.
(5.18) Definition. A class g of finite groups is called a Berger class if it is a Fitting
class with the property that for all primes p and for all (g, p)-active groups G, we
have XS e g whenever G e g, X e Sf«(U, G), and S e Sylp(Nc(X)).
Thus, when g is a Berger class, (P/Po, dv) is an g-Fitting pair for all primes p and
for all (g, p)-active groups G, and this is precisely what we shall require to prove our
version of Berger’s theorem.
5. Examples of Fitting pairs and Berger’s theorem
751
(5.19) Remark. A Fischer class is a Berger class, but not conversely.
Proof. Let g be a Fischer class, and recall from IX, 3.5(c) that s S = X If X sn G
and 5 e Syl,(Nc(X)), then (XS)91 <j X sn G. Therefore XS e sf(G) s g, and g fulfils
the requirements of Definition 5.18. In Example 5.34(a) below we shall describe an
example of a Berger class which is not a Fischer class. □
The following property of the transfer Fitting pairs plays a crucial part in the proof
of Berger’s theorem.
(5.20) Lemma. Let pandqbe primes, not necessarily distinct, let % be a Berger class,
and let U be an (g, practice group with completion UP. Assume that V is an (g, qfactive
group with completion G = VQ such that | V\ < 1171, and if p = q, assume further that
V£U. Then Gdc’p = 1.
Proof. Since V is q-active. we have V = [Ц Q] and therefore G/G91 s Q. If q * p,
it follows that OP(G) = G and hence that Gd%p e Sp nO1 = (1). Now suppose
that p = q. If X e G), then U S X = OP(X) < OP(G) = V. But the hypotheses
exclude the possibility that U is isomorphic with a subgroup of V; hence .77(6', G) =
0, and consequently Gd^,p = 1 by the definition of in (5.15). □
We have now completed the first stage of our preparations for Berger’s theorem,
namely the construction of the transfer Fitting pairs with their stated properties. We
now embark on the second stage, which is concerned with the choice of special
representatives for cosets of the subgroup T(g) of A(g) defined in (4.2). In order
to simplify the notation, for the rest of this section we shall adopt the following
convention, which clearly involves no loss of generality.
Convention. Let p be a prime and g a Fitting class; let U e Set(g), and let
UP (< [17] Aut(l7)) be an (g, p)-completion of U. Our convention states that if
U = [и, P], then one of the (g, p)-completions of 17 actually belongs to Set(g). (The
presumption that U = [17, P] here implies that if UP is an (g, q(-completion of some
subgroup V, then q = p and V = 17.)
(5.21) Lemma. Let pePJetUe Set(g), and let G = UPbe the (g, pfcompletion of
U in Set(g). Then {P; Aut(l7)}ec < F(g).
Proof. Bearing in mind that P is regarded as a subgroup of Aut(L), we see that a
typical generator of {P; Aut(l/)}ec has the form (Г‘«’ /Wg with a e Aut(L) and
fi, a~'fa e P. We assert that the map a: U</i> -> L'<a Pa) defined by
a: (P‘, u) -► (a lP‘a, ua} (u e U),
. • and since a is obviously bijective, we have only to show that it
««A ‘°,r<
and the image of this element under a is
752
X. Fitting classes—the Lockett section
(a~lp,+Ja, (uflja)(u*a)) = (a~'P‘aa '/Pa, (uaa fija)(u*a)}
= (a 'fi‘a, ua)(a~'PJa, u*a),
the product of the images. Thus our assertion is justified. Since 17 and 17 (a1 Pa}
are subnormal subgroups of G, it follows from the final sentence of the statement of
(4.11) that T(g) contains the element (P~l(pa))eG = (P~'a~'Pa)eG. □
(5.22) Lemma. Let p e Char(g), and let H be a group in Set(g). Assume that
H = H'}>(h}, where h is an element of H of p-power order. Let UP be the group in
Set(g) with the following two properties:
(i) Я91 U e Set(g);
(ii) UP is an (g, p)-completion of U.
Then there is an element x in P such that heH = xi:Gp (mod F(g)).
Remark. The hypothesis that H = H^fh} implies that 17 = [17, P], and so we can
apply our convention and suppose that UP e Set(g).
Proof. Write Г for T(g), let h denote the inner automorphism of H induced by
conjugation by h, and form the semidirect product [Я]<й>. Within this group
the element /i-1A induces the trivial automorphism on H, and so is a
normal p-subgroup of [Я] <h}. By (5.7)(b) we have [Я] <h} e g, and so we can find
a group L e Set(g) such that there exists an isomorphism а: [Я] </i> -» L. Since
a|H e Nemb(ff -+ L), we have
hr.H = has, (mod Г)-
Because p e Char(g) by hypothesis, the nilpotent subnormal subgroup of L
belongs to g, and it therefore follows from (4.18) that (</i-1/i>a)«L < Г. Consequently
we have
haeL = haeL (mod Г)-
Next, let ф denote an isomorphism from Я91 to 17 and ф: H -► Aut(l7) the associated
homomorphism defined in Step (a) of Stage 1 of the construction of (P/Po, dv} on
page 744. As remarked in the subsequent analysis of Step (b) on page 744, there is an
isomorphism from H'Jfh} to [17] <htp> sending h to Ьф: moreover, there is a further
isomorphism pf : [17]</i^> -+ [17] (Ьф)11 which maps 1гф to the element x = (Ьф)? of
P. On composing a-1 with these two isomorphisms, we obtain an isomorphism from
the subnormal subgroup (Я9|(й»а of L onto the subnormal subgroup U <x> of UP
that maps ha to x. Hence by (4.11) we have
haeL = xr.LP (mod Г).
and now putting the three displayed congruences together, we obtain the desired
conclusion. □
5. Examples of Fitting pairs and Berger’s theorem
753
ре?, wedefine”8’ C”(5) °f Se‘<®>’ Let n e For «**>
•^f(S) - [t' e Set(g): U is (g, p)-active, 1171 < n}.
Lra л г den°te ‘Ьа‘(5’ p,-comPletion of L' which by convention lies in
aeUo)- *ve then define
C„(S) = {UP: p e P, U e Л£(Э)}.
Note that if G = UP e C„(S), then |G| < 1U||Aut(l/)| < n(nl), and therefore the set
C„(S) is finite.
(b) (The subgroup 0„ of A(S)). Let n e M, let | C,(®| = m(n), and set C„(g) =
{Gn..., Gm(„)}, where G; = UtPt is the (S, pj-completion of Ц. If ef denotes the
natural embedding of G; into Д(д), we define 0„ to be the following nilpotent
subgroup of A(S):
(lifC.(8) = 0, and
Vt®i x • X P^e^ otherwise.
The next result shows that each coset of Г(Э) m A(8) contains an element of 0„ for
some n 6 M.
(5.24) Proposition. Let % be a Fitting class, and letgeGe Set(S). Let n = IG91!- Then
geG e Г(8)®„-
Proof. We proceed by induction on n, writing Г for T(g). If n = 1, then G e 91 n 5,
and Gcc < Г by (4.18), and certainly the proposition is true in this case. Therefore
suppose thatn > 1, and set g = where the elements fcfc, are generators
of the Sylow subgroups of <p> and, in particular, have prime power order. For i = 1,
..., t let K, = <A;G>, the subnormal closure of kt in G. Then Kt e s„S = 5, and we
can find Hj e Set(5) and a,- e Snemb(H, — G) such that Я,а, = К,. Let h,- = k:cf'.
Since Ki is the subnormal closure of A, in Kt and ap -» K, is an isomorphism, Ht
is the normal closure of ht in Ht. Because А,сс = /1,а,«с = h^g. (mod Г), we have
(5.л)
0®c = П ^ieh. (mo<f П-
We now focus our attention on a particular ht in this product and, to simplify the
notation, we suppress the suffix i. Thus H = <hH> e Set(S) and o(h) is a power of a
prime, p say. If H is perfect, then He„ < Г by (4.3)(b) and certainly he„^;Г©
Now suppose that H' < H. In this case Я” is a proper subgroup of H and it is
supplemented in H by <Л>. Let U be the group in Set(S) such that U s H , and let
754
X. Fitting classes—the Lockett section
VP be the (g, p)-completion of V. Since [Я91, /1] = Я'л, we have [V, P] = 17, and by
convention we can suppose that VP e Set(g). Since |H9>| < |G”1, we certainly have
1171 < n. By (5.22) there is an element x in P such that
he„ = xeUF (mod Г).
If x 6 P\P0, then V is (Jy, p)-active; therefore VPe C„, and in this case we conclude
that heH e Г0„. If, on the other hand, x e Po, by (5.4) on page 743 we can write x = yw
with у e r(P) and w e {P; Aut(l7)}, and hence by (5.21) we have
xeuf = yeVF (mod Г).
By definition of r(P) we can find elements у,,..., yr in P such that у = у, y2 • • • yr and
[17, y,] < Cfori = 1, ...,r. Let i;\ = [17, y,] <y,>. Then(X)51 < [UyJL91 < 17, and
so |(У<)’|| < 117| < n. However sn VP e g, whence Yt e g and we can find groups
X e Set(<5) and maps Д- e Snemb(5< -» VP) such that ХД = Yt. Let yf = J’;/?;1, and
let Bj denote the natural embedding of Yt into A(g). Then
heH = Уеив = П YiEvr = П № (mod r)-
i=l i=l
By induction we have y,e( e Г0„_, for i = 1,..., r, and therefore heB e Г0„_, < Г0„
in this case. Thus, in any case, we have shown that each term hjg. in the right-hand
product of Equation 5.л belongs to Г0„, and therefore so does geG. □
In the final paragraph of the preceding proof we have shown the following:
(5.25) Lemma. If VP e C„, then PoeVF < Г0П_,.
The following consequence of Proposition 5.24 is obvious.
(5.26) Corollary. Let 0 = (J“=i ®n- Then A(g) = L(^)0.
Our final preparatory lemma provides a better fix on the location of a suitable
coset representative of Г(<5) in A(8).
(5.27) Proposition. Let ne N, and let d e Г0„, where Г = r(g) and 0„ is the subgroup
of A((5) defined in (5.23)(b). Then either
(i) d e Г, or
(ii) there exist distinct groups 17, P,,..., V,P, in the set C„ and elements x, e Pf\(P,)o
such that
(5.p) d = (x,e,)(x2e2)...(x,e,) (mod Г),
where s, denotes the natural embedding of VJ} into A($5)-
5. Examples of Fitting pairs and Berger’s theorem 755
X '"г ЬУ ““T r. U. . г 2,
existetoem./tae.ih'ta''’ of®- ““
m(n)
= П У& (mod Г).
j-i
Let 5 denote the (possibly empty) set of suffices J in this product for which IGI - n
andy.e PAWo-Ifim < norif|UJ = nandУ1 e(Pj)o,itfollowsfrom(5 "janfc)
that ^еГе^. Hence d = (П^.О’лИ* with rf’ere,.,. By induction either
d e Г or Assertion (n) with C„_, in place of C„ holds for d*. If S = 0 and d* e Г
then d 6 Г; otherwise Assertion (ii) clearly holds for d because {UP- ieSlr,C =
0. 1 J 1 "-1
Ю □
We now come to the promised theorem of Berger. The proof which we present is
due to Brison [4], as is the foregoing preparatory material.
(5.28) Theorem (Berger [3]). Let g be a Berger class, and let Geg. Letn = |G”|,
and for p 6 P, let 4J(g) denote the subset of Set(g) defined in (5.23)(a). Then
Gs. = Cl {Ker(d''-<-): p11G/G911, U e 4*(g), U e s„(G)}.
Proof. Let dL-A(g) P/Po denote the homomorphism derived from dL'’ according
to Equation 4.6 in Definition 4.6, and put
К = П {Ker^'): р 11G/G911, U е 4₽(g), U e S„(G)}.
By (4.7)(b) this К is a normal subgroup of A(g) containing the subgroup Г = F(g).
We assert that
(М
G«c п К = Geg п Г,
and for this it will be enough to show that the set (Gsc r~\ K)\(Geg n Г) is empty. Let
g e G such that geG ф Г. By (5.24) we have geG e Г0„, and therefore according to (5.27)
there exist distinct groups Ц ..., U,Pt in C„(g) and elements x, e РД(Л)о such that
geG = (xie1)(x2e2)...(x,e,) (mod Г),
where Sj is the natural embedding of Urf into A(g). For 1 < i < t suppose that P, is
a p—group. Then Uf e 4£'(g) by definition of C„(g), and if Pi = Pj, for 1 < i * j < t,
then Ц f Vj because the groups {ЦР,} are distinct and therefore pairwise non-
isomornhic. We now suppose without loss of generality that | Ц| < I GJ for 1 < i < t,
and to simplify notation we write p, V, and P in place of p„ U„ and P, respective y.
By (5.20) we have x.eZ" = 1 for i = 1,. -,t - L and by (5.17) we have x • -
xk(e Р/РЛ Since x,t Po and Г < Ker(d1'we conclude that (0ac)d * 1- In
particular, the definition of dllp implies that gd%" * 1, and so dGpis a non’tri™
homomorphism from G into P/Po. Consequently G has a non-tnvial p-quotient
756
X. Fitting classes—the Lockett section
group, and hence p| | G/Gя|. Furthermore, the definition of г/д '’ in (5.15) implies that
,77(U, G) 0, in other words, that U e s„(G). Since geG </ Ker(d1' '’), we therefore
conclude that gcG ф К. Thus we have justified Equation 5.<r.
With the help of this we now obtain the following sequence of equations:
(G8.)«g = G«g n Г (by (4.15)(b))
= Gee n К (by Equation 5.o)
= П {GccnKer(dL'-'’):p||G/G'Jj, U e Л'(55) n s,(G)}
= П {Ker^ 'kc: pIlG/G91!, U e 4J(g)ns„(G)} (by (4.7)(c)).
Since eG is a monomorphism, it has a right inverse, which respects intersections. If
we now apply this to the first and last terms of the preceding chain of equations, we
obtain the conclusion of the theorem. □
Thus, if (5 is a Berger class, the section GR/GR> of an arbitrary finite group can be
computed just from knowledge of the subnormal (^-subgroups of G and their auto-
morphism groups. We now state some consequences of this theorem and its proof.
(5.29) Corollary. Let 55 be a Berger class. Then
Г(55) = Q {Ker(dv’'’): p e P, 17 is (55, p)-active}.
Proof. Write Г for Г(55), and let K* denote the intersection the right-hand side of
the equation in the statement. Since K* is also the intersection (over all choices of 17
and p) of the groups К which appear in Equation 5.о in the proof of (5.28), it follows
that Г < К* and that
(5.<r*)
Geg oK* = Gec n Г
for all G e 55-
Consider a coset fT of Г in K*, and for simplicity of notation suppose that the
groups in Set(55) = {Gj, G2)...} have been indexed by an initial subset of FJ. Then
there exists an m e FJ such that /(i) = 1 for all i > m, and we can find a group G„ in
Set(55) for which there is an isomorphism
6: Gj x • • x Gm -♦ G„.
Let g(e G„) denote the image of (/(1),..., f(ni)) under 0, and let f* be the element of
A(55) defined by setting f*(n) = g and f*(i) = 1 for I e FJ\{n). Using the obvious
normal embeddings of Gf into G„ (i = 1,..., m) we deduce from the definition of Г
that f*T = and if c„ denotes the natural embedding of G„ into A(55),
we obviously have f* e G„e„ n K*. From (5.a*) it then follows that f* e Г; hence
/Г = Г, and we conclude that Г = К*. □
5. Examples of Fitting pairs and Berger’s theorem 757
(5.3°) Corojary (Berger [3]f If g is a Berger class, then its Lausch group A(g) is a
{restricted) direct product of finite cyclic groups. W
Pr°qfLvll (®’ P>-active ^“P for some prime p, by (4.7)(b) and (5.16) the roof
map d : A(g) - P/Po Is an eptmorphism. If G e Set(g), then obviously SW, G) is
non-empty for only finitely many g-active groups U, and it follows that if fe A(g)
then the set of pairs (U, p) for which fd1^ * 1 is also finite. Thus, according to the
prescription described in A, 4.2 we obtain a homomorphism from A(g) into the
restricted direct product D of the groups P/Po (taken over all g-active groups U).
By (5.29) the kernel of this homomorphism is T(g), and so A(g) = A(g)/T(g) is
isomorphic with a subgroup of D. Since each P/Po is a finite abelian group, it is clear
that D is the restricted direct product of finite cyclic groups, and hence so also is A(g)
by a theorem of Kulikov (see Fuchs [1], Theorem 18.1). □
The following obvious reformulation of Berger’s theorem is often more apt.
(5.31) Theorem. Let g be a Berger class. For an (g, p)-active group U, let p denote
the kernel of the Fitting pair (P/Po, dv") (see Definition IX, 2.13(a)). Then
g* = Q {В' ’1’: p e P, U is (g, pfactive}.
In conclusion we apply the preceding ideas in conjunction with the deep result B,
12.19 of Bryant and Kovacs to obtain the following theorem.
(5.32) Theorem (Laue, Lausch, and Pain [1]; Bryant and Kovacs [1]). Let p and q
be distinct primes. Then G,SP £ <5t.
Proof. By the cited theorem of Bryant and Kovacs there exists a q-group U such that
Aut(L') induces on U/Q(U) a group of automorphisms having order p and acting
irreducibly on U/Q(U). By A, 12.7 the centralizer of U/<I>(U) in Aut(Lj is a q-group,
and therefore Aut(L') = O,(Aut(G))P, where |P| = p. First we observe that [L']P is
an (G, p)-completion of U, which is clear from the fact that P e Sylp(Aut(Lj) and that
[U]Pe6. Next we assert that, in the notation of Definition 5.8(c), the subgroup Po
of P is trivial. Since P acts faithfully and irreducibly on Р'/Ф( L'), we have [I/, a] = U
for all 1 a e P, and consequently r(P) = 1. Moreover, since Aut(D)/O,(Aut(D)) s
P e 91, we have {P; Aut(L)} < Pn(Aut(G))' < Pn O,(Aut(Lj) = 1, and therefore
Po = r(P){P;Aut(D)} = 1, as asserted.
Let d denote the map dv"-6 defined in (5.15). From (5.16) we deduce that (P d) is
an S-Fitting pair, and if Я denotes its kernel, then (UP)* = U by (5.17). It follows
from (5.31) that the <St-radical of UP is contained in U (in fact, equals G), and so we
conclude that UP e GeGp\®*-
From (1.18), (!.23)(a), and the preceding theorem we can draw the following
conclusion.
(5.33) Corollary. If it £ P, then the class <S„ has trivial Lockett section if and only if
|я| < 1.
758
X. Fitting classes—the Lockett section
(5.34) Example, (a) (Brison [3], [4]). Here we use the method of Variation II in
Section 6 of Chapter IX to construct a Fitting class containing non-nilpotent groups
which has a trivial Lockett section. (The only previously-known Fitting classes with
this property were the subclasses of 91.) The same method also yields an example of
a Berger class which is not a Fischer class.
In Theorem IX, 6.26 we characterized the Fitting class g generated by a group S of
the form S = PQ, where the groups P = 0p(S) and Q e Syle(S) satisfy Hypotheses IX,
6.22. In Example IX, 6.27 we indicated how to construct one such group P with these
properties; it is a metabelian 7-group of class 3, exponent 7 and order 77, and Aut(P)
has a Sylow 19-subgroup Q of order 19 which acts irreducibly on P/P' and P'/K3(P)
and trivially on Z(P) = K3(P}.
In Proposition 8.7 of [4], Brison shows that the Fitting class g = Fit(S) contains
only one (g, recompletion UR which satisfies r(R) < R, namely the group S with
U = P and R = Q (see Definitions 5.8). However, if P has the properties described in
Case 1 at the end of Example IX, 6.27, then {Q\ Aut(P)} = Q, and so Qo = Q in the
notation of(5.8)(c). Hence there are no g-active groups in this case, and so the Lausch
group is trivial and Fit(S) = Fit(S)t. But in the Proposition of [3], Brison proves that
Fit(S) is always a Lockett class whenever S = PQ satisfies Hypotheses IX, 6.22, and
therefore Fit(S) has a trivial Lockett section.
Now suppose that P is in Case 2 of IX, 6.27. Then S is evidently the unique g-active
group in g = Fit(S), the Lausch group has order 19, and Locksec(g) = {g, gt}. If
P = X sn G e g and R e Syl4(Nc(X)), then |R/Cr(X)| divides q (= 19) by IX, 6.22(d),
and so either XR = X x Re 9i|p e) or X R/OP(XR) = PQ. In either case we have
XR e Fit(S), and therefore g satisfies (5.18), the definition of a Berger class. (Of course,
(5.18) is also satisfied vacuously if P is in Case 1.) However, it is obvious that the
subgroup H = PQ of the g-group S (= PQ} is a nilpotent extension of a normal
subgroup of S. Since Hypotheses (b), (c) and (e) of IX, 6.22 imply that (k(H} = )
O'1 (H/0р(Н}} = H, and since H is evidently not in the class I defined in IX, 6.24, we
conclude that H £ g and hence that g = Fit(S) is not a Fischer class. Since S/Z(S) is
obviously not in g, it is clear that g is not Q-closed. Furthermore, if M denotes a
copy of P' (viewed as an Fp(S/P')-module), then it is straightforward to verify that the
semidirect product [M]S belongs to Rog but not to g. Thus g is an example of a
Lockett class which is not closed under any of the operations q, r0 and s, (compare
with Proposition 1.25).
(b) (Brison [7]) Here we aim to show that any Fitting class g satisfying
s^g^eA
is not a Berger class.
Let p be an odd prime, and recall from B, 9.17 that the quaternion group Quat(8)
of order 8 has a faithful simple module of dimension 2 over Fp. Therefore by Theorem
B, 12.19 we can find a p-group U with the following properties:
(i) |1//Ф(С)| = p2, and
(ii) if С = САи,(П(С/Ф(С)), then Aut(C)/C s Quat(8).
Since C is a p-group by A, 12.7, we can therefore write Aut(C) = CQ with Q s Quat(8).
Now consider the semidirect product Any non-identity element \ of Q has
5. Examples of Fitting pairs and Berger’s theorem
759
a ikT therefore involution */ C, which acts by .„version on U/*(U},
and therefore (L, <x>] = U, tn particular, we obtain (UQ)‘ = up = U<z) »nd
therefore G<z> e (S„S2)t s g. n follows lhat {VQ ' 17 <z>’
№ n Q 3 Q. Next we identify the subgroup Po (defined in Equation ЗД for the
at ° ЛГ**’' ee ПО,е ‘ha* * = ' * ‘he 0П|У Л in Q f°r which
[L', <x>] U, and hence t(P) - 1. Secondly we consider the subgroup
{P-, Aut([/)} = {x *x’: g e Aut(G) and x, x« e P}.
Since g e Aut(L') may be written
9 = cq
with с e C and q e Q, we obtain
x lxe = [x, </] = [x, q] [x, c]’.
If x e P (< Q), we have [x, q] e P, and so [x, g] e P if and only if [x, с] e P. However,
[x, c]eC < Aut(L) and C n Q = 1. Thus [x, g] e Pif and only if [x, c] = 1,and this
holds if and only if [x, g] = [x, q], Thus [P; Aut(L')[ = {P; Q] = [P, Q], and we
conclude that Po = [P, QJ. But P is a non-trivial normal subgroup of the nilpotent
group Q, and therefore Po < P. Thus UP is an (g, 2)-active group. Suppose, for a
contradiction that g is a Berger class, that is to say, Hypothesis 5.14 holds. Then
Theorem 5.16 applies and allows us to deduce that (dL, P/Po} is an g-Fitting pair.
By IX, 2.11 the class
B = (GeSet(g):GdE=l)
is a Fitting class satisfying
g,(=e,Kfts&
and by (5.17) the Я-radical of UP is UP0. Since (L'P),, < UP0 < UP eft. it follows
that UP/(UP)S has even order, which contradicts the hypothesis that g e S,S2-
Hence g cannot be a Berger class.
As Brison points out in [7], the fact that Berger’s theorem cannot be applied to g
to calculate the gt-radical is not a serious problem; this is because g„ - <=*, and we
can apply Berger’s theorem instead to the Berger class S. However, it is not known
whether every Lockett section contains a Berger class (which can then be used to
compute the “lower star’’ radical). As a test case, Brison suggests looking at the class
X = (Ge S|2,31: Soc2(G) < Z(G)).
He points out that if g is a Fitting class satisfying
I,cgcJEni,®3>
760
X. Fitting classes—the Lockett section
then the argument used above in this example shows that ft is not a Berger class. It
follows from Remark (3) of IX, 2.9(a) that X = R0T, and so X is a Lockett class by
(1.25). Therefore ft* = X and we would like to know the answer to the following
question.
Open Question. Is The Fitting class 3E defined above a Berger class? More generally,
does every Lockett section contain a Berger class?
Postscript
Hawthorn [1] (see also [2]), in his doctoral dissertation of 1990, has constructed a
Fitting class ft with the property that ft* is not determined by the set of transfer
Fitting pairs used in Berger’s Theorem 5.28; thus the equation in the statement of
Theorem 5.31 does not hold for this Fitting class ft. Hawthorn makes good this
deficiency by developing more sensitive versions of the transfer Fitting pairs: whereas
the definition, in (5.15), of the map dG ranges over all conjugacy classes of subnormal
copies of U in G. those defined by Hawthorn are dependent on a restricted set of
subnormal copies of U, determined by the so-called ‘embedding type’ of U in G. These
new Fitting pairs cope with the pathology of Hawthorn’s example ft, in that ft* is
indeed the intersection of all their kernels; they also work for Berger classes. However,
it is not clear at the time of writing whether Hawthorn’s improved version of Berger’s
theorem is applicable to all Fitting classes.
Exercises
1. (Blessenohl and H. Laue [1]) Let G be a group and n e N. Show that an inner
automorphism of G induces an even permutation (a) on the elements of G and (b)
on the subset {geG: o(g) = n and g2 ф G'}.
2. Let ft and T be Fitting classes, and for each G e ft let U(G) denote the set of linear
characters of Gz. Regard U(G) as a G-set by defining
X9: x-> x(<?x<? ’) (xeU(G))
for x e Gj and g e G. If a: G ->• G° is an isomorphism, show how to define a map
a: U(G) -► O(G“) so that Equation 5.a on page 738 holds. If TV < Я e ft and x e U(/7),
define ^(x) to be the restriction of x to Nx. Are Hypotheses 5.4(b) satisfied for this
map
3. (Zappa [2]) A subgroup R of a group G is called a radical of G if R = Gs for some
Fitting class ft. If X and Y are radicals of G such that Y < X and no radical of G
lies strictly between them, then X/Y is called a Fitting factor of G. Show that if G
is soluble, then a Fitting factor of G has prime power order. If X and Y are radicals
of G, then for g e G let dG x r(g) be the product of the determinants of the linear
transformations induced by g on the chief factors of a partial chief series of G
passing between Y and X. Show that the class
6 = (G e S: dG x.r(g) = 1 for all Fitting factors X/Y of G and all geG)
is an S-normal Fitting class.
6. The Lockett conjecture
761
4 япНкТкС .I1' L»Ct к dcenOtC thC Fi‘ting Class describcd in the preceding exercise
and let § denote the Fitting class of all soluble groups G all of whosf elements
Example’s.^ Prove Zt"5™“'°'“ on the °^) (a special case of
5' !няГг/г [}6t A Ье 3 eiVCn fin'te abel‘an grOUp' Find a soluble 8r°uP G such
inm v/vg =
6‘ JWA £;normal FittinS classes «1 and Я2 are said to be independent if
,n the notation of (4.5). Investigate the mutual independence
of the various (S-normal Fitting classes which arise as kernels of the S-Fittine
pairs defined in Examples 5.3 and 5.6.
7. (Charnes [1]) Let V be a finite soluble group with a unique maximal normal
subgroup M. For G e <S, let 11(G) denote the set ofelements x in G such that (i)
o(x) is odd, (ii) x e U < G with V s U, and (iii) x M <• U, and regard 11(G) as a
G-set by conjugation. If n e H(N), where n e N < H e 6, let </>(«) = n e Q(H).
Verify that these definitions satisfy Hypotheses 5.1(a) and (b), and show that the
corresponding (S-Fitting pair (A, d) defined in (5.2) is non-trivial (i.e. that A # 1).
8. (Brison [4], Theorem 6.4) If g is a Fitting class, a group G is called ^-constructive
if 6 = VP, where UP is the (5, p)-completion of a group U e g for some prime
p and if т(Р) < P (see Definition 5.8(b)). Show that g is generated by its g-
constructive groups, together with its single-headed perfect groups and its groups
of prime order.
6. The Lockett conjecture
Let g and ® be Fitting classes of finite groups with g £ ft. We recall from Section
1 that
(6.a)
J -> In 8 (I e Locksec(6))
defines a map p from Locksec((5) to Locksec(g), and according to Definition 1.19 we
say that g satisfies the Lockett conjecture with respect to 6 if p is surjective; in other
words, if the Lockett section of the smaller class is determined by that of the larger.
Whether this always happens was a question first raised in 1974 by Lockett [4] for
the case 6 = 6. It was answered positively for primitive saturated formations g by
Bryce and Cossey [5] in 1975, and negatively for general g S £ by Berger and Cossey
[1] in 1977 Then in a paper devoted to the concept of normality for Fitting classes
which are not necessarily soluble, H. Laue [1] explicitly asks whether £ = 6„
which is equivalent to asking if the Lockett conjecture holds for the pair £ s G. The
following year in 1978 Berger [2] was able to answer this case affirmatively by
exploiting the transfer map in order to extend the domain of definition of a Fitting
pair. His method also provides another proof of the result of Bryce and Cossey
mentioned above. Later in this section we will summarize this approachofBergei s
but first we attack the Bryce-Cossey theorem from yet another direction “° *
Beidleman and Hauck [1]. Although their line of argument is designed essentially
762
X. Fitting classes the Lockett section
for a soluble universe and, for example, cannot handle the proof of the Lockett
conjecture for the pair 6 £ G, it has the advantage of giving information about
class-membership properties inherited by injectors. We shall close the section with
an account of the Berger-Cossey counterexample to the Lockett conjecture. The only
limitations placed on the universe G in this section are those stated in the hypotheses
of the results.
(6.1) Proposition (Bryce and Cossey [5]). Let 55 and ® be Fitting classes with 55 £ G.
Then 55 satisfies the Lockett conjecture with respect to ® if and only if 5» = 55* n ®»-
Proof. According to Theorem 4.19, the map p defined in (6.a) sets up a lattice
isomorphism between Locksub«G*, 55*» on the one hand and the lattice of Fitting
classes that lie between 5* r> ®» and 55* on the other. Evidently this map is onto
Locksec(5) if and only if 5* n 6, = 55*- □
The approach of Beidleman and Hauck is based on the study of the following concept,
which is a weak form of subgroup-closure for a class of groups.
(6.2) Definition. Let 9) be a class of groups, and let X be a Fitting class. Then 9) is
said to be ^.-injector-closed if Inj j(G) £ g) for each Ge f).
(6.3) Remarks, (a) In the notation of Construction В in Chapter IX, Section 2, the
fact that ?) is X-injector-closed may be expressed thus:
9)eXf9).
(b) The property of being X-injector-closed obviously says nothing about groups
G in the class for which Injx(G) = 0. The value of the concept is therefore largely
confined to a soluble universe.
(c) Each of the following conditions evidently guarantees that a class ?) is X-
injector-closed:
(i) X £ ?); (ii) 9) = s9) and X is arbitrary; (iii) 9) = 55®, where ® is an <s, Q>-closed
Fitting class and 55 is a Fitting class contained in X.
(6.4) Lemma (Beidleman and Hauck [1]). Within the universe 6 let ft be a permutable
Fitting class, and let G be a Fitting class which is Lpffi)-injector-closed for all primes
p. Then 6 is ^f-injector-closed.
Proof. Let GeG, and let V e Inj R(G). We proceed by induction on | G|. If G = 1, then
the conclusion is certainly true. Therefore suppose that G > 1, and that the 55"
injectors of all G-groups of order less than |G| belong to G. Let peP. Since 55 is
permutable by IX, 1.16, there exists a Gp. e Hallp.(G) such that FGp. is an Lp(55)-
injector of G, and by hypothesis FGp. e G. By IX, 1.5(c) we have V e Inj я( FGp.), and
so if VGp. < G, it follows by induction that V e G.
Thus we can suppose that FGp. = G, in other words that |G: kj is a p'-number.
Since this holds for all p e P, it implies that V = G e G, and the induction step is
complete. □
6. The Lockett conjecture
763
Another preparatory lemma is the following.
I’rT™ ,Hntl]l ? <s -1
Ve(X^,S,;
in particular, we have £ n Q)* c (J n Q))*g .
Proof. We first observe that the hypothesis: £gp = £ implies by IX, 2.3 that
£ f (£ n 9))t (Sp. is a Fitting class. Furthermore, if we substitute ft = (£ n QI) and
© = ©* = Sp- in (1.27) (c), we obtain *
(£n?))*6p. = ((£n?)),Sp.)*.
If W is an £-injector of a group in Q), then by hypothesis ITe £ n Q) s (£ n Q))*g .,
and consequently We((£n ?)),,£,,.)*. It follows that Q) e £ f ((£ n '!))„£„.)*, and
hence that 0) £ (£ f (£ n 5))t'3p )’ by (1.36). From (1.18) we then conclude that
9, (In ?))» Sp., which is simply a reformulation of the main assertion of the
lemma. The last assertion is obvious. □
(6.6) Proposition (Beidleman and Hauck [1]). Let g and ® be Fitting classes of finite
soluble groups. Assume that g is normally embedded and that 6 is Lffy-injector-closed
for all primes p. Then 6* is also Lp(tfyinjector-closed for all primes p, and consequently
6* is fi-injector-closed.
Proof. Let p e P, let G e G„, and let W be an Lp(g)-injector of G. Since Lp(8) =
Lp(g)gp. by IX, 1.15(b), the hypotheses of (6.5) are satisfied with p = p', £ = Lp(g),
and 0) = (f>, and therefore by that lemma we have H'e(Lp(g)nG)#Sp s ®»Sp.
However, if L denotes the Lp(g)-radical of G, by IX, 1.19(a) and (b) we have W/L e &p-
Consequently W = W^L. and because L e s.(G) S ©,, it follows that We n06, =
6t. Thus we have shown that 6, is Lp(g>injector-closed for all primes p, and since
a normally embedded Fitting class is permutable, we can apply (6.4) to conclude that
®* 's g-injector-closed. □
Remark. A special case of (6.6) arises when 6 is subgroup-closed. Then, if 5 is
normally embedded, we have
Ge®, and V e Inja(G) => V e 6*.
This bears comparison with (1.29). which suggest a duality, admittedly imprecise
between Q-closure and s-closure in this context. It would be interesting to know if
the requirement that g be normally embedded is really necessary here. Further
specializing by setting © = S, we obtain the following corollary.
764
X. Fitting classes —the Lockett section
(6.7) Corollary (Hauck [3]). If ft (£ 6) is a normally embedded Fitting class, then
G* |S t$-injector-closed.
The next result is the key for the Beidleman-Hauck approach to the proof of the
Lockett conjecture for primitive saturated formations.
(6.8) Theorem (Beidleman and Hauck [1]). In the universe 6 let fjbea Fitting class,
let it £ P, and let X = ftG„ Further, let 9) be a Fitting class which is ^fQn-injector-
closed and Lp(X)-injector-closed for all primes p. If V is an X-injector of a f)t-group,
then V e (X n 9))t; in particular, X r,'If* = (X r\ 9))*.
Proof. In Example IX, 3.7(c) we showed that X is a Fischer class, and so X is certainly
normally embedded by IX, 3.4(a). It therefore follows from (6.6) that 9)* is X-injector-
closed, and we can apply (6.5) with p = it' to conclude that if V is an X-injector of a
?) „-group, then l'e((Xn 9))* G„) n ?)»
Let R denote the ftG„-radical of V, and observe that R e s„9)t = '()„ Since 9) is
ft6„-injector-closed by hypothesis, we can apply (6.5) again, this time substituting it
for p and ftG„ for X. Since R e Inj^JLj, the final conclusion of that lemma yields
R e (ft6„ n 9))*6„.. Because ft6„ n 9) £ X л 9), it therefore follows from (1.18) that
R e (X n ?))*£„., and since V/R e X/ftG„ £ 6,., we deduce that
Ие(Хп9),6,п(1п®Д=(1г<. □
In order to explore some of the consequences of the preceding theorem, we shall need
the following observations.
(6.9) Lemma. Let r £ P, and let be a Fitting class of finite soluble groups.
(a) (Beidleman and Hauck [1]) If 9) is Q,-injector-closed, then (6tn9))„ =
etn?)„.
(b) (Brison [2]) If Q, £ 9), then satisfies the Lockett conjecture withrespect tof).
Proof, (a) First recall from IX, 2.2 that f ft is a Fitting class for any Fitting class
ft. Thus from (1.36) we have (6, f (Gt n 9)))* = G, f (Gt n ?))* = 6, f ((G, n 9)),)* =
(Gt f (G, n 9))*)*, and consequently (6, f (G, r> 9)))* £ G, f (6, n 9))#. Since 9) £
5, f (6, n 9)) by hypothesis, it follows from this that 9)„ £ Gt f (6, n 9))* by (1.18).
Hence 6,n9)t £ (6,n?))t, and since the reverse inclusion follows directly from
(1.18), equality holds.
(b) If G, £ 9), then certainly 9) is Gt-injector-closed. In this case we have (£,)„ =
<5T n 9)# by Part (a), and so the Lockett conjecture holds for the pair G, £ 9) by (6.1).
□
The following result is a modified part of the proof of Korollar 3 of Beidleman and
Hauck [1].
(6.10) Proposition. Let I be an index set, and assume that to each i e I there is
associated a set rq of primes and a Fitting class X, £ 6. If I is infinite, assume further
6. The Lockett conjecture
765
that for each finite subset a of P, the class « conM in alt but a fMte number
tas'o^ofthefon FUtin9 C,ass>and “sume that at
least one pf the following two hypotheses is satisfied:
(A) Each of the classes 3q, i e I, is s-closed;
(B) There is a Fitting class 5 such that 1. = g for allied
Then
(M) f П S„A n ?), = f Q 3E.-SW16,. n Q)
\*eJ / \ie/
Proof First we deal with the case where I is finite, say I = {1,..., и}, and prove (6./J)
by induction on n. Since 4) is subgroup-closed, the truth of (6.0) when и = 1 follows
directly from (6.8). Now suppose that n > 1, and set
i=l
for m = 1, 2,..., n. Then we assert that our class n Q) fulfils the requirements of
the class denoted by ?) in the statement of Theorem 6.8. This is certainly true when
Hypothesis (A) holds, for then n 4) is s-closed.
Therefore assume that Hypothesis (B) holds. Then £„,£„;), and since
8 £ n ^(8®» 35„) f°r all primes p and for all i,j e f, it follows from Case (iii)
of (6.3)(c) that is closed under taking and Lp(g6„.G„.)-injectors; hence so
also is the class n 9) since 9) = s9) by hypothesis. Thus we are in a position to
apply (6.8) with n ?) in place of 4). With i = n and m = n — 1 we therefore have
I„n?), = X„G„„G..n(T„_1n?)J
= 3E„e,nG„.n(T„_1n?)K
= (3EnG,„G<nT„_1n?)),
(by induction)
(by (6.8))
= (*„"?)).•
This completes the induction step and hence the proof of the proposition for the case
|/| <00.
Now suppose that the index set I is infinite. By (1.18) we have
A 3E.-6,,2 (,Q A‘S-<S"in ?))/
and it only remains to prove the reverse inclusion. To this
belonging to the left-hand side of Equation 6.0, and set a = <r(G).S no ЬУ
G„ is contained in all but a finite number of the clasps
ni....n,, of P such that GpCMAie/SiSp,®»!) ° 4 I- 1 < •
have
766
X. Fitting classes—the Lockett section
G e n Q S,; J n 'l)t
= ( C )n (£« n "S)» (by (69)(a))
\i=l /
= (( C I n n 'Э I (by the case I finite with
'^'=1 ' '* G„ n 9) in place of 9))
«=(п 3E,e„e..n?)) (by (1.18)),
Vel /»
and we have shown that G belongs to the right-hand side of Equation 6./J. □
First we draw the following conclusion from (6.10).
(6.11) Corollary (Beidleman and Hauck [1]). In the universe 6 let g and 9) be Fitting
classes with 9) subgroup-closed. Then
(6.y) g«n?\ = (5«n?)V
In particular, g91 satisfies the Lockett Conjecture with respect to 6.
Proof. In (6.10) take I = N, and for each i e I set 3E; = g and л( = {p,}, where p, is
the ith prime. Note that if a is a finite subset of P, then £ ffZpZp- for all primes
p except possibly those in a, and that with these substitutions the hypotheses of (6.10),
in particular Hypothesis (B), are satisfied. In view of the fact that T, =
Cpe p = g91, we can therefore conclude from (6.10) that Equation 6.y holds.
Since g91 is a Lockett class by (1.27) (b), the last assertion now follows from (6.1).
□
(6.12) Theorem (Bryce and Cossey [5]). Let 7f be a primitive saturated formation, and
let 6 £ 6 be an s-closed Fitting class. Then
gn©, = (gn®V
In particular, then g satisfies the Lockett Conjecture with respect to ®.
Proof (Beidleman and Hauck [1]). We begin by proving the theorem in a special
case, namely under the assumption that 5 has bounded nilpotent length. In this case,
by VII, 3.9 a primitive saturated formation g has the form
S = (d
\Я=Р /
where p IP, where each 3En is a primitive saturated formation, and where for each
6. The Lockett conjecture
767
finite subset о of P the class 6 is contained ,n к , r •
T S S Set 9) — G n (6 Sinee m • u u 1* but a finite number of the classes
wheahavlhyP°theSeS °f (610) °" Particular’ HyPothesis (A)) are fulfilled, and therefore"
8n®* = ( А W. n(6,n(5,)
\Я£Р /
= („C'p3£"S»S”)n^* (by(6.9)(a))
= ( A n?)) фу (6.Ю))
\7tSP Л
= (8n®),.
Hence the theorem is true in this special case.
Now, in the general case, set g(i) = g n 9? for i e N. Since g(i) has bounded
nilpotent length, we can now appeal to the special case to conclude that
5 n 6, = ( 0 8(i)) n ®. = (j (30) П 6 J
\i=l / i=l
co
= U (3(<) 6).
i=l
(by the special case)
£((U8(i))n©
= (Sn©),.
(by (1.18))
Since Proposition 1.18 also implies that (3 r> ©)» — 8 n ©» the two classes are
equal. □
Another approach to the Lockett conjecture has been developed by Berger [2].
His starting point is the following question: Given two Fitting classes 3 and 6 with
3 c © and a Fitting pair (A, d) for 3, can one find a Fitting pair (A, d*) for 6 which
agrees with (A, d) on g? He gives an affirmative answer for a general set of hypotheses,
which includes the following as a special case.
(6.13) Hypotheses. Let 3 £ © be Fitting classes, and let (A,d) be a Fitting pair for
3- Assume that
(i) A is a p-group for some prime p, and
(ii) if G e 6 and P e Sylp(G), then PGS e g.
768
X. Fitting classes—the Lockett section
(6.14) Theorem (Berger [2]). If Hypotheses (6.13) are satisfied, then there exists a
(^-Fitting pair (Л, d*) such that
do — dG
for all G e g.
We will not prove this theorem, but will simply describe the construction of d*
from d and refer the reader to Berger [2] for the detailed calculations, many of which
recall those used in the construction of the transfer Fitting pairs of Section 5.
Therefore suppose that (A, d) is an ^-Fitting pair which we wish to extend to 6 and
that Hypotheses 6.13 are satisfied. For a given GeG the homomorphism d*: G -» A
is then defined as follows: Let S/Gs denote a Sylow p-subgroup of G/GB, and let
n = n(G) be an integer such that n|G: S| = 1 (mod |G|p). Then set
(6.<5) gdG = (gvG^s)"ds,
where vG~s denotes the transfer from G into S/S". By hypothesis S e g; consequently
ds is a homomorphism from S to A, and since S' < Ker(ds), we can lift ds to S/S' to
give meaning to the right-hand side of Equation (6.<5).
Berger shows that the definition of d* given in (6.<5) is independent of the choice
of both the integer n and the Sylow p-subgroup S/G% of G/Gs; he further shows that
d* has Property FPI by comparing its effect on two isomorphic groups and by
computing the restriction of a map dG to a normal subgroup of G; and finally he
verifies that dG = d* for all G e g, from which it follows, of course, that (A, d*) also
has Property FP2.
Armed with this theorem, which he proves under hypotheses more general than
those of (6.13), Berger gives a new and shorter proof of Theorem 6.12 of Bryce and
Cossey. As a further application he also proves that 6 satisfies the Lockett conjecture
with respect to 6, a result which is out of reach of the methods of Beidleman and
Hauck developed above because they depend on the existence of injectors in the
larger of the two classes under consideration. However, our formulation of Berger’s
Theorem 5.28 allows to deduce this case of the Lockett conjecture very easily.
(6.15) Theorem. The Lockett conjecture holds for the pair 6 £ G; in particular, =
SnG„.
Proof. By (6.1) it will suffice to show that G* = 6 n (£*, and by (1.18) we already
know that £ Gn To prove the reverse inclusion, let G be a soluble group in
Let n = JG9*], and let p be a prime divisor of |G: Ся|. From Definition 5.8 it is
clear that (S, p)-active groups are (G, ppactive. Therefore, if V sn G and V e AP(S),
then U e /1J(G), and, because G e (£*, it follows from Theorem 5.28 that GdG-p = 1.
But then that theorem implies that G = Gs e and we have shown that
Gn(f,cet. ’ □
We now come to the Berger-Cossey counterexample to the Lockett conjecture. It
falls within the compass of the Dark construction described in Chapter IX, Section 5.
6. The Lockett conjecture
769
(6.16) Example (Berger and Cossey [1]). Our aim will be to construct a group Y and
fulfi Ы S er°UPS °f AUt(K) SUCh tha* the hyP°theSES Of Corollary^, 5.17 are
Let £ be an extraspecial group of order 27 and exponent 3 By В 9 16 the eronn
£ has a faithful, absolutely irreducible module, M say, of dimenrion 3 over th! field
F7. Let Y be the semidirect product [M]£, a group of order 3’7’. In order to identify
the set .о/ we must first calculate Aut( Y). 3
(1) The structure of Aut( Y)
We determine this in two stages: first we construct a certain subgroup of Aut(Y);
then we show that this subgroup is in fact the full group of automorphisms of Y
By A, 20.11 the group Aut(£) is the semidirect product of the elementary abelian
group Inn(£) of order 9 with a subgroup H s GL(2, 3). Let S = H'. Then C„(Z(£)) =
S S SL(2, 3). Since therefore S s [Quat(8)]U with |U| = 3, it follows that Z(S) is the
unique nontrivial elementary abelian 2-subgroup of H centralizing Z(£), a fact which
we shall cite later. Let T denote the semidirect product [£]S. Since [Z(£), S] = 1,
the F7 £-module M is T-invariant and therefore by B, 7.12 extends to an F7 T-module,
which we also call M, and which is clearly irreducible and faithful for T. Now the
subgroup Y = [M]E is the G, , 7|-radical of the semidirect product [M]T and is
therefore characteristic; furthermore CMT( Y) < Z(Y) = 1, and consequently MT in-
duces a faithful group of automorphisms on Y. We therefore identify MT with the
corresponding subgroup of Aut(Y).
Next consider the ‘inverting’ automorphism of M (in additive notation, that is the
linear map sending each me M to — m). Since it clearly belongs to Z(Aut(M)) and
therefore commutes with the action of T on M, it can be regarded as an element of
Aut(Y), and as such we denote it by z. Set Z = <z> and L = (MT, Z>. Then M is
the unique minimal normal subgroup of L and is complemented in £ by the subgroup
TxZ.
We now assert that L = Aut(Y). Since M (= O,(Y)) is characteristic in Y, we have
M < Aut( Y). Let C, = CAul(n(M); then M < Q < Aut(Y) and C, n £ = 1. Hence
[M£, CJ < M£nC, = M, consequently [ME, Q, CJ = 1, and it follows from A,
12.4(a) that Ct is a {3, 7}-group. We want to show that С, = M. First suppose that
7||C,/MI, and let x be an element of 7-power order in C,\M. Since the 3-group £
centralizes Ct/M, it acts trivially on the 7-group <x>M/M, and so from A, 121 we
can conclude that <x>M\M contains an element, у say, of C(£). Since <x>M is
central-by-cyclic and therefore abelian, it follows that у centralizes ME = Y and then
we have the contradiction that 1 # у e СА11ИЛ(У) = 1. Hence CJM isa 3-group, and
since M < Z(C,), we have С, = M x 0 with 0 e Syl3(C). But then C'char C,
<Aut(Y), whence Q < Aut(Y), and it follows that [M£,0<M£n()S
ME r^CIr,Q<MnQ=L Therefore Q < CAul(n( Y) - 1, and consequently
С. = M, as desired. Thus Aut(Y)/M acts faithfully on M.
Next consider C2 = CAul(n(Y/M). Obviously M and«wnutes wh
the action of £ on M. Since M is an absolutely ^reducible IF7 E-module it follow^
from Definition B, 5.5 that C2/M acts in scalar fashion on M and hence that |C2/M|
irom uenmuo , Д ... r d । z(g\ x Z| = 6, we therefore conclude
di vides | F7 | = 6. Since M(Z(£) x Z) _ 2 11 nf AuttYl and so bv
that C2 = M(Z(£) x Z). Now M and Y are normal subgroups of Aut(Y), and so by
770
X. Fitting classes—the Lockett section
a familiar argument there is a homomorphism p from Aut(T) into Аш(У/М) with
kernel C2. Thus Aut(T)/C2 is isomorphic with a subgroup of Aut(Y/M), and since
Aut(17Af) = Aut(£) S [£/Z(£)]GL(2, 3) and L/C2 [£/Z(£)]SL(2, 3), it follows
that £ has index 1 or 2 in Aut( У). However, in the faithful action of Aut( Y)/M on M,
the subgroup Z(E)M/M has scalar action and is therefore central. Consequently the
image of Aut(L) under ц centralizes Z(E)M/M, and it follows that Aut(T)/C2 is
isomorphic with a subgroup of the centralizer of Z(E)M/M in Aut(L/M) s Aut(£).
But this centralizer is isomorphic with [£/Z(£)]S£(2, 3), and so by order considera-
tions we have Aut( У) = L, as claimed.
(2) Satisfying the hypotheses of IX, 5.17
In order to construct a Fitting class of the form D*(Y, si) described in Notation
(b) after IX, 5.7, we need to identify л, r, and ,oZ and to check that the hypotheses of
IX. 5.17 are then satisfied. With this in mind we define К to be the normal subgroup
К = YZ(S)
of Aut( Y), recalling that, in the identification of MT (= YS) with a subgroup of Aut( Y),
the group У itself is identified with Inn( Y). We also recall that S = H' = S£(2, 3), and
that Z(S) is a group of order 2 whose generator inverts Y/Y' and centralizes Y'/M.
We set
л = {2}' and r = {7}',
and observe that Y is a soluble л-group with Z( Y) = 1 and that K/Y is a non-trivial
normal л'-subgroup of Aut(Y)/Y. In the notation of IX, 5.17 we set
si = {A: Y < A sn K}.
Thus si = {£}, and we now verify that Conditions A — D in the statement of IX,
5.18 hold.
Condition A: The only central chief factors of К are the 2-chief factor K/Y and the
3-chief factor Z(E)M/M, and certainly 2, 3 e r.
Condition B: In our case it is Condition (ii) that holds: Suppose that У < YK <
Aut(Y) with У = К By way of contradiction, first suppose that К < YK and note
that YK/K is then a non-trivial {3, 7}-group. Since Aut(Y)/K is soluble and has a
Hall {3, 7}-subgroup of order 3, it follows that YK/K has order 3. Thus | Y: Yr> K| =
3, and we conclude that У о К therefore has index 3 in the unique Hall {3, 7}-
subgroup Y of K. Since Z(S) acts by inversion on Y/(Z(£jM) and centralizes
Z(E)M/M, it follows that Z(S) inverts (У n K)/(Z(E)M). Thus, in particular, the
generator of Z(S) induces on Y/M (S £) an automorphism of order 2 which centralizes
the centre Z(Y/M) = Z(E)M/M and satisfies [Y/M, Z(S)] < (Yn K)/M < Y/M.
However, Aut(£) has only one conjugacy class of involutory involutions centralizing
6. The Lockett conjecture
771
fredv on £/Z(£) Th “volutions tn Z(S) Inn(£)) and these all act fixed-poini-
freely on £/Z(£). This y.elds the desired contradiction, which therefore mphesftat
К - YK. Hence Y ,s contained in the unique Hall 13 71-suberoun Y of к
consequently Y = Y ( = Inn(F)), as required ’ P ’ and
M°”-XlCFCrainly Y haS a UniqUe minimal ^group, namely
M - O7( У). Furthermore, M ts a r'-group and satisfies the requirement
[M, F] = [M, AfZ(£)] = M # 1.
Condition £>: Since У = 1пп(У) and К is the only group in X we have only to verify
on У/F (~ £/E')and thlS follOWS easily from the fact that Z(T*acts fixed-point-freely
(3) The definition of the Fitting class g.
Now let ЗЕ = %"(У, зУ), the class defined in IX, 5.7, and let 8 =
(G e S . О (G/O,(G)) e X), which is the class 3 n /)"(у .</) in the subsequent notation.
Then from IX, 5.17 we know that g is a Fitting class containing K. We aim to show
that
K6(e,n8*)\g„
for this implies that S, n 8* / 8» and hence justifies by (6.1) the announced inten-
tion of this example by exhibiting a Fitting class 8 which fails to satisfy the Lockett
Conjecture with respect to S. Because К = YZ(S) < FS' = (FS)' and (FS)' e by
Remark (c) after (3.7), we have К e s„S* = S,. Therefore, since Keg — 8*, it only
remains to prove that К ф g*, and this we do by constructing a suitable transfer
Fitting pair for g, based on the group У.
(4) The Fitting pair (Z2, dr-2,s)
The first step in this direction is to show that К is an (g, 2ycompletion of У. From
our previous calculations we know that a Sylow 2-subgroup of Aut( У) has the form
P* = Q x Z, where Q (sQuat(8)) e SyL(S). Let R = (YP*)f, and write P = Z(S)
(= Z(Q)). Since К = YP e g, we have К < R. Because R/K is a 2-group, it follows
easily that R 6 C” (in the notation of IX, 5.1(c)) and hence that R e X But because
У 6 Hall» 71(R) and СК(У) = 1, it follows from the definition of X in IX, 5.7 that
Re л/ (in fact, t = 1 and <h (R) = R in the notation of that definition). Hence R = K,
and so P = P*n(FP*)s, thus justifying our claim that К = УР is an (8,2)-
completion of К
Next we calculate the subgroup Po = r(P){P; Aut(F)} of P. Since P has order 2
and acts fixed-point-freely on Y/Y, it is clear that r(P) — (a e P. [ , aj <
P = (1 «} and oT'ga e P for some a e Aut(F), then «~lga = g, and consequently
[g a] = 1 Thus {P; Aut(F)} = 1, and hence Po = 1; in particular УР is (g 2Facttve
and Hypotheses 5.10 are satisfied with U = Y. Thus by (5.16) we obtain an g-Fitting
pair (P/Po, d1,2, #) provided that (5.14) is satisfied.
772 X. Fitting classes—the Lockett section
(5) Verifying Hypotheses 5.14 for g
Let Geg, let X' be a subnormal subgroup of G isomorphic with Y, and let
Te Syl2(Ac(X)). Then we must show that XT e g.
Let N = O7.(G) and L/N = O2 (G/N). Thus L/N is the key section k(G) of G and
belongs to the class X*(Y, .</). In this case Ol}1j(L/N) is a direct product of copies of
Y on each of which a Sylow 2-subgroup IV of L/N induces an automorphism group
of order 2 such that the corresponding group extension yields a copy of K. Since by
Condition 14 of IX, 5.7(b) the subgroup W acts faithfully on this direct product, it
follows that W is elementary abelian. Furthermore, since 211G: L|, we have T < L,
and we can therefore assume that TN/N < W. Because M ( = 07(X)) is the unique
minimal normal subgroup of X, the normal 7'-subgroup X n N of X must be trivial.
Hence [X, Tr.N] < XgA' = 1, and consequently Tn N < CT(X) < Or(XT). By
IX, 5.2(a) we have XT e g if and only if XT/(TriN)eg. But XTnN<
CXT(X)nN <Tc,N <XTc\N, and it follows that XT eg if XTN/N eg.
Therefore without loss of generality suppose that N = 1.
Now set V = 07(L) (e Syl7(L)). If Iri X = 1, then L n X is a normal 7'-subgroup
of X and is therefore trivial. In this case [T, X] < L n X = 1, and so XT = X x T
But then «(XT) = 1, and we have XT e g, as desired. Therefore suppose henceforth
that V r>X 1, and note that in this case Fr>X = M, which is the socle of the
primitive group X. Since the subgroup L of G belongs to ХДУ, j/), by Definition
IX, 5.7 it has a normal subgroup D = У, x • • • x Y, such that У = У, < L and
[ЩИУСи^У))) = К for i = 1,..., t; furthermore, the Sylow 2-subgroup W comple-
ments D in L. Let 1 e e E e Syl3(X). Since e normalizes D = 02(L), by A, 4.10 it
permutes the direct components {!)} of D. If Y‘ I) for some 1 < i < t, then D
contains a subgroup of the form Yt x Y‘ x l<e2, which contains an <e>-subgroup
isomorphic with the direct sum of three copies of the regular module F7(e>. In this
case [У, e] has order at least 76 because [F,(e>, e] has order I2. But [V, e] = [V, e, e]
by A, 12.4(b), and therefore we have [V, e] < [G, X,..., X] < X because X sn G.
This contradicts the fact that |X|7 = 73; consequently Y‘ = Yt for all e e E and i = 1,
..., t, and in particular each O7(Y/) is X-invariant. Since 1 M = [M, Z(E)] <
[K Z(E)] and V = П‘=107(К)> follows that there is a je{l,...,t} such that
[O7(I<), Z(E)] # I. By A, 12.4 the subgroup Z(E) acts faithfully on the E-invariant
subgroup [O7(I)), Z(E)], which is consequently faithful for the extraspecial group
E. Hence [0,(1'), Z(E)] has order at least 73 and must therefore coincide with
07(I)); consequently Z(E) has fixed-point-free action on 07(l<). Therefore 07(I)) =
[07(l<), Z(E),Z(E)] < [G, X,..., X] < X, and we can conclude that 07(I)) =
M = O-AYjX).
Suppose that E n Yj = 1. Then there is a Sylow 3-subgroup of YjE of the form EE
with E_e Syl3(L’). Since 07 (G) = 1 and YjX sn G, we have OT(YjX) = 1 and hence
Ker(EE on 07(YjX)) = 1. But this means that Aut(M) contains a subgroup of order
at least |ЕЁ| = 36, against the fact that |GL(3, 7)|3 = 34. Hence Er> Ij 1. Now
E n Yj -Er.E for suitable EeSyl3(lj), and since EnE<jE, we conclude that
Z(E) E. However, easy calculation shows that Z(E) is the only subgroup of order
3 in E which, like Z(E), acts fixed-point-freely on O7(Yj), and so we conclude that
Z(E) = Z(E).
Since [ij](W/Ciy(Yj)) s K, the Sylow 2-subgroup W of G centralizes
6. The Lockett conjecture
773
Z(E)O7(l<)/O7(Ij). Because M = О-ЛУЛ and T is a Svl v
у ow 2-subgroup of К ads fissd-poiur-fredy on О,Щ j, af«, (Оц^, u,w
(6.C)
[M, T] < M.
If [Al, T] = 1, then CXT(M) = M xT, and so
XT — X x T e g. Therefore suppose that
T — O2(CXT(M)) < XT; hence
(6.0
[M, T] * 1.
Let To = CT(EM/M). Then MT0 < XT, and so [M, To] = [M, MT0] < XT. If
[M, 7J # 1, then [M, To] = M because M<i XT. Since this contradicts (6.t), it
follows that To centralizes M and hence by A, 12.4(a) that To centralizes X. Thus by
(6-0 the quotient T/T(l is a non-trivial elementary abelian 2-group of automorphisms
of EM/M centralizing Z(E}M/M. But, as mentioned earlier, CAut(£)(Z(E)) contains a
unique non-trivial elementary abelian 2-subgroup, namely Z(5) in our earlier nota-
tion, and YZ(S) = K. Thus XT/Ta s [X](T/T0) s YZ(S) = K. Since To = 0T(XT),
it follows that XT e {J- This completes the proof that 5 satisfies Hypothesis 5.14.
(6) Conclnsion
Let Я denote the kernel of (P/Po, dr’2’s), which we now know to be an g-Fitting
pair. By (5.17) we have Y = (YP)S{ = (K)B, and by (5.31) we have g* £ Я. Therefore
Kg < Y < KeS, which implies that К ф g*. Since we have already shown that
К e S* n S*, it follows that
g**g*^e*>
and so g fails to satisfy the Lockett conjecture with respect to S.
In Theorems 1.27 and 1.28 we gave some sufficient conditions (due to Hauck) for
the equation (I 0 V))* = I* 0 to hold. With the help of Example 6.17 we can
now describe Hauck’s example of a pair of Fitting classes 3E and '!) such that
3E* (10 $)* £ 0
(6.17) Example. We will continue with the notation of Example 6.16. Thus, setting
H = MT = MES we recall that H is a primitive group with socle M of order 7- and
stabilizer ES, which is the semidirect product of the extraspecialgroupEoforderS-
and exponent 3 by S = SL(2, 3). We also recall the notation Y = MEand К - YZ(5>).
Let I = g where g is the Fitting class defined in Example 6.16. Then we saw at
the end of (6.16) that Kx < Y < К eX*. Since H has a unique chief senes and since
by (1 21) the normal section Hx./Hx must be central, it follows easily that
Hx= Y and Hv = K.
ПА
X. Fitting classes—the Lockett section
We recall from IX, 2.9(a) that the class 32 of soluble groups with central 2-socle is
an R0-closed Fitting class. Thus the class
® = 32nto|2 3)
is also R0-closed and is therefore a Lockett class by (1.25). Set
?) = (®e7e|2.3.51 f ®e3e5e(2.3.51.),
the class of soluble groups whose ®S7S(2 3 51-injectors belong to ®S3<55S(2 3 5)..
Then 9) is a Fitting class by IX, 2.3, and since the class ®S3S5S(2 3 =
G О (®з55®(2.з.5| ) is a Lockett class by (1.26)(b), it follows from (1.36) that 9) is
also a Lockett class.
We will show first that (ЗЕ 0 9))* £ ЗЕ* 0 9) ( = 1*0 9)*). Let H be the group
defined above, and let G = Hrb„g Z2- Since H/Hx = H/Y s SL(2, 3) e ®, we have
H e ЗЕ 0 ® and therefore G e ЗЕ 0 (®S2) = ЗЕ 0 ®. But clearly ® £ 9), and hence
Ge3Eo9)£(3Eo 9))*. It follows from (2.1)(a), since H £ ЗЕ*, that Gj* = and
hence that G/G3. = Alt(4)Qjreg Z2. Hence it will be enough to show that the group
W — Alt(4)0jreg Z2 is not in 9). Now evidently W e ®S7Si2 3 5( and it follows easily
from the definition of 9) that the 9)-radical of W coincides with its ®S3S5S(2 3>5).-
radical. Since = 02(W), we can therefore conclude that кИ{| = (Alt(4))11 and, in
particular, that Thus we have shown that (ЗЕ 0 9))* £ ЗЕ* 0 9)*.
Finally, we will prove that ЗЕ* О 9)* £ (ЗЕ 0 9))*. Set J = H'bZ-, and L = JQjZ5
(where the wreath products are regular), and note that by (2.1) (a) again we have L =
(Jv)" = ((Hj»)")11. Thus L/Lj. S (Alt(4)Qj Z7)Qj Z5. It is straightforward tocheck that
(Alt^y^Zj is a ®<57S(2 з jj-injector of (Alt(4)rbZ7)rbZ5 and that this injector
belongs to the class ®S3S5<5(2 3 5).. Therefore L/L^eY and L e I* 0 9) =
ЗЕ* 0 9)*. On the other hand, since H e ЗЕ 0 G, we have L = (HrbZ7)rbZ5 e
ЗЕ О ®S7Si2 3i5|, and again by the definition of 9), the ЗЕ 0 9)-radical of L coincides
with its ЗЕ 0 ®S3<55(5(2 3i51-radical. SincefH11)11 iscertainly the3E 0 ©-radical ofL, it
follows that J" is the ЗЕ 0 ®e3S5S,2 3 5|.-radical of L and, in particular, that
L£ ЗЕ О 9). Because L = JQjZ5 and J e ЗЕ 0 9), we can deduce from (2.9)(b) that
(ЗЕ о 9))*. Hence ЗЕ* О 9)* £ (ЗЕ О 9))*, as desired.
Exercises
1. Show that each of the following conditions is sufficient to ensure that a soluble
class 3E is 'J-injector-closed: (i) 3E = s„3E and 9) is 3E-normal; (ii) 3E = <5* and 9) is
in the Lockett section of some normally embedded Fitting class.
2. (Brison [2]) A class 3E (£S) is said to be Hall-closed if all Hall subgroups of
3E-groups belong to 3E (i.e. if ЗЕ E | j for all it £ P). If a soluble Fitting class g
is Hall-closed, show that g* and are also Hall-closed.
Chapter XI
Fitting classes—their behaviour as classes of groups
1. Fitting formations
In 1982, in a work of considerable technical bravura, Bryce and Cossey proved the
following remarkable fact.
(1.1) Theorem (Bryce and Cossey [8], [9]). A subgroup-closed Fitting class of finite
soluble groups is a saturated formation.
It follows that s-dosed Fitting classes are just the primitive saturated formations
described in Chapter VII, Section 3. Bryce and Cossey’s proof of this theorem is a tour
de force in the subtle application of deep results from representation theory and ideas
from the theory of varieties. As a full account of this proof, together with essential
background material, is too long to include in this book, we have to be content with
a descriptive outline, given towards the end of this section. Nevertheless, we are able
to treat fully an important related result, which is the first stage in the proof of
Theorem 1.1. It was published a decade earlier (see Bryce and Cossey [l])and already
breaks new ground in the application of representation theory to the study of classes
of soluble groups.
(1.2) Theorem (Bryce and Cossey [1]). A subgroup-closed Fitting formation of finite
soluble groups is saturated.
Since s. < s and Ro < sd„ < sn0 by II, 1.18(c), this theorem is equivalent to showing
that in the universe S we have
(La) E®<<S,N0,Q>.
Throughout this section (apart from Example 1.6 and the Exercises), we shall work
in the universe S.
In outline the proof cf (1.2) runs as follows: Let 3E be an <s, No, Q>-closed class of
finite soluble groups, suppose that 3E is not ^-closed, and choose a group G of
minimal order in Let К be a minimal normal subgroup of G. Since < e.Q
by II, 1.17(iii), it follows that G/K e E.I and hence that G/K e 3E by the choice of G.
Because 3E is R0-closed, К is the unique minimal normal subgroup of G, therefore
FIG) is a p-group for some prime p and К < 1>(G).
Now, corresponding to a given group H, ъ field E, and a non-empty class ?) of
groups containing H, we define a class Mod(£, H, $)) of EH-modules thus.
ПЪ XI. Fitting classes—their behaviour as classes of groups
(If} Mod(£, H, '!)) = (M: M is a £H-module with [M] H e '.!)).
Since H e 8 by assumption, Mod(£, H, 91) contains the zero module and is therefore
non-empty. We shall now show that, in the special situation described above, the
following conditions are satisfied when we set 'Di = Mod(Fp, H, I) and H — G/K.
(a) If 17 is a submodule of M e ®i, then 17 e 'Di;
(a*) If 17 is a submodule of M e 'Di, then M/17 e 'Di;
(b) If M,/V e'Di, then M @/V e'Di;
(c) If M,/V e'Di, then M /V e'Di;
(d) If an £H-module 17 contains a submodule F e 'Di such that
[17, H] < V, then U e ®i;
(e) There exists an Me®! such that Ker(H on M) = Op(H).
[In the language of closure operations applied to the category of EH-modules
Conditions (a), (a*), and (b) correspond respectively to the s-, Q-, and d0-closure of 'Di.]
Having shown that Conditions (l.y) are fulfilled with 'Di = Mod(Fp, G/K, X), we
then prove the following theorem.
(1.3) Theorem. Let H be a finite group, E a field of positive characteristic p, and 'Di a
class of EH-modules satisfying Conditions (b) (e) and at least one of (a) and (a*) in
(l.y). Then SH consists of all EH-modules.
This done, we can then conclude that the base group of the wreath product W =
К rLrcg(G/K), regarded as an Fp(G/K)-module, belongs to Mod(Fp, G/K, X), and this
means that W belongs to X. But then by A, 18.9 we have G e s(PF) S sX = X,
contradicting the choice of G. Our initial supposition must therefore be false, and so
X = ЕфХ, as desired.
We will now fill in the missing details. In proving that Mod(Fp, G/K, X) satisfies
the conditions of (l.y), it is convenient to isolate the following result.
(1.4) Lemma. If H e 9) = <s, No>9), then the class 'Di = Mod(Fp, H, ф) satisfies
Conditions(a),(b)and(c)of (l.y), and if pe o(9)) (in particular, if p||H|), it also satisfies
Condition (dj.
Proof. If A is a submodule of Meffli, then it is clear that [iV]H may be viewed as
a subgroup of [M]H, and so the s-closure of 'Di follows from that of 9).
Next, let M, N e ЯИ, and observe that the group
D = [M]H x [A]H
1. Fitting formations
777
я„wi,h л‘'™‘ “*°“p »•=
JS?"F'^“M® •«« T.ma.»H X H-module
(m®n)(hl,h2) = mhl®nh2
(see B, 1.12). If H is identified with H x 1 in the obvious way, the restricted module
THxl is isomorphic with a direct sum of Dim.JN) copies of M. Hence [T] (H x 1)
belongs to V) by the D0-closure of ЯП. Similarly'!) contains [T] (1 x Я), and therefore
[T] (H x H) e n„9) = 5). Now regarding T as an H-module in the conventional way
we have [T]H s [T]H* e s9) = 9) and can conclude that ЯП satisfies Condition (c)
of (l.y).
Finally, let U and V be FpH-modules as described in Condition (d) of (l.y). Then
evidently [C]/f is a product of normal subgroups U and VH. If p||H|, it follows
from IX, 1.7 and IX, 1.9 that U e 9). Since Fe ЯП, we have VHe 9) and therefore
[G]H e n„9) = 9); hence U еЯП, as desired. □
Now set
ЯП0 = Mod(Fp, G/K. X).
Since К < Ф(6) < F(G), which is a p-group, we have p 11 G/K |, and therefore by (1.4)
Conditions (a) -(d) of (l.y) are satisfied with ЯП = ЯП0. (It iseasy to verify that ЯП = ЯП0
also satisfies Condition (a*), but this fact will not be needed.)
Our next goal is to prove that Condition (e) of (l.y) is satisfied with ЯП = ЯП0
and H = G/K. Set M = F(G)/®(G), an elementary abelian p-group centralized by
K, and view M as an FpH-module in the usual way. By A,10.6(c) we have
Ker(G on M) = F(G) and by A, 9.3(c) also F(H) = F(G)/K = O„(G)/K = 0p(H).
Therefore Ker(H on M) = 0„(H). and consequently 0p(H) can be regarded as a
normal subgroup of the semidirect product
X =
Moreover, Х/ОЛН) is isomorphic with [F(G)/®(G)] (G/F(G)) and therefore also with
G/<D(G) e q(G/K) S qX = F by A, 10.6(c). (This is the only place where the Q-closure
of X is needed.) Since X/M H = G/K e X and OP(H) n M = 1, we conclude that
X e r X = X. Hence M belongs to ЯП0, and we have proved that ЯП0 also satisfies
Condition (e) of (1 .y). Therefore, to complete the proof of Theorem 1.2, it only remains
to prove (1.3).
The Proof of Theorem 1.3. Let ЯП be a class of EW-modules satisfying Conditions
(b) (e) of (l.y) as well as either Condition (a) or Condition (a ). Let
P Ker(HonM),
M e SR
778 XI. Fitting classes—their behaviour as classes of groups
and observe that since H is finite, there exists a finite subset {Л/,,..., of ®1 such
that
N = Q Ker(H on Mj).
«=1
By (l.y) (b) the class ®i contains the module D = Mo (±) ф • • • ф M„, where Mo is
the trivial simple EH-module; clearly N coincides with the set of elements of H which
have scalar action on D. By B, 10.17 there is a finite direct sum T of certain tensor
powers of D which contains a regular E(H//V)-module R regarded as a EH-module
by inflation. By (l.y) (b) and (c) we have Te ®1, and since R is projective and hence
a direct summand of T, we can appeal to either of Conditions (a) or (a*) of (l.y) to
conclude that ReW. But an arbitrary EH-module M with N < Ker(H on M),
viewed as a E(H//V)-module, is isomorphic both with a submodule and with a
quotient module (see B, 2.6, B, 2.12(a), and B, 4.10(a)) of a suitable free E(H//V)-
module. Therefore M is either a quotient module or a submodule of a direct sum of
finitely many copies of R, and by Condition (b) in conjunction with (a) or (a*) once
more, we conclude that M contains all EH-modules which have N in their kernels.
Therefore, to complete the proof of (1.3), it will suffice to show that N = 1. We
suppose that N 1, let N/L be a chief factor of H, and derive a contradiction by
producing a module В in ®i such that Ker(H on B) = L. Therefore, without loss of
generality, suppose that L = 1, and hence that N is an elementary abelian p-group
because N < Op(H) by (l.y) (e). By B, 11.3 there exists an FpH-module В which is
faithful for H and has a submodule В such that
(i) B/B = (Fp)H, the trivial FpH-module, and
(ii) Ker(H on B) = N.
Since we have proved above that В ®Fp E e 9)1, by (l.y) (d) we have В E e ®l.
But as Ker(H on В ®F E) = 1 < N, this contradicts the definition of N, thereby
completing the proof of (1.3) and with it the proof of Theorem 1.2. □
We recall that 9l(p) denotes the class of elementary abelian p-groups.
(1.5) Corollary. Let g be a subgroup-closed Fitting class containing 9I(p)S,. Then g
contains SPS,.
Proof. Let G e SPS,, and assume inductively that for all H e SPS, with |H| < |G|
it has already been shown that H e g. Since 9l(p)S, S g, suppose that Op(G) is not
elementary abelian; it therefore contains a minimal normal subgroup К of G such
that p||G: K|; furthermore, the group H = G/K belongs to g by induction. Thus by
(1.4) the class ®1F = Mod(Fp, H, g) satisfies Conditions (a), (b), (c) and (d) of (l.y). Let
M be the regular Fp(H/Op(H))-module, viewed as an H-module. Then Ker(H on M)
= OP(H), and if L denotes the semidirect product [M]H, we have L e Rog because
L/M s H e Rand L/Op(H) [M] (H/OP(H)) e 9I(p)S, c g. Therefore Leg, and so
the class Hi, also fulfils Condition (e) of (l.y). Consequently, by Theorem 1.3 the group
W = KlJreglG/K) belongs to g; hence by A, 18.9 we have G e s(W) £ sg = g, and
the induction step is complete. □
1- Fitting formations
779
(1.6) Example. Let J be
H < J, then H is soluble
form
a non-abelian finite simple group with the
(e.g. J = Alt(5)). Let g denote the class of
property that if
groups G of the
(1.Й)
G = Jt x x J„ x H
for some n e N with J( s j (i = 1, 2,.... n) and H e S.
Let g0 = d„( J) By II, 2.13 the class g0 is a Fitting formation. Let L be a subgroup
о the group G of (l.<5), and forii = l,...,n + 1 let л, denote the projection of G onto
the i th component of the direct product (l.<5). Let .</ denote the subset of (1 n)
defined by i >•••• t
ie.7 о л,(Ц = J;,
and set
K = А Кег((л.)г.) and K* = A Кег((л;),).
Then L/K e RC(J) = So and L/K* e R0S = S. Since Kr>K* = 1, it follows that
K*( = K*K/K)es„g0 = g0 and that Kes,3 = 3. Since L/KK* e Qg0 n Q® =
go n S = (1), we conclude that L = К* x К e g. Thus we have shown that g is
s-closed.
Now if g, and g2 are Fitting formations with the property that g, n g2 = (1), it
is easy to check that the class of all groups G which have the form G = G, x G2 with
G,- e g; (i = 1, 2) is also a Fitting formation. Therefore the class g defined by (l.<5) is
an s-closed Fitting formation.
Now specialize g by taking J = Alt(5). Let G = SL(2,5). Then G/<D(G) s J e g.
However, G is directly indecomposable and consequently G e r^gXg. This example
therefore shows that g is not Е»-closed and hence, in particular, that Theorem 1.2 is
false without the hypothesis of solubility.
(1.7) Theorem (Bryce and Cossey [1]). The s-closed Fitting formations of finite
soluble groups are precisely the primitive saturated formations (defined in VII, 3.1).
Proof. By Theorem VII, 3.10 it will be enough to verify Hypotheses (i)- (iv) of that
theorem when a = <s, n0>. Since (i), (ii) and (iv) have already been verified in VII
3.11, it only remains to observe that Hypothesis (iii) is a direct consequence of
Theorem 1.2 above.
For metanilpotent Fitting formations the same characterization holds without as-
suming s-closure.
(1.8) Theorem (Hawkes [5]). A Fitting formation g gf metanilpotent groups is a
primitive saturated formation; in particular, g = sg.
780
XI. Fitting classes—their behaviour as classes of groups
Proof. By (1.7) it is sufficient to prove that g is subgroup-closed. Let H < G e g.
Then G < 912, and so F(G)H e s,(G) £ s„g = g. From IV, 1.14 it follows that He g,
and therefore g = sg. □
In Chapter IX, Example 2.21(a) we describe a Fitting formation contained in 913
which is not saturated. The following characterization shows that this example is also
not s-closed.
(1.9) Theorem (Bryce and Cossey [1]). Let ft be a Fitting formation contained in 9l3.
Then g is subgroup-closed if and only if it is saturated.
Proof If g is s-closed, then it is saturated by (1.2). Now suppose that g is saturated.
Then g is local by IV, 4.6, and by IV, 3.16 the canonical local definition F satisfies
F(p) = <Q, Ro, S„ No >F(p)
for all primes p.
Let p e Char(g), define = (G/O„.„(G): G e g), and set g(p) = <q, r0, s„, n0>X„.
Since £ F(p), it follows from IV, 3.10 that
(1.8) ftp) £ g(p) £ F(p).
If p Char(g), set g(p) = 0. In this case ftp) = g(p) = F(p), and so (Le) holds for all
p. Therefore g = LF(g) by IV, 3.11. Since 3Ep is a subclass of 9l2, so also is the Fitting
formation g(p); consequently g(p) = sg(p) for all primes p by (1.8), and hence g = sg
by IV, 3.14. □
We now offer a short essay outlining the main ideas and methods of Bryce and
Cossey’s proof of Theorem 1.1.
A survey of the proof of Theorem 1.1
By Theorem 1.2 it is enough to prove that an s-closed Fitting class g is Q-closed, and
for this it suffices to show that each of the classes g л 9lr (r = 1, 2,...) is Q-closed
since g = IJ’=i (8 n ^r)- Thus, if the theorem is false, there exists a natural number
r(> 1) and an s-closed Fitting class g £ 9lr such that
(a) g is not Q-closed, and
(U)
(b) all s-closed Fitting subclasses of 9Г1 are Q-closed.
Let и denote a sequence (pit p2,..., pr) of primes with p; # pi+1 for i = 1,..., r — 1,
and set
' ®v=®P.®₽/"®^
In Section 3 of [7] Bryce and Cossey prove the existence of a sequence и of this type
1. Fitting formalions
781
such that
(1'1)
n 9F 1 C g <z Go.
Let G denote the class of 9Г ‘-critical groups contained in g. (Recall that G is said
in sxi -c ard я \ ° (rG) -x) since *=s5- a®roup °r ™nimai °rder
-cnticaUnd therefore G is non-empty. Using Theorem 1.3 and(LC)
(a) and (b), Bryce and Cossey prove in Lemma 2.9 of Г7Т that if G e G then
<S, No> (G/<D(G)) = G„, and hence deduce that
(1.0)
if G e G, then G/O>(G) ф g.
They devote the rest of the proof to finding a group G in G with G/<D(G) e g, thereby
obtaining a contradiction. Starting from one unspecified group in G, they invent ways
of building from it a sequence of further groups in G with ever more tractable
properties.
A group G in Gu is said to be in expanded form if it has p,-subgroups /7(1 < i < r,
where и = (pt, p2,..., pr)) such that the subsets
L, = LfG) = P, P2...Pi_l, and
Rf = Ri(G) = P,Pi+t...Pr
are subgroups of G satisfying
(i) Pf < R,, (ii) G = LjRj, (iii) L, n R, = 1
for 1 < i < r. Bryce and Cossey show that G contains groups in expanded form: in
fact, if H is any group in G, let 'P be the variety generated by H, and let G denote the
'P-projective cover of H. Then it turns out that G is again in G and has expanded
form. From the structure of ’«’“‘-critical groups described in VII, 6.21 it follows easily
that a group G in G in expanded form G = P, P2... P, has the following properties:
(i) Pi has a unique maximal R,(G)-invariant subgroup,
denoted by Л/,, say (1 < i < r);
(ii) [P;, Pj+1] = Pi (1 < i < Hi
(iii) [Л/j, P;+1] < С (1 < i < r), where C, = 1 and C, = CpJ/’ ,) for i > 1,
(iv) C, < Mi and P./C, is a special p.-group with Ф(Р,/С) - M,/C; (1 < i < r).
A group G in 6 is said to be in standard form if it is W'1-critical, in expanded
form and if C. = 1(1 < i < r) in the notation of (l.i) (iii). Ш this case each building
block P of the expanded form is a special p,-group which acts faithfully on the G-chief
XXI/H- Jminediately below. The next major step in the proof .s to prove that
782
XI. Fitting classes—their behaviour as classes of groups
G contains a group in standard form, and the key to this is a theorem which allows
a given group G = Pi P2 • • • Pr (expanded form) in G to be transformed into another
group H = Qi Q2 . -. Qr (expanded form) in 6 such that, for a given i e {1,..., r — 2},
(i) Ki+1(G) S Л1+1(Я),
(ii) V = P-JM, is replaced by V = Qi/Nh where V may be chosen to be any
irreducible submodule of (l/Pi i)/t‘*,,ci, viewed as an P,+l(H)-module via the iso-
morphism in (i), and
(iii) if Pk/Ck is abelian (2 < к < r), then so is the corresponding section of H.
This result. Lemma 7.1 of [8], is used to slide an unwanted centralizer C( down past
each lower P2 (j < i) in turn and, when it reaches the bottom, to cast it out with the
help of Lemma 7.2 of [8], which states that if N < H = QkQ2 ... Qr, the expanded
form of a group in G, and if N < Qt for some i > 2, then H/N belongs to 5 (and hence
to 6). This procedure, applied repeatedly, transforms the original group G = Pk P2... Pr
into another group H = Qk Qi ... Qr in 6 which is now in standard form and retains
the property that Qi is abelian whenever PJC, is abelian.
The proofs of these transformation theorems require skillful use of the twisted
wreath product for the construction of new groups, together with a deep knowledge
of representation theory with which to analyse their properties; in fact, nearly half of
Bryce and Cossey’s paper [8] is devoted to establishing requisite facts about induc-
tion, extension, and tensor products of modules, and it is a useful source of basic
information about finite modules for soluble groups not available elsewhere in the
literature.
Similar techniques also dominate the last stage of the proof, which includes two
further transformation lemmas, having the following consequences:
(1) If G = PiP2 .. Pr is a G-group in standard form with Pk abelian for some
к e {2,..., r — 1}, then G contains a group H = QkQ2 ... Qr in standard form with
Qk, Qi+i> all abelian (see Lemma 8.2 of [8]).
(2) If G = P,P2 ... Pr is a G-group in standard form with PkPk_k non-abelian
for some к e {2,..., r}, then G contains a group H = Qk Q2 ... Qr in standard form
such that either Qj is abelian for some j < к — 1 or else Q,, Q2, ..., Q4_[ are all
non-abelian, Rk(G) S Rk(H), and Lk(H) centralizes d>(Q,) (see Lemma 8.1 of [8]).
A combination of these two results shows that G contains a group G* = PkP2 ... Pr
in standard form such that for some к e {2,..., r}
(i) Rk(G*) is a polyprimitive group, with unique complemented chief series
1 < Л < PkPk+i < < Rt(G*), and
(ii) fj, P2,. ., Pk_k are all non-abelian and centralize Ф(Р,) (= Ф(С*)).
Let N be a minimal normal subgroup of G*, necessarily contained in Ф(С*). Then,
on setting P = Pt, L = R2(G*), A = CJN) = P2P3 ... Pj-i (for some j > k) and
В = Rj(G*), we see that G contains a group G( = G*) with the following properties:
(a) G has a normal p-subgroup P complemented by a subgroup L of nilpotent
length r — 1,
(b) L has a normal subgroup A complemented by a polyprimitive subgroup В of
nilpotent length at most r — 2, and
(c) G has a minimal normal subgroup N with CL(N) = A.
In the culminating result of their paper (Lemma 9.1 of [8]) Bryce and Cossey show
that if § contains a group G having properties (a), (b) and (c), then G/N belongs to
2. Metanilpotent Fitting classes with additional closure properties
Я and hence to G. Repeated application of this result in r* m .
“Jcomply Ф<С*)’ С*/Ф(С*) 6 5' th,S “Si If the prZ
Exercises
1. Suppose that g, and g2 are Fitting formations and that g, n g2 = (1). Show that
8i x g2 - (G G - G, x G2, G,e gf(i =1,2)) is also a Fitting formation.
2. Let 3 be a non-empty class of non-abelian finite simple groups, and let л be a set
of primes containing <r(3). Assume further that if H < G e 3, then He S. Let g
denote the class of all finite groups G which satisfy the 3 properties- (i) the
composition factors of G belong to {Z„: p 6 n} и 3; (ii) G induces only inner
automorphisms on each insoluble chief factor; (iii) G induces only soluble groups
of automorphisms on each abelian chief factor. Show that g is a subgroup-closed
Fitting formation which is not saturated.
Now specialize g by taking 3 = (J) with J — Alt(5). Let A denote the stabilizer
of 5, so that A = Alt(4) and |J : A \ = 5. Let U denote the 1-dimensional trivial Fs
Л-module, and let V = U1. Then by B, 6.16, the quotient F/Rad(F) is the trivial
simple f5 J-module. Since V is non-trivial, it is faithful for J by the simplicity of J,
and as J is not a 5-group, it follows that V has a composition factor which is
faithful for J.
Therefore the semidirect product S = [F] J does not belong to g. However,
S/d>(S) = J x Zs e g; therefore S e E„g\g.
3. (Hauck (for § = U), Pense—unpublished) Let § be a saturated formation of meta-
nilpotent groups, and let h be an inclusive local definiton of § with h(p) £ 91 for
all p 6 P. Let g = Fit(§). Then the Lockett class g* is a saturated formation and
is locally defined by f, where
ftp) = Fit(h(p)) = 91, for n = Char(h(p)).
In particular, g* is s-closed and is a primitive saturated formation.
4. Find an example of a saturated Fitting formation in 9t4 which is not subgroup-
closed.
2. Metanilpotent Fitting classes with additional closure properties
If the goal of finding all Fitting classes of finite groups is hopelessly beyond the reach
of present knowledge and methods, then a more modest objective, such as classifying
all Fitting subclasses of 9l2 *. would seem to offer a better prospect of developing new
ideas and gaining insight into the general problem. But even to understand meta-
nilpotent Fitting classes appears to be very difficult. For example, the ^t 1probtem
of finding an effective description of the Fitting class generated by Sym(3)sril
unsolved Furthermore, the family of examples contained in GpGe which are de
.. я IX 6 26 suggests that a daunting complexity can anse, even within Ji .
r„T tx: «= ~ .»=
classifications.
784
XI. Fitting classes—their behaviour as classes of groups
The subject matter described by the title of this section has been the almost
exclusive preserve of R.A. Bryce and John Cossey. Their main results in this area may
be summarised as follows:
(A) Each of the following conditions is both necessary and sufficient for a Fitting
class g £ 'JI2 to be a primitive saturated formation:
(•) 8 = <28; (ii) 8 = еф8; (Hi) 8 = s8-
(B) If a Fitting class 8 is Ro-closed and properly contained in SpS,, then 8 is a
subclass of the class of groups with p-socle central.
(C) There exist metanilpotent Fitting classes which are Ro-closed but not s-closed
(Example 2.17 below) and also metanilpotent Fitting classes which are not closed
under any of the operations Q, Ro, еф, or s (Example IX, 2.14(b)).
In Theorem 1.7 we showed that a metanilpotent Fitting class is s-closed if and only
if it is a primitive saturated formation and hence that (iii) => (i) and (iii) => (ii) in (A).
In the sequel Theorem 2.1 shows that (i) => (iii) and Theorem 2.16 that (ii) => (iii).
Statement (B) is precisely the content of Theorem 2.21.
(2.1) Theorem (Berger, Bryce, and Cossey [1]). If 8 ix a Q-closed Fitting class of
metanilpotent groups, then 8 is s-closed.
We shall prove this theorem in two stages. First we exploit some general methods
for Q-closed metanilpotent Fitting classes to show that Theorem 2.1 follows from
Theorem 2.5. Then, after presenting some elementary facts about verbal products of
groups associated with a variety, we finally establish the truth of Theorem 2.5, which
is at the heart of Berger, Bryce, and Cossey’s artful proof of Theorem 2.1. Although
the next proposition helps with the implication: (2.5) =>(2.1), its main application
comes later, namely in the proof of Theorem 2.16, which is the second main result of
this section. Its hypotheses are based on the supposition (from which contradictions
are derived) that Theorems 2.1 and 2.16 are both false.
(2.2) Proposition. Let g be a Fitting class of metanilpotent groups. Assume that 8 is
either Q-closed or f^-closed, but not s-closed. Then 8 contains a group G with the
following properties:
(a) G = F(G)C, with F(G) a p-group and C a non-trivial cyclic q-group (p and q
distinct primes):
(b) G has a subgroup H = (F(G) n H)C not in 8-
Proof. Let G be an g-group of minimal order such that s(G) £ 8, and, among
subgroups of G not in 8, let H be one of minimal order. This choice of H clearly
implies that H is single-headed and hence that H = Н'ЯС, where C is a non-trivial
cyclic q-group for some prime q. Observe that H'x is a q'-group since СУ(Н^) = Я91 e 91.
Since F(G)H e s„(G) S 8, the minimality of G forces the conclusion that F(G)H = G.
Consequently Я/(Е(6) n Я) S G/F(G) e 91, and so Я 91 < F(G); therefore Я =
(FIG) n Я)С and G = F(G)C.
2. Metanilpotent Fitting classes with additional closure properties 785
It remains to prove that F(G) is a p-group for some prime p / q Let 0 e Svl (Ftcn
»d s. нл Wo» s™ ,1101 0«S. th.
the quast-R lemma (IX, 1.13) SC S G/Q e g. Since H91 < S, we have H < SC, and
so SC - G by the choice of G. Therefore Q = 1 and F(G) is a q'-group
Let p be a prime dividing |F(G)|, and let P e Syl„(F(G)). Let Те Hall (F(G)) and
suppose for a contradiction that T / 1. The assumption that g = ^implies that
G/(<I>(P) x T) e g (see Lemma 2.15 below), and because СР/Ф(Р) s С/(Ф(Р) x T)
and Ф(Р) < Ф(СР), it follows in this case that CP e E»g = g. Thus, in any case, the
hypotheses of the Proposition imply that G/Те g, and therefore by the quasi-R0
lemma CT ~ G/P e g. Since by supposition CP and CT are proper subgroups
of G, the minimality of G yields C(P n H) and C(T n H) e g. But clearly F(G) n H =
(P H)(Tc\ H), and so H e g by the quasi-R0 lemma once more. This contradiction
forces the conclusion that T = 1 and that F(G) is therefore a p-group. □
(2.3) Lemma. If a Q-closed Fitting class g contains a group G of the form described
in (2.2) (a), then E(q/p) e g.
Proof By hypothesis G = 0p(G)C, where 0p(G) = F(G) and C is a non-trivial cyclic
q-group. Let Z = f2j(C) and M = [0p(G), Z]. Since Cc(F(G)) < F(G), it follows that
M ф 1, and furthermore, since MZ < 0p(G)Z < G, that MZ e s„g = g. Let M/N be
a chief factor of MZ. From the fact that [Л1, Z] = M by A,12.4, it follows that
E(q/p) = [M/7V] Z S MZ/N e Qg = g. □
The next proposition is the key to the implication: (2.5) =» (2.1).
(2.4) Proposition. Let g be a metanilpotent Fitting class which is closed under at least
one cf the two operations Q and Ro. If g contains the class &p(Zq), then it contains
<SP6,.
Proof Any group in GpG,\Gp is evidently generated by subnormal subgroups X
satisfying
X/0p(X) == Z,„ (л =1,2,...).
Therefore it will suffice to show that all such groups X belong to g, and this we do
as a subgroup of IF = Ti regZ,. Let
у = РгЪгИ'=[Р"']И/
Ь. th. .»»« »r«l, XX*
“ 1'eKS ’ ""d
786
XI. Fitting classes—their behaviour as classes of groups
therefore that P"Z e s.(F) c g. But by B, 6.20 we have
(2.a) S X
seS
where S is a complete set of (Z, Z)-double coset representatives in IV and 1 e S.
Therefore (Ри )z = P x K, where К denotes the product of the components on the
right-hand side of (2.a) taken over the non-identity s e S; clearly К is Z-invariant. It
follows that
PZ = (P x K)Z/K = PWZ/K e Qg,
and so if g = Qg, the induction step is complete.
Finally, suppose that g is R0-closed. Let TV, = P x 1 x 1 and N2 — 1 x P x 1,
both normal subgroups of the group L = [P x P x K]Z. Since L/Ni ~(Px K)Z e
g for i = 1, 2, it follows that Le Rog = g. Then by the quasi-Ro lemma (IX, 1.13) we
have PZ s L/(l x P x K) e g, and again the induction can proceed. □
The following theorem is the core of Berger, Bryce, and Cossey’s proof of Theorem
2.1.
(2.5) Theorem. Let % be a Q-closed Fitting class which contains the group E(q/p). Then
g contains the class <5p(Zg).
The proof that (2.5) =>(2.1):
If Theorem 2.1 is false, then by Proposition 2.2 there is a group G in gr,
satisfying (2.2) (a) which has a subgroup H not in g, and so by Lemma 2.3 we have
£(<?/p) e g. But then by (2.5) the class g contains <Sp(Zg), and from (2.4) we conclude
that H e Sp6q £ g, which is a contradiction. Thus Theorem 2.1 follows from Theor-
em 2.5. □
Before we can prove Theorem 2.5 we shall need some facts about the verbal product
of groups associated with a variety Ф. Our account will be limited to the bare
essentials required in subsequent applications. A reader seeking further information
should consult Hanna Neumann’s book [1], pp. 32-37, or, for more comprehensive
details, the relevant papers of Moran [1], [2], and [3].
As we saw in Chapter II, Section 2, a variety 53 is a class of groups defined by a set
IV of words, which, when equated to 1, give the laws identically satisfied by just the
groups in 53. The verbal subgroup 53(G) of an arbitrary (not necessarily finite) group
G is the subgroup generated by all elements of the form w(gl,..., g„), where w runs
through W and for each word w we allow all possible substitutions xf -+ g: of elements
of G for the variables x,,..., x„ appearing in w.
Let X and Y be arbitrary groups, and let
F = X* Y
denote their free product. This can be written F = X У[Х, У]; indeed, each element
2. Metanilpoient Fitting classes with additional closure properties
got F has a unique expression in the form
9 = xyc,
with x 6 X, у 6 Y and с e [X, Y], It then turns out that
(2-^) ®(F) = ®(X)23(Y)([X, Y] n 18(F)).
(2.6) Definition. The verbal product X *8 Y of two groups X and Y is defined thus:
X *s Y = (X * Y)/([X, Y] n 23(X * Y)).
Denoting images under the natural homomorphism
~:X* Y->X.SY
by bars, it is easy to see that the verbal product of X and Y contains subgroups
X(=X) and Y(= Y) such that X n Y= 1. What is more, since
(X *s Y)/[X, Y] s (X* Y)/[X, Y] = X x Y,
it has the direct product of X and Y among its epimorphic images. In fact, the verbal
product actually becomes the direct product when 21 = 21 and coincides with the free
product when 21 is the class of all groups. We also remark that when X and Y are in
23, then 23(X * Y) < [X, Y], and so in this case we have X*s Y = (X* Y)/23(X * Y)e23.
(2.7) Lemma. Let Hbca locally finite variety, and let X and Y be finite groups. Then
the verbal product X *s Y is finite.
Proof. Since the free product X * Y = <X, Y> is finitely generated, so is the verbal
product
G = X *3 Y = (X * Y)/([X, Y] n 23(X * Y)).
Let N = [X, Y]/([X, Y] n 23(X * Y)) < G. Since |G:N| = |X*Y:[X, Y]| =
|X x Y|, it follows that |G: N| is finite and that [X, Y] is finitely generated. Since, by
hypothesis, ® is locally finite, we conclude, that [X, Y]/23([X, YJ) is finite and hence
that N is finite because obviously 21([X, YJ) < [X, Y] n ®(X* Y). It follows t ere
fore that G is finite.
Since a variety generated by a finite group is locally finite (see II, 2.14), we obtain the
following consequence of (2.7).
(2.8) Corollary. Let H = <X, Y> and 23 = Var(H). If His finite, then so is the ver bal
product X *3 Y.
788
XI. Fitting classes—their behaviour as classes of groups
The next result describes an important property of verbal products.
(2.9) Theorem. Let YR be a variety containing a group J = <_H, K), and let a: X -»H
and p. Y -» К be group epimorphisms. Then there exists an epimorphism
such that the restrictions 0X and 0r coincide with a and p respectively.
Outline of proof. Let F denote the free product F = X * Y. It is a basic property of
free products that there exists an epimorphism в: F -> J such that вх = a and = p.
Since J e 53 by hypothesis, we have 53(F) < Ker(0) and so can find an epimorphism
v, an epimorphism p, and an isomorphism ф between the following groups as shown:
F F/([X, У]п 53(F)) r F/Ker(0)7 J
such that в = ф ° p ° v. It is then straightforward to verify that the map в = фо p has
the desired properties stated in the theorem. □
(The reader is referred to Theorem 18.42 of Hanna Neumann’s book for further
details.)
(2.10) Lemma. Let YR be a variety and X, Y groups. Then the verbal product G — X *s У
admits Aut(X) x Аи1(У) as a group of automorphisms in such a way that the restric-
tions of (a, /?) e Aut(X) x Aut(y) to the subgroups X and Y of G are a and [i respect-
ively.
Proof. Let (a, p) e Aut(X) x Aut(y). By (2.9) there is a homomorphism r(a): G -» G
such that r(a)x = a and t(a)y = tr, the identity map on У; similarly a homomorphism
r(p): G -» G with t(p)x = tx and v(p)x = p.
If <r(a): G -* G is the homomorphism satisfying <r(a)x = a'1 and c(a)y = ir, then
the homomorphisms <r(a)r(a) and r(a)<r(a) each fix X and У elementwise and hence
coincide with ic because G = <X, У>. Thuso(a) = r(a)_*, and therefore r(a) e Aut(G).
Similarly t(p) e Aut(G), and it is straighforward to verify that
(i) т(ос)т(/?) = v(p)v(a), and
(ii) r(a)r(/J) = tc if and only if a = tx and P = ir.
Thus (a, p) -»r(a)r(/J) is a monomorphism from Aut(X) x Aut(T) into Aut(G), and
by identifying each (a, p) with its image in Aut(G), we obtain the desired conclusion.
□
(2.11) Proposition. Let YR be a variety containing a group J = (A, B), and let G denote
the verbal product A *SB.
Set
S = {a e Aut(F): a(A) = A, a(B) = B},
and for each a e S let a = (aA, aB) e Aut(A) x Aut(B). If a is regarded as an element
2. Metanilpotent Fitting classes with additional closure properties
f?Ut(G)?/ 2-Ю, then the epimorphism 0-G^J defined in
Theorem 2.9 extends to an epimorphism of the following semidirec. product :
0: [G]S - [J]S,
where S = {a: a e S}.
Proof. Define в: [G]S -> [J]S in the obvious way thus:
0:ga = (0g)a,
noting that themap a —> a is a bijection from S to S and therefore has a unique inverse.
It is clear that в is onto, so we have only to show that it respects group multiplication.
Let a e S and h = e G(af e A, b, e B). Using the notation from the proof of
(2.10), in the semidirect product [G]S we have:
a lha = (П«,Ь,)"“>|,,|“',) = Па°'1ЬГ* =
Thus 0(a-1ha) is the element = а-1(ПаД)“ = a-10(h)a of [JJS. Hence
0((»a)(¥)) = 0(g(aha~lMf)
= в(д)в(аНа~')ар = в(д)ав(Ь)а~‘ар
= (0(g)a)(0(h)p) = O(gS)0(hp),
and so в is an epimorphism, as claimed.
□
This completes our short survey of some pertinent properties of verbal products.
Now, suitably equipped, we can return to matters more directly concerned with the
proof of (2.5).
(2.12) Definition. If X is a group, define Ф‘(Х) recursively by
Ф°(Х) = X. and
ф‘(Х) = Ф(Ф' '(X))
for i = 1,2,.... If X is a finite group, then Ф"(Х) = 1 for some n e N. The least such
n we call the Frattini length of X and denote it by 2Ф(Х).
contains all groups X in ^fZ} with Д^ОДХ)) <n.„„
(1) К has Z*-invariant subgroups R and Q such that R < O„(K) - RQ,
(2) RZ* and QZ* belong to g, and
790
XI. Fitting classes—their behaviour as classes of groups
(3) 2e([R, Q]) = m < n.
Then К eg.
Proof. Let ® denote the variety generated by RQ, and V denote the verbal product
F = R, *«61.
where R, and Qx are isomorphic copies of R and Q respectively. By (2.6) there is an
epimorphism
a: V-rRQ
such that the restrictions of a to R, and Ci are the prescribed isomorphisms (r, -» r,
etc.) onto R and Q. If Z* is viewed as a group of operators for R and Q (and hence
for R, and Qi in the obvious way), we know by (2.10) that Z* x Z* is a group of
operators for И and furthermore that, if Z = {(z, z):zeZ*J < Z* x Z*, then a.
extends by (2.11) to an epimorphism
(2.)-)
a:[VJZ->K.
We write P — RQ, and prove the proposition by induction on the central Frattini
length 2®(R, P) of R in P. This is the shortest length of a series
1 = R, < R,_! < < Ro = R
in which R,-) < P and R^/R, is an elementary abelian subgroup of the centre of
Р/R,- for i = 1,..., t. Formally, it is defined thus: for X < P set Л(Х) = Ф(А') [X, P];
let A0(R) = R, and for i > 1 let A,(R) (relative to P) = A(Af_i(R)). Then 2®(R, P) =
Min{t: A,(R) = 1}. It should be observed that Af(R) is invariant under all automor-
phisms of P which leave R invariant.
If /®(R, P) = 1, then R is an elementary abelian group contained in the centre of
P. Since К then induces a p'-group of automorphisms on R, by A, 11.5 there is a
normal subgroup R of К complementing R n Q in R. In this case we have К/Р = QZ*,
which belongs to Й by Hypothesis (2), and K/Q = RZ*, which belongs to {J because
2e(0p(RZ*)) = 1 < n by hypothesis. Since RQ = P, the quasi-R0 lemma (IX, 1.13)
applies, and we can conclude that К e {J. Thus we have a point of departure for the
induction.
Now suppose that 2&(R, P) = t > 1. Our aim will be to show that a suitable
quotient group of [V]Z belongs to 3, because then the existence of a. in (2.y) and
Q-closure will force К in J. Define
A, = A^rifR;, Qi]) relative to V,
N2 = ^Rl,Ql]),and
N = NtN2.
2. Metanilpotent Fitting classes with additional closure properties 791
Evidently Nt, N2 (and hence N) are normal subgroups of Ц invariant under the action
of the operator group Z* x Z* on И Since epimorphisms preserve Л( ) (because for
example, Ф(Х) is the residual for the formation of elementary abelian p-groups when
[R,P]SA(RX whence A,_1([R,e])<A,(R)=l, and consequently A<
Ker(a)< Ker(a) Since by hypothesis <D”([R,C])=1, it similarly follows that
N2 Ker(a), and therefore N < Ker(a).
Adopting the notation
X e g (mod У)
to mean that X notuializes У and XY/Ye g, we now assert that
(2A) [Ri.CJRJZ* x l)eg (mod N).
To prove this we use induction, first checking that hypotheses of the proposition
are fulfilled with [K„ Qf]N/N in place of R, RtA/A instead of Q, and (Z* x 1)N/N
in the role of Z* in its statement. It is obvious that Hypothesis (1) is satisfied.
Hypothesis (2) holds because by definition of N we have >.Ф( [R j, Qf}N/N) <m<n,
and because (R;A/A)((Z* x 1)N/N) s RZ* e gby hypothesis. (Here we are appeal-
ing to the fact that N < [Rt, Qfj and to the property of a verbal product that
R, r> [Rj, Cil = 1.)Moreover, ^([[R,, gj, R,]A/A) < /^([Ri. < m,and
so Hypothesis (3) is satisfied with the given substitutions. Furthermore, by definiton
of Nt(<N) we have 4([Hi> QilN/N) <t - 1, and so by induction Assertion (2.A)
holds. Similarly we have
(2-<5)
[R„ Ci3Ci(l x Z*) e g (mod N).
Evidently VZ is normal in F(Z‘ x Z*), which is the normal product of the subgroups
in (2.<5) and (2.<У). Therefore
FZeS.NjR',, R2)<= g (mod A),
and since A < Ker(a), it follows that К e q(FZ/A) £ g.
□
We need one further preparatory result before we can prove (2.5).
(2 14) Lemma. Let В be an injective KG-module, where К is a finite field whose
т „de, Ц, a «1 Tte. В i „d °
decomposition
B = E®E*
a, „ dbee, •*«**> Л -d Г ""
a induced by 8 on E has order q' and satisfies aT - y.
792
XI. Fitting classes—their behaviour as classes of groups
Proof. By B. 2.13 there exists a module-automorphism 6 of В such that 6T = y. Since
o(y) — q'. by replacing 6 by a suitable power, we can suppose that o(5) = qs for some
s > t. Let 5* = <5’‘. and set £ = CB(<5*) and £* = [B, <5*]. Clearly £ and £* are
Л-invariant submodules of B. and by A, 12.5 we have В = E @ E*; moreover T < £
because = y*' = iT by hypothesis. Since (<5£)’‘ = acts trivially on £, evidently the
automorphism a = <5£ has the properties stated in the lemma. □
The proof of Theorem 2.5. Let £, = (X : |X : Op(X)| = <7, Лф(Ор(Х)) < i). Since
obviously IJ/Ei £,• = Sp(Z,), it will be sufficient to prove that the given Q-closed
Fitting class g contains V, for i = 1,2,.... Suppose this is not so, and let
и = Min{i: £; £ g}.
Since by hypothesis g contains E(q/p) and hence Zp,, repeated application of the
quasi-Rc, lemma (IX, 1.13) yields V, s g; thus n > 2.
Let G be a group of minimal order in C„\g, and let Г be a minimal normal
subgroup of G. Since g contains 5JI!p 4, by IX, 1.9, the subgroup £ is a p-group;
therefore G/Te and consequently G/Те g by the choice of G. Denoting OP(G) by
P and the elementary abelian group ФЯ1(Р) by A, we have 1 # N о G by the choice
of n and may therefore suppose that T < N r> Z(P) by a well-known property of
p-groups. Form the wreath product
w; = n^g/n = BJG/A),
where Bt denotes the base group of . By A, 18.9 there is a monomorphism
0: G - IF,
such that fVj = B;0(G) and B, n 0(G) = 0(A). For notational simplicity we identify
G with 0(G) to obtain
G < Wt = BrG and Bt r> G = N.
Since Bt is abelian and T< An Z(P), observe that T < Z(OP(1V1)) r> Bt. Also note
that ФЛ-1(В1Е) = В, and hence that лф(В1Р) = n.
Let Z e Syl,(G); then Z = <z>, where z’ = 1. Since B,(ZN/N) e C, S g and G/N e
d(Q/T) <= g, we can apply (2.13) with R = B,, Q = Op(G/N) and Z* = ZN/N to
conclude that Wt e g. (Note that here Ze([R, £>]) = 1 < n.) Let <7-+<7, denote an
isomorphism from G to a copy G, of G, and form the direct product
H = Wr x G,.
Let A = {(t, Г,): t e T} < H and C = (z, zj e H. Clearly A is a normal subgroup of
the subgroup Ор(Я)<0 = (B^P x £г)<0 of H. We assert that
(2.e)
0p(H)<O/A e g.
2. Metanilpotent Fitting classes with additional closure properties
To prove it, we apply (2.13) with the following substitutions:
В = В,РД/Д,
Q {(*> *i): x e P}/A, and
z* = «>Д/Д.
Condttion (1) of the hypotheses of (2.13) dearly holds. Moreover, we have RZ*
‘ p p р\С . 2V1.!R’CZ*sPZ/r=C/7’6 8. and so (2) also holds. Finally [R,Q]
r [ Ht ’ f]ini^<l - S° tha‘ 4([R’ - 1 < "• a"d fond,-
tion (3) is fulfilled. Thus 5 contains the group RQZ* = (B,P x P, )<(>/Д, as claimed
in (2.c).
Next form the wreath product
w2 = 7'Ч8(И'1/Т) = В2(ЫХП\
where B2 is the base group of W2. As before, regard Wt as a subgroup of W2 such that
wi = B2Wt and B2 n Wi = T, and once again apply (2.13), this time with R = B2,
Q = OplfP,), and Z* = Z, to deduce that IV, e g.
Let у: T -> T be the map defined by
ty = z~'tz (teT).
Because T < Z(Op(Wj)), the map у is an automorphism of Tas an Fp(H/l/T)-module,
and since the Kj/T-module B2, as the direct sum of Dimr (T) copies of the regular
module Fp(Wj/7’), is injective, by (2.14) there is a module-automorphism 6 of B2 such
that
B2 = E® E*.
where E(> T) and E* are ^-invariant submodules, and such that a = <5r is an auto-
morphism of E of order q whose restriction to T is y. Clearly
EfV, s W2/E* e Qg = g,
and from this isomorphism we also obtain a complement W( = Wt /Т] to £ in ЕИ-j.
Since the action of W on E commutes with the action of a, we can form the semidirect
product
K = E(W'x <a)) = EWi<o)
and deduce that the element c = z-a centralizes T. (Here we have as umed withou
loss of generality that a conjugate of W has been chosen to contam ZJSince Msthe
product of the normal subgroups EW, and E<a>, and since the latter group belongs
to С, £ g, it follows that Keg.
794
XI. Fitting classes—their behaviour as classes of groups
Finally we form the direct product
L = К x G) = Eff1<a> x G,.
Observe that H = Wt x G, may be viewed in the obvious way as a subgroup of L,
that then L = B2H<a> and En H = T; furthermore Д, the diagonal of T x 7], is
normal in the normal subgroup
Г = Op(£)«, (c, 1)>
of L since c centralizes T We aim to show that
(2.0
Г/Д e g.
Since Г is the product of the normal subgroups Op(L)<(c, 1)> and 0р(Е)<£>, it will be
enough to show that
(2-0
Op(L)<(c, 1)> e g, and
(2.0)
0,(£)<C> e g (mod Д).
Now Op(L)((c, 1)> = 0p(K)<c> x Pj, and since 0p(K)(c) < К e g and P, eSp £ g,
it follows that Op(E)<(c. 1)> e Nos„g = g, giving (2,r;).
Next observe that 0p(L)(£> = E0p(H)<£>, and so a further application of (2.13)
with the substitutions R = ЕД/Д, Q = 0р(Н)/Д, and Z* = <£>Д/Д yields (2.0), once
it is noted that (2.e) can be used to verify Hypothesis (2) of that Proposition. Hence
(2.Q holds.
Now (a, Zj) = f(c, 1) and [И7,, (a, z,)] = [PF,, a] < T < B2. Therefore
(E x PJfla, z,)> sn (EO^JTJ x P,)<(a, z,)> = 0p(E)<(a, z>)> sn Г,
and consequently (E x Pj)<(a, Zj)> e g (mod Д). By Maschke’s theorem there
is an <(a, Zj)>-invariant complement V to T in E, and so g contains
(E x PJ <(a, z,)>/(U x 1)Д s (T x PJ <(a, zJ>/A = &P, <(a, zJ>/A = P, <Z1> =
G, = G. This contradicts our supposition that C„ £ g for some n and hence completes
the proof of (2.5). □
Theorem 2.1 is now proved, and thus we know that the only Q-closed metanilpotent
Fitting classes are primitive saturated formations. Our next main goal will be to
show that these are also the only e*-closed Fitting subclasses of 912. Before em-
barking on the proof we need the following preliminary result. As it was already
cited in the proof of (2.2), we should draw attention to the fact that nothing used in
its proof depends on (2.2) or its consequences; thus no circular arguments are
involved.
2. Metanilpotent Fitting classes with addilional closure рВДй
X'ZAX'.A' XZAS Str *“ ™' *»•»
* L.,
p о. мд л л/а0, and tor a e A write a = aA„ e A- recard Я «
: ,ог а* “"“е Л4«-р- -—* -
(X, >’)'’ = (хь, у”),
where x, у e A and b 6 B. Let P = <0> Zp. It is straightforward to confirm that the
map
(х,у)-(х,у)’ = (х,ху)
defines an action of Р on N which commutes with the action of B, and so we can
form the semidirect product
G = [N](B x P).
If P* denotes a Sylow p-subgroup of N, then A = [/V, P] = [P*, P] < (P*P)' <
Ф(1УР) < Ф(С). Since G/A = H x P e g, it follows that G e E®g = g, and hence that
NB e s„g = g. But NB/A = AB e g, and so by the quasi-Ro lemma (IX, 1.13) we have
NB/A e g. But NB/A [Л]В s H/Ao, and therefore H/Ao eg. □
We are now in a position to prove the second main theorem of this section.
(2.16) Theorem (Bryce and Cossey [3]). Let g be an Enclosed Fitting class of meta-
nilpotent groups. Then g is s-closed.
Proof. Suppose that the theorem is false. Then by Proposition 2.2 the class g contains
a group G of the form G = F(G)C which has a subgroup of the form H = (F(G) r\H)C
not in g; here F(G) is a p-group and C is a non-trivial cyclic «/-group for distinct
primes p and q. .
Let M = F(G)/®(G), viewed as an FpC-module; by A, 10.6 (c) it is faithful tor C.
Let Mo be a maximal submodule of M containing QdfiJC)). Since C is cyclic, the
irreducible module V = M/Mo is also faithful for C, and by (2.15) we have [G]C s
G/Mo eg Let D < C and by Maschke’s theorem set VD = W® W with IF irre-
ducible. Then [ИЗО S [U]D/1V‘, and since [G]D sn[G]C e g, it follows from
(2.15) once again that [IF]Deg. If V is any irreducible F.C-moduJe ±en
Ф(ГИС) = Сс(ПпФ(С) and so [ПСМПО^да f°r som® ° 5 C.thus
me e E.s - 8. C0«4«eMl, « Г i. W —*
796
XI. Fitting classes—their behaviour as classes of groups
consequently, since Ф(К) < Ф(Н), we have H e Еф([К/Ф(К)]С) £ еф8 = g. This con-
tradiction shows that our initial supposition is false and that the theorem is true.
□
We now turn our attention to r0-closed metanilpotent Fitting classes.
(2.17) Example. We recall from IX, 2.9 (a) that the class 3P of finite soluble groups
whose p-socle lies in the centre is an R0-closed Fitting class. Therefore the class
is an r0-closed metanilpotent Fitting class. Furthermore we have
* 8 5 ®₽®«;
for clearly E(q/p) e SpS,\8, and if Q e Syl,(E(q/p)), it is not hard to check that the
wreath product Zp Qj0 E(q/p) is a non-nilpotent group in g. Since there is clearly no
primitive saturated formation lying strictly between 9llp and SPS, (such a forma-
tion would have to have a canonical local definition with F(q) = (l)and F(p) a Fitting
class strictly between (1) and SQ!), and since closure of a metanilpotent Fitting class
g under any one of q, еф or s forces g to be a primitive saturated formation, it follows
that SPS, <~i 3P >s not Q-, еф-ог s-closed.
Our final objective in this section is to show that any R0-closed Fitting class
properly contained in SpS, is already contained in 3P, in other words, that SpS, <~i 3P
is the unique maximal <R0, s„, N0>-closed subclass of Sp$,. This fact was first proved
by Bryce and Cossey [2] under the hypothesis that p|(q — 1), but it is Bryce’s
subsequent proof, dispensing with these prime restrictions, that we give below.
(2.18) Lemma (Bryce [2]). Let g be an R„-closed Fitting class containing E(q/p), and
let H be a group in <5p(Zp). Further, let V be an ^pH-module with a submodule V such
that V/V is semisimple. Finally, let d be the order of p modulo q. Then
(a) [17] H e g implies that [F© • © F]He g, and
(b) [F] H e g implies that [17 ® © 17] H e g.
Proof. Let E(q/p) = TVZ, where |N| = pd and |Z| = q. If H e Sp, then {UH, VH} £
Sp £ g. Therefore suppose that H = KQ, where К = Op(H) and |(?| = q. Regarding
N as an Fp Z-module, we can form the outer tensor product F ® N over Fp and regard
it as an (H x Z)-module in the usual way. Since V/V is semisimple, we have К <
Ker(H on V/U}, and so К (identified with К x 1) acts trivially on the quotient module
M = (F® N)/(U ® N). Hence H x Z induces an elementary abelian q-group of
automorphisms on M, which is therefore semisimple by Maschke’s theorem. We can
therefore write M = @'=г X,, where each summand Xt = X^U ® TV) is simple. Set
Ki = Ker(H x Z on X;). Since К <, Kt, it follows from B, 9.8 (a) that (H x Zf/Kj is
cyclic and therefore of order 1 or q. But because [TV, Z] = N and [F, Z] = 1, we have
[M, Z] = M and hence [X;, Z] = X,-. Consequently |H x Z: K,-| = q, whence
H x Z = K,Zioti= l,...,r.
z r,M„J dle, ль 1Л,0ои1| с|ито wrtfci
K"d х х-ш-1.....s b,
YtKi+1 <g Yt+lKi+i eN0{i;.Ki+1, yc+1}.
Since g is a Fitting class, we therefore conclude that
(2.i)
*1*4+1 eg if and only if Yj+1K1+1 e g.
By Maschke’s theorem for all j we have Y,Z e R0(£(<?/p)), which is a subclass of g by
hypothesis. Therefore, using (2.i) and the fact that H, K} < H x Z = KtZ for all j, we
obtain the following implications: 1
YiHe'S=>Yi(H x Z)eNog = g=> yc(Kj+lZ)e g
^f<Ki+1es,g = g,?„K+1Ki+i6 8
=> Yi+i(Ki+1Z) e Nog = g => Yi+lH e s.g = g.
Similar reasoning shows that if Yi+1H e g, then У;Н e g. By induction we then
infer that [17 ® N]H = Y0H e g if and only if [V ® N]H = YrH e g, and since
(U ® N) H = (U ® @ U)H e Ro(UH) etc., the conclusions of the lemma are clear.
□
(2.19) Corollary. Let ft, H be as in (2.18), and let W be an К pH-module having a chain
of submodules
0 = 1VO < Wt < < Wt = W
with WJWi-i semisimple (i = 1,..., t).
(a) If H e g, then [_W ® - ® IT] H e g.
(b) If [И7] H e g, then He%.
Proof. Let nM denote the direct sum of n copies of a module M, and let i e {1,. >«}•
Apply Part (a) of Lemma 2.18 with V = d^WM and V = d-‘tVt to denve the
following induction step:
[di-1 e g=> e 5-
If н e g, then [d^W^H e g for i = 1. Therefore by induction Id'W.JH e g, as
asserted in Part (a). Part (b) follows from (2.18) (b) in the same way.
. tv i S the splitting theorem of Barnes and
The following lemma brings to mind IV, 1.5, tne sp
Kegel for formations.
798
XI. Fitting classes—their behaviour as classes of groups
(2.20) Lemma. Let G be a group contained in an Ro-closed Fitting class J. Let N be
an abelian normal subgroup of G centralized by G "n. Then g contains the semidirect
product [TV] (G/Cg(1V)).
Remark. If N is a minimal normal subgroup of G, then F(G) < CG(N) by A, 13.8 (b),
and so if G is metanilpotent, G 91 centralizes N.
Proof. Let N* be an isomorphic copy of N as a G-module, and let n -+ n* denote a
G-isomorphism from N to N*. Form the semidirect product
X = [IV*] G.
Then the diagonal D = {(и*, n)-.ne N} is clearly a minimal normal subgroup of S
(isomorphic with N and N*), and DrG = Dr>N* =1. Thus S/N* £ S/D = G e g,
and S e RC(G) £ g. Set C = CG(N). Since S/N*C = G/C e 92 and N* n C = 1, it
follows from the quasi-R0 lemma (IX, 1.13) that g contains S/C, which is evidently
isomorphic with [/V] (G/C). □
(2.21) Theorem (Bryce [1]). The class cf 'ZfZq-groups with p-socle central is the
unique maximal Re-closed Fitting subclass of ®p®,-
Proof. Let g = <R„, s„, N„>g £ ®p®,. and assume that g £ 3P- We must show that
g = ®p®,. Under these assumptions, g contains a group X which has a minimal
normal subgroup N satisfying [/V, X] # 1, and since X/Cx(N) is a q-group and
8 = s,g, we may suppose without loss of generality that |X/Cx(/V)| = q. But then
[IV] (X/Cx(/V)) s E(q/p), and so by Lemma 2.20 and the subsequent Remark we can
conclude that E(q/p) e g.
By (2.4) it will be enough to prove that
(2.K)
Sp(Zq) £ g.
Suppose that (2.k) is false, and let G be a group of minimal order in ®p(Z,)\g. Let
M be a minimal normal subgroup of G, and set H = G/M. Since H e ®p(Zq), we have
1 # H e g by the choice of G. Let R be a regular FpH-module, and let t (> 1) be the
Loewy length of R. If d is the dimension of a faithful simple module for Z, over Fp,
and if A denotes the direct sum of d‘ copies of R, it follows from (2.19) (a) that g
contains the semidirect product W = [A]H. Since [M, Op(G)] = 1 and |G: Op(G)| =
q (we cannot have G = Op(G) because G £ g), it follows that the Fp-dimension of M is
1 or d. Since the base group of M QjTegH is isomorphic, as FpH-module, to the direct
sum of 1 or d copies of R, it follows that A has a submodule Bo which is isomorphic
with B. Since B0H = M QjrcgH, it follows from A, 18.9 that there exists a monomor-
phism p: G -> W such that W — Zp(G) and A n p(G) = p(M). Consider the semidirect
product
S = [A]G,
з. Further theory of metanilpotent Fitting classes 799
where A is viewed as a G-modnle Ev о-
S/M e g. It is straightforward to check that thTmarU $ S AH’WE have
epimorphism from S onto W and so if M* • P °’9 6 '‘MQ is an
But evidently MnM*-l’Zt denotes its kernel, we have S/M* s ive g.
2 19(bLm£ tha?G e <7and * 6 R°5 = % Bu‘ ^en Corolla?
2.19(b) implies that G g g, and we have the desired contradiction, showing that (2.K)
□
The metanilpotent Fitting class ®({3}) n S3S2, described in Example IX 2 14 (b) is
shown there not to be closed under any one of the operations Q r„ r or s The d
alS° fallS ‘° haVe ‘heSe d0SUre properties- For we ^owed in Example X,
Э.j (a} mat
^(2.3} Fit(Sym(3)) S362.
As we pointed out inJ2J 7), there are no Q-, eo-, or s-closed classes lying strictly
between 9i|p.41 and topto,, and since Sym(3)£33, we conclude from (2.20) that
Fit(Sym(3)) is also not r0-closed.
Open Question. If G g S\91, can Fit(G) ever be s-closed? If the answer here is negative,
and if G g врЭДЗ”. then Fit(G) is not Q-, Ro-, e»-, or s-closed.
3. Further theory of metanilpotent Fitting classes
In this section we consider another approach to the problem of classifying certain
types of metanilpotent Fitting classes. This is due to Johnsen and H. Laue [1] and
can be described in terms of the following general philosophy: Choose a ‘small’ fixed
class of groups, call it 3E, and given G in X, determine all groups H in X such that H
belongs to the Fitting class generated by G. In this way one obtains the subclasses
Xo of X such that Xo = X n g for some Fitting class g and ‘ classifies’ Fitting classes
according to their intersection with X. In the case of Johnsen and Laue’s study, the
class X in question consists of all extensions [PJQ of a homocyclic abelian p-group
P of exponent pr by a q-group Q, where p and q are distinct primes.
Notation. We recall that, if X is a class of groups, Fit(X) denotes the Fitting class
<s„, n0> X generated by X, and that £(q7pr) denotes the unique extension of a faithful
indecomposable Zg,-module over the ring /p>- by Zg, (see B, 12.4).
(3 1) Lemma. Let p and q be distinct primes, and let r,neN. Let V be a homocyclic
abelian p-group of exponent p', and let Qbea cyclic group of operators on V of order
q”. Then [nCeFit(£(q/p9, £(<?7/), £(<?"//))• or
iei|(P - 1). then [F](> g Fit(£(q7₽'H
Proof Since q" > 1 the class Fit(£(<?7p'l) contains By A, 11.6 the group
fne мм » • diren pred“ *
800 XI. Fitting classes—their behaviour as classes of groups
form [!<](). where V] is an indecomposable submodule of V. By B, 12.6 we have
S E(q‘lp') for some j e {0, 1,..., n), and it follows from IX, 1.13 that
each [V,]Q, and hence [I'JQ, belongs to Fit(E(q‘/p'): i = l,...,n). If n = 1 or
<?"|(p — 1), it is clear that Fit(E(q‘/p'): i = 1,..., n) = Fit(E(q"/p')), because in the
second case E(q"/pr) has a normal subgroup isomorphic with E(q4p’} for i = 1,..., n.
□
(3.2) Corollary. Let Q be a q-group operating on a homocyclic abelian p-group V of
exponent pr, where p and q are distinct primes. Then
(a) [1'32 e (Fit(£(<?/pr))*, and
(b) if [K # 1, then (Fit([F]0)* = Fit(£(q/p'))*.
Proof (a) Let 5 = Fit(£(<?/pr)). By (3.1) we have FQjreeZ4 e 3, and therefore by X,
2.7 the group W = FQjree Q is in g*. But W has a quotient group isomorphic with
[!']() by A, 18.9 and A, 11.4; hence from X, 1.24 we conclude that [I'JQ e g*.
(b) Because Fit([l'](2) = Fit([I ](Q!Cq(I'))) by IX, 1.13, we can suppose that V
is faithful for Q. Since Q / 1 by hypothesis, Q contains a subgroup Qo of order q. Let
F= Ц x ••• x Vs
be a decomposition of V into indecomposable Qo-submodules (see A, 11.6). Since V
is faithful, there is а Ц faithful for Qo, and by B, 12.6 we have [K]Q0 S E(q!jf). Since
[l']Qo e s„([l'](2), it follows from X, 1.24 that E(q!pr) e (Fit([F]6))* and therefore
by X, 1.8(b) that g* S (Fit)([V]Q))*. Since [VJQ e g* by Part (a), the reverse inclu-
sion is clear, and equality holds. □
The next result shows, in particular, that Fit(Sym(3)) contains non-nilpotent groups
of considerable complexity. For the meaning of the notation g/91 and 591 the reader
is referred to Definitions X, 3.4 on page 706.
(3.3) Theorem (Hawkes [12]). Let p, q e IP, p / q, and r e M. Let ft be a Fitting class
which contains E(q/p'). Then (a) S, S 8/91, and
(b) ep e ад®91); if r = 1, then Gp e s,®91).
Proof (a) Set Wt = E(q/pr), and for i > 2 define a group Wt inductively as follows:
^=^_1rbre,Z,.
If Uj e SylJWj) and P, = Ор(И<), then it is straightforward to verify the following
properties:
(i) O, s (...((ZerbZq)rljZ,)rb...)rbZ4, the iterated regular wreath product in
which Zq appears i times;
(ii) U, is generated by elements of order q;
(iii) Pf is a homocyclic abelian p-group of exponent pr, W{ = PiV, and CVl(Pj) = 1.
Let 1 / хе Ц with x’ = 1. Then by (3.1) we have Pf<x> e Fit(£(<?/p')) S 8- Since
Wi is generated by subnormal subgroups of the form Pj<x> by Properties (ii) and (iii),
it follows that e n08 = g for i = 1,2,....
з. Funher theory of metanilpotent Fitting classes
propeny we h.« P_ . f(,_a „„„ Q-^p “ 8 Ьу
c ‘ X ' fc'( f' * Г"’ 101 = • X’- адвд L« о -
1 , ".’ . •• ’ K}. and let Я denote the Hartley group Н(У) whose
“weft'rm’the s" V ’ ’’ ‘° °f G °" » “Tn
’r . Л T Semidirect Product s = t«]G, claiming that Se g.
e m e { identify Gm with the mth component of the direct product G
and set r ’
Sm = HGm.
Since S is the product of its normal subgroups S,,..., S„, it will be enough to show
that Sm e 5- By B, 12.16 we have H = HmKm, where Km is the centralizer in H of Gm
and H„ is a Gm-invariant normal complement to K„ in H. Thus Sm is the product of
normal subgroups f/mGmand H. Now by B, 12.16 the subgroup Hm is a direct product
of copies of Vm; in particular, Hm is homocyclic of exponent pr, and so by (3.1) we have
e Fit(E(qlpr)) e g. Since H e Gp £ g by IX, 1.7, we conclude that Sm e Nog =
g, and hence S e g, as claimed. Since [F„ G] = t;, it follows that [Я„, G] = Я„, and
because H = Я,H2 ...H„ by B, 12.16, therefore [Я, G] = H. As S/H e91, we deduce
that Я = S91 e g91.
To complete the proof of Part(b) it will suffice to show that an arbitrary p-group
P is isomorphic with a section of H(7j for some n. By writing the entries of Я(тР')
corresponding to Ц modulo Rad(f<), we obtain a homomorphism from Я(7) to
Я(1С*), where T~* = {H/Rad(I<)}”=1. It is easy to see that Я(7*) contains a copy of
U (n, p), the group of upper unitriangular matrices over Fp. The regular representation
gives an embedding of P into GL(|P|,p), whose image, by Sylow’s theorem, is
conjugate to a subgroup of a Sylow p-subgroup of GL(|P|, p). Since G(|P|, p)has the
order of a Sylow p-subgroup of GL(|P|, p), it therefore follows that P is isomorphic
with a section of Я(У ) and, indeed, with a subgroup of H(i ) when r = 1. □
Although the test problem of describing the groups in the Fitting class generated by
Sym(3) has not yet been solved, the preceding theorem indicates that it is a large
subclass of G3G2.
(3 4) Corollary. Let g = Fit(Sym(3)). Then g/91 = <S2 and g91, a subclass of <S3.
contains groups of arbitrary large nilpotency class, derived length and exponent.
If x is an automorphism of a homocyclic abelian p-group V of exponent P'-'"e ™
choose a -basis of V and represent x as a matrix with entries in Z,, The determi-
nant Det(x on F) is a unit of the ring and is independent of the choice of basis.
(3 5) Lemma. Let p qe P, p * 4, and reN.Let^ = Flt(E(q/p')). Further, let V be a
q-power order. „
(a) If (q, p - 1) = 1> then СИ <*> e S* and
802
XI. Fitting classes—their behaviour as classes of groups
(b) Assume that Det(x on F) = 1. Then each of the following conditions ensures that
fL]<x> e 5,:
(i) o(x)|(p - 1);
(ii) p = 3 (mod 4), q = 2, and o(x)|(p + 1).
Proof (a) First suppose that V is indecomposable as an <x>-module, and let W =
F/Rad(F). By A, 11.7 the group <x> acts irreducibly on W, and since it acts faithfully
on F, it also acts faithfully on W by A, 12.7. By B, 9.8 we can identify W with
the additive group of Fpk, where к is the smallest natural number such that o(x)|(p‘ — 1),
and can take for the action of x on IV field multiplication by a suitable element A of
the multiplicative group ГД. We shall use the fact that the normalizer of Fpl in
Aut(li ) contains the Galois group Г of F^. Since <2> is the unique subgroup of
order o(x) in the cyclic group F^., it is normalized by Г. Moreover, the hypothesis
that qi(p — 1) implies that the g-group <2> meets the fixed field Fp of Г in {1}, and
therefore
(З.а) [<Л>, Г] = <2>.
Let C = CAul(n(W). By B, 12.3(a) the group C is a p-group, and there exists an
isomorphism в: Aut(F)/C -»Aut(fF) sending xC to f Since (o(x), |C|) = 1, by the
Frattini argument we have МАи)(И(<х>О = Клы(п(<х))С. Consequently NAut(r>(<x>)
contains a subgroup G such that 6(GC/C) = Г, and applying 0'1 to Equation З.а, we
obtain [<x>C, ГС] = <x>C, whence [<x>, Г] = <x>, and therefore
[F<x>, Aut(F<x»] = F<x>.
By (3.2) (a) we have [f ] <x> e F*, and so by X, 1.3 it follows that [f ] <x> e (8*)* =
8* In particular, E(q'lp') e 8* for i = 1,2,..., and thus 8 = 8*- Finally, for the case
where F is not necessarily indecomposable, we apply (3.1) to conclude that [f ] <x>
belongs to Fit(£(q‘/pr) •’ 1 < i < n) = 8*-
(b) First suppose that o(x) = <7'|(p — 1). Then by A, 11.7 and B, 9.8(d) the in-
decomposable I r<x>-modules are cyclic abelian groups, and by B, 12.6 there is a
normal embedding g. [[/] where E = E(q'/p'\ such that 0(F) = Op(£m).
Write £ = PQ with P = Op(E) = Z r and Q e Sy 1,(£); then without loss of generality
we may suppose that 0(x) = (Xj,..., xm) belongs to Q". Thus 1 = Det(x on F)
= Det(0(x) on 0(F)) = fj7=1 Det(x, on P) = Det(f[” t x. on P). Since P is cyclic,
the map g -► Det (г; on P) from Q to the group of units of 1 r is a monomorphism,
and consequently J I™ ! x, = L Now by (3.2) we have £e 8* and by IX, 1.7 we know
that P e 8». Hence by X, 1.11 we have F<x> O„(Em)<.(xt,..., xm)> <(£m)s.,and
since £m/Op(£'") e SI, we conclude that F(x> e s„8* = St-
Now suppose that p = 3(mod 4). Let 2s||(p + 1), and suppose that o(x) = 2“ with
2 < a < s.Let W be an arbitrary indecomposable 7L r<x>-module. Then W7Rad(lF)is
irreducible by A, 11.7 and therefore has Fp-dimension 2 by B, 9.8(d). By B, 12.3(b) a
Sylow 2-subgroup T of Aut(W) is isomorphic with a subgroup of GL(2, p), and hence
T' contains an element у of order 2“ by A, 21.5(b). By (3.2) (a) we have [IF] T e 8*-
Hence [W JT' = ([WJT)' e 8» by X, 1.3, and consequently, appealing to B, 12.4, we
3. Funher theory of metanilpotent Fitting classes
803
conclude that [W] <x> s E(2“/p') S [iq <y> e s (W1r) c , n. u
yeV, which is isomorphic with a subgroup of GW м 2 7
De^°n t//Rad<t/»»= 1 hence from B, 12.7 thaton « foll°ws tha‘
anXriteW C° thE ееПеГа1 CaSe’WhEre <*>-™dule * таУ be decomposable.
with each V, indecomposable. If Xj denotes the automorphism induced by x on Fj, then
[F] <x> [F] <X1x2 ... x,> 3 ft [lj]<x;>.
If o(xf) = 1, then K]<x,>ESpEg, by X, 1.20, and if o(x,)>4, then again
[KJ (xt) e g* by the argument of the previous paragraph. On the other hand,
if o(xf) = 2, then Ц is cyclic and Detfx, on FJ = - 1. Since 1 = Det(x on V) =’
ni=t I—»eton lj), it follows from the earlier observation that there must be an even
number of Fj for which ofxj = 2. If we pair them off and denote by the sum of
such a pair, then Detfx on H<) = 1, and by the case o(x)|(p - 1) already dealt with,
we have [WJ] <y7> e g*, where y; is the automorphism induced by x on FFj. Thus
[F] <x> is isomorphic with a normal subgroup of a direct product of groups in g4
and therefore belongs to g*. □
(3.6) Remarks. Let p, q e IP, p A q,let r e M, and set g, = Fit(£(q/pr)). Then
(a) g, = (gr)* if and only if (q, p - 1) = 1, and
(b) if ql(p - 1) and s < r, then gr £ gs.
Proof First a preliminary observation. Suppose that g|(p — 1), and let </||(p - 1).
Set £ = £(q'/pr), and let U = Op(E) S ZPr and Q e Sy 1,(£), so that E = UQ. Reverting
to the notation and terminology of Chapter X, Section 5, we claim that UQ is an
(G, ^completion of U and that Qo = 1. By A, 21.1(b) Autf U) is abelian of order
(p - l)pr-1. Since CQ(U) = 1, we can regard Q as a subgroup of Aut(L') and conclude
that {Q- Aut(G)} = 1. Moreover, since ПДС) acts fixed-point-freely on L//Rad(L/),
we have r(0 = 1. Therefore Qo = 1. Obviously Q = Qn(UQ*)s where Q<
Q* e Syl (Aut(l/)), and so it follows from X, 5.16 that (Q. d1'-") is an G-Fitting pair.
Thus, if R denotes its kernel, by X, 5.17 we have £S1 = V. It follows that £\ = U,
and, in particular, that £ta/pr)s, = Op(£(q/pr))- , , , . , n
We can now justify Remark (a). If (q. p - 1) = h then gr = (&)t b5' (3-5) aOn
the other hand, if q|(p - 1). then the preceding observation shows that E(q/p)
g \6. c g \(g,L. and therefore gr A (g,)»- - =
Г ;*eSо
similarly E(q/ps) e R\R, and it follows easily that E(q/p ) e g,\gs-
804
XI. Fitting classes—their behaviour as classes of groups
It is not known if (3.6) (b) also holds when qj(p — 1). As a test case one could
investigate the relationship between the Fitting classes generated by Alt(4) and
£(3/22).
(3.7) Lemma. Let p and q be distinct primes, let r e M, and set g = Fit(£(q/pr)).
Further, let V be a homocyclic abelian group of exponent p', and let x be an automor-
phism of V of q-power order. If Det(x on V} = 1, then [F] <x> e gt.
Proof. The case where (q, p — 1) = 1 has already been dealt with in (3.5) (a).
Therefore suppose that q|(p — 1), and choose an integer к such that qk > n =
DimF (F/Rad(F)). Let W be a direct sum of (qk — n) copies of Z.r, view IT as a trivial
<x>-module, and form the semidirect product [F@ IF]<x>. Clearly we can now
regard x as an automorphism of V © W with IL-determinant equal to 1. By Sylow’s
theorem and B, 12.3(b) we can find a q-subgroup Q of Aut(F © kF) such that x e Q
and Q is isomorphic with a Sylow q-subgroup of SL(qk, p). Let q'||(p — 1), and first
suppose that q is odd. By A, 21.3(b) the group Q can be generated by elements
Л], ..., xs such that o(x,) = q or q‘ for i = 1, s, and by (3.5) (b) (i) we have
[F © IP] <x,-> e g*. Next suppose that q = 2. Then by A, 21.3(b) and A, 21.6(b) the
group Q can be generated by elements xt of order 2, 22 or 2'. If p 3 (mod 4), then
t > 2, and so [F| <x;> e g* by (3.5) (b) (i). On the other hand, if p = 3 (mod 4), then
t = 1 and 22|(p + 1), in which case by (3.5) (b) (ii) we again have [F © IV] <x,> e g*.
Thus, in any case, F<x> sn[F © IV] <x> sn [F © W]Q e Nog* = g*. □
(3.8) Lemma. Let qs|(p — 1) for some s > 0 (p, q e IP), and let Qs denote the unique
subgroup of order qs of the group cf units of Let G e <=>„<=>,, and assume that
F = Op(G) is a homocyclic abelian group of exponent pr > 1. If{Det(g on F): g e G} =
Qs, then Fit(G) = Fit(E(qs/pr)}.
Proof. By the quasi-R0 lemma we can suppose without loss of generality that the
Sylow q-subgroups of G act faithfully on F and that G < VW with IF e Syl,(Aut(F)).
Let S’ = {x e W: Det(x on F) = 1}. By B,12.3(b), A,21.3(b) and A,21.6(b) we have
IF = <S, w> with o(w) = q', where q'||(p — 1).
Set g = Fit(G), and let x denote the q'-sth power of w, so that o(x) = q5 and
FSG = FS(x> by hypothesis. By (3.2) (b) we have Fit(£(q/p’’))* = g* = Fit(F<x»*,
and by (3.7) we have FS e Fit(£(q/pr))*. Therefore VS e g n Fit(F<x>), and it follows
that
g = Fit((FS)G) = Fit((FS)<x>) = Fit(F<x>).
Since qs|(p — 1), there is a natural number m and a normal embedding v: F(x> -»• D =
(£(qs/pr))m such that t>(F) = Op(D). Thus g <= Fit(£(qs/p')). Let t>(x) = (x,,..., xj
and У = Op(E(qs/[f)). Then t>(F<x>) = ^"«x,,..., x„)> < Ds, and therefore
by X, 1.11 we have x, ... xm e E(<f/p')^. Since У is cyclic and ]~[” i Det(Xj on У) =
Det(x on F) generates Qs, it follows that x j... xm has order q5 and hence that E(qs/p')
[У] <x, ... x„> e g. Hence g = Fit(£(qs/pr)). О
3. Further theory of metanilpotent Fitting classes
805
A°'a,iOh n ^₽<S’ Md 'f °'(G) is a homocydic abelian
denote by the homomorphism (written exponentially) from
ot U-r, denned by
group of exponent pr,
G to the group of units
= Det(g on Op(G)),
where g is the automorphism induced by g on Op(G).
(3.9)Th eorem(JohnsonandLaue[l]). Let G,He Sp<S,\9l, where p and q are distinct
primes. Assume that OP(G) and OP(H) are homocyclic abelian groups of exponent pr.
Then G and H generate the same Fitting class if and only ifGDc = HD“.
Proof. Let q ||(p — 1) and for 0 < s < t let Qs denote the unique subgroup of order
cf in the group of units of Zpr. Further, let gs = Fit(E(^/p')).
Suppose that GDc = Qs. If s > 0, by (3.8) we have Fit(G) = gs. If s = 0, then
Fit(G) £ (J, )* by (3.7). But by (3.2) (b) the classes Fit(G) and belong to the same
Lockett section, and therefore Fit(G) = (KJ*. Thus GD'- = HD" is a sufficient condi-
tion for Fit(G) = Fit(H).
To prove that it is necessary, suppose that GDc HD", so without loss of generality
let
gDg = Q,< Q. = nD"-
In the proof of (3.6) we saw that (Qt, dv-9) is an S-Fitting pair (with U = Op(E(q‘/pr)).
Thus by IX, 2.11 the class
Я, = (GeS: Gd?" = Qf
is an S-normal Fitting class, and clearly E(qwlfT) 6 Я„,\Я5. Therefore from (3.8) we
conclude that E(qw!f) e Fit(H)\Fit(G), since Fit(G) = gs £ Я, if s > 1 and Fit(G) =
(Я,)„ £ Яо if s = 0. Consequently Fit(H) Fit(G) and the condition is necessary.
(3.10) Corollary. Let G e ^Р<5Я, p and q distinct primes, and assume that 0„(G)is
homocyclic. Then the Fitting class radicals of G are precisely the subgroups 1, O„(G),
O,(G), Op(G) x O,(G) and the subgroups containing Ker(DG).
Proof. Let .T be a Fitting class. If Gj is nilpotent, then C,_= 1. «
О (G) x O„(G). If Gj is not nilpotent, then Gj - Gs, where S Fit( j). У ( • )
groups Gj and Gj Ker(Dc) generate the same Fitting class and so KHC) Gg It
is also clear from (3.8) that any subgroup of G containing Ker(Dc) is actually Fitting
class radical.
Exercises
1. Show that Z3 %aI Sym(3) e Fit(Sym(3)).
2. (O.J. Brison—unpublished) Let E be an
extraspecial group of order 33 and
806
XI. Fitting classes—their behaviour as classes of groups
exponent 3, let 5 e Syl2(Aut(E)), and let T denote the semidirect product of
Z3 x Z3 by an inverting automorphism (of order 2). Then
(a) [E]S e Fit(T), and
(b) Fit(T) = <5Ф n Fit(Sym(3)) = Fit(Sym(3))*.
3. (Brison—unpublished) Fit(Sym(3)) contains all extensions of elementary abelian
3-groups by 2-groups. (This result is stronger than Corollary 3.4.)
4. We know from Camina’s example (see X, 5.3(a)) that Fit(Sym(3)) does not contain
Dih(18) and hence cannot be subgroup-closed by (1.5). Show this explicity as
follows:
(i) If W = Z3 rb,esZ3 and S' e SyljfAutfH')), then S s Z2 x Z2;
(ii) Fit(Sym(3)) contains the semidirect product [IF]S;
(iii) [WjS has a subgroup isomorphic with Dih(18).
4. Fitting class boundaries I
We recall that a Schunck boundary ® is a class of groups satisfying
SB1: If Я is a proper epimorphic image of a group in ®, then И $ ®;
SB2: ® consists of primitive groups.
In our treatment of Schunck classes in Chapter III, boundaries played a central
pan because we could exploit the one-to-one correspondence between Schunck
classes and Schunck boundaries given by the maps b and h( = b~') defined in III, 2.9.
Boundaries for Fitting classes have already made a brief appearance in IX, 3.20,
where they were used to describe strong containment between certain Fitting classes.
It is clear how the analogous maps b and h for Fitting classes must be defined:
(4.a)
' b(3£) = (В e ® : В ф JE and s„(B)\(B) e X);
h(«)) = (He®:s„(H)r1«) = 0).
[Here, and throughout this section, ® denotes some fixed <s, n„, q, Enclosed uni-
verse, which from time to time we make specific. For simplicity, proofs are usually
written with the universe ® = G in mind, and the reader is left to check that the proof
holds for a more general ®.] The maps b and h just defined bear the same relation to
the closure operation s„ as the maps b and h of Chapter III bear to the closure
operation q, and to be unambiguous, we ought to qualify them with notation such
as b0, h^. Instead we will rely on the context to distinguish them; in particular,
unadorned they will have their (4.a) meanings for the rest of the chapter.
The following observations are immediate from the above definition of the Fitting
class boundary map b.
(4.1 j Lemma. Let X be a class of groups, and let '[) = b(3E).
FCB1: If К is a proper subnormal subgroup of a group in 'Й, then К
FCB2: If I = n„3c, then '[) consists of single-headed groups.
4. Fitting class boundaries 1
807
(4.2) Definitions. Because we wish to reserve the term ‘boundary’ for classes of the
Or?1i7^o-»where ®1S a Flttine class’ we CaM a class satisfying Properties FCB1
a™* , 2 a Flt„tmg ClaSS Preboundary (or simply a preboundary). A class satisfying
FCB1 alone will be called subnormally independent.
Lemma 4.1 states that b(g) is a preboundary when g is a Fitting class.
(4.3) Lemma. If f) is a class of groups, then
(a) h('D) is s„-closed, and
(b) if 'll consists of perfect single-headed groups, then ВД) is N0-closed.
Proof. Assertion (a) follows at once from the definition of the map h. To see that (b)
is true, suppose that Nob('|l) fi(9)) and let G be a group of minimal order in
^oh^)\h(y). By (a) and II, 2.10(b) the group G is the product of maximal normal
subgroups Nj and N2 in МФ)- By definition of b(?)), the group G has a subnormal
subgroup KeT) such that У -f Nj for i = 1, 2; therefore TN, = G. Consequently
G/Nj = Y/(Y r> Nj) is perfect, and it follows that G/(Nt n N2) is the direct product of
non-abelian simple groups Nj/(N\ n N2) and N2/(Nt nW2). Moreover, since У is
single-headed, Y(Nj n N2)/(N\ n N2) is a proper subnormal subgroup of G/flV, n N2).
But {A'/fNj n N2): i = 1,2} are the only maximal normal subgroups of G/(N, n N2)
by A, 4.13 (b), and so either У < Nt or У < N2. This contradiction proves that
N0h((P) = h(?)). °
We shall soon see that a preboundary of perfect groups is, in fact, the boundary of a
Fitting class, and that by suitably restricting the domains of the maps h and b, they
behave like their Schunck class analogues. But (4.3)(b) may fail when contains
imperfect groups—for example, the group Z2 x Sym(3) belongs to Noh(J)\b('J|)
when ?) is the preboundary (Z2)—and this explains why the analogy with Schunck
class boundaries breaks down in the universe of soluble groups.
The following theorem is in two parts. The first shows that there is a one-to-one
correspondence between Fitting classes and their bounties (within ajiven «mi-
verse) The second part shows that under this bijection Fitting classes of the form
g = ge correspond to preboundaries of perfect groups.
(4.4) Theorem. _
(a) (i) If 5 is a Fitting class, then h(b(?f)) - »• _
(ii) If ® is the boundary of a Fitting class, then ( ( ))
(b) The maps b and h, restricted to the following domains-.
808 XI. Fitting classes—their behaviour as classes of groups
{Fitting classes 5 = gS} Д {preboundaries of perfect groups}
are inclusion-reversing, mutually inverse bijections.
[Of course. Part (b) of this theorem has no content unless our universe is sufficiently
large and, in particular, contains S.]
Proof, (a) (i) Let 5 be a Fitting class, and let К sn G e g. Since К e g, we have
К b(g), and therefore s,(G) n b(g) = 0. Consequently g s hb(g).
Suppose that the class hb(S)\S is non-empty. Then it contains a group, H say, of
minimal order. Since the class hb(g) is s„-closed by (4.3)(a), it follows from the
n0-closure of 5 that H is single-headed with Cosoc(H) e g (see footnote). But then
H e b(g) by definition of b. which contradicts the choice of H in Zi(h(g)). Therefore
hfe(S) = S-
(ii) Let g be a Fitting class with ® = b(g). Then h(®) = hb(g) = g by Part (i), and
so b(h(®)) = b(S) = ®.
(b) By Part (a) it will suffice to show that the restrictions of b and h to the stated
domains have the stated codomains. If 5 is a Fitting class, by (4.1) the class b(g) is
a preboundary of groups В satisfying В ^5 and Cosoc(B) e g. Therefore, if g = $G,
this preboundary clearly consists entirely of perfect groups.
Let ® be a preboundary of perfect groups. We first show that Wt(®) = ®. Let
Be®. Since proper subnormal subgroups of В are not in ® by FCB1, it follows that
Cosoc(B) e h(®). Hence В e hh(®) since В is single-headed. Thus ® s bh(®). Now
let G e bli(®). Since h(®) is a Fitting class by (4.3), we can conclude from (4.1) that G
is single-headed and that all its proper subnormal subgroups belong to h(®). Thus
(s, — 1 )(G) n ® = 0. But then, if G $ ®, we have s„(G) n ® = 0 and in this case
G e h(®), contrary to the choice of G. Hence b(h(®)) S ®, and equality holds, as
asserted.
Finally, set 5 = h(®)- A group of minimal order in 3<=>\3 would be an imperfect
group in b(g) = Wi(®) = ®. Therefore g = $G when the preboundary ® consists of
perfect groups. □
We wish to isolate the differences in behaviour between perfect and imperfect groups
in a Fitting class boundary by studying their influences separately. With this aim in
mind we introduce the following notation.
(4.5) Definitions, (a) If I is a class of groups, we define
6(3E) = (Be b(X.): G' < G), and
b(X) = (Be b(X): G' = G).
Thus b(X) = b(X) и b(X) and b(3f) n b(I) = 0.
(b) Let л s IP. For the remainder of this chapter the suffix л on the boundary map
b and its derivatives will have the effect of restricting its range to groups В for which
We recall that the cosocle of a group is the intersection of its maximal normal subgroups.
4. Fitting class boundaries I
B/Cosoc(B) is a л-group. Thus, for example, we have
\(Л) = (В e ®\F; (s„ _ i)(B) E j |B. Cosoc(B)| ,s a л_питЬег)
Note that b(T) = Upepbp(3E)
(c) If Ж is a class of groups, we define
££ = Fit(CosocfG): G e b,(3E)),
where Fit denotes <s„,n0>, the Fitting class closure operation. Classes # and
*„=аге defined analogously, and the suffix n will be omitted when and only when
We note that when F is a Fitting class, then the classes FJ, FJ and F‘ are Fitting
subclasses of x. 6
By (4.3) the class g = h(<B) is a Fitting class when ® is a preboundary of perfect
groups. The next result gives a description of all those Fitting classes F which share
the same perfect boundary, that is for which b(.F) is equal to a fixed ®.
(4.6) Theorem (Pense [ 1 ]). Let i8bea preboundary of perfect groups, and let g = /1(23)
(a Fitting class by (4.4)(b)). Denote by St the Fitting formation of all groups with no
abelian chief factors, and set
ffi = gng»R
Then
(a) b(ffi) = b(g) = ®, and
(b) if F is a Fitting class with b(X) = ®, then ffi S F s g.
Proof, (a) By (4.4) we have ® = bh(S3) = b(g) = b(g). Suppose, if possible, that
b(ffi)\® contains a group, В say. If В ф g, evidently В e h(g) = ®, contrary to
assumption, and therefore В e g. By definition of (ffi), we have Cosoc(B) e ffi c gbR
and В e (g\4)Я = g\«. But then В e g n g”« = © and we have a contradiction.
Hence b(ffi) e ®. On the other hand, if now Л e 23, we have Cosoc(^) e g n g6 s ffi.
Since A $ g, certainly А ф ffi, and so A e £>(©). Therefore b(ffi) = S, as asserted.
(b) Let b(F) = ®. Then clearly F <= b(b(F)) = /1(23) = g. Suppose, for a contradic-
tion, that © £ F, and choose a group G of minimal order in ffi\.F. Then G/Cosoc(G)
is simple by II, 2.10 (a). If G' < G, it follows that G has no non-trivial quotient in Я
and hence that G e gb by definition of ffi. But from b(g) = b(F) S b(X) we deduce
that 'Xb £ Fb and obtain the contradiction Ge Ps F. Therefore G is perfect, and
we conclude that G e b(F) = b(g), against the fact that G e © s g. This final contra-
diction forces the desired conclusion that © s F. u
The Fitting subclass gb of a Fitting class g is a catalyst in the preceding theorem
and in several later proofs. Next we characterize gb as the smallest Fitting class whose
boundary contains b(g).
810
XI. Fitting classes—their behaviour as classes of groups
(4.7) Proposition. Let g and X be Fitting classes. Then
(a) b(g) £ b(X) if and only if g* £ X £ g, and
(b) (S'? = g‘
Proof, (a) Assume first that b(g) — b(X). Then g* £ Хь £ X by definition. A group
of minimal order in X\g would belong to b(g) and hence to b(X) by assumption.
Since X n b(X) = 0, we must have X £ g. Conversely, assume that g* £ X £ g, and
let В e b(g). Then В $ g and certainly В ф X. But Cosoc(B) e g‘ £ X, and since В is
single-headed, we have В e b(X), as desired.
(b) Since (gb)t’ £ g* £ g, by the sufficiency of the condition in Part (a) we have
b(g) £ b(g1’) £ hence gb £ (g'1)'1 by the necessity of that condition, and so
equality holds. □
Our next objective is to show that a Fitting class g is uniquely determined by its
imperfect boundary b(g), provided this is non-empty. The following lemma is funda-
mental.
(4.8) Lemma (Doerk [6]). Let p be a prime and g a Fitting class. Let G, be a group
in bp(g) and G2 a group in g satisfying G2/Cosoc(G2) = Zp. Then bp(ef) contains a
group X with normal subgroups Nt and N2 such that
(i) Nt n N2 = 1, and
(ii) X/Ni = Gifori = 1,2.
Proof. By A, 19.1 and A, 14.17 there exists a single-headed group X with normal
subgroups A, and N2 satisfying Conditions (i) and (ii) and such that X/NtN2 is a
cyclic p-group. If X were in g, by the quasi-R0 lemma (IX, 1.13) we should have
G, £ X/N2 e g (since X/N2 = 62 6 g), contradicting the hypothesis that G, e b(g).
Therefore X ф g.
To show that X ebp(g), let M = Cosoc(X). Then M/Nt <1-X/Nl = G, e hp(g),
whence M/Nt e g. Also M/N2 e s„(X/N2) = s„(G2) £ g, and since M/Nt N2 e Gp, we
can again apply the quasi-R„ lemma to conclude that M eg. Hence X e hp(g). □
We can now deduce that a non-empty imperfect boundary b(g) of a non-trivial
Fitting class g must contain infinitely many isomorphism classes.
(4.9) Corollary. Let pbe a prime and g a Fitting class.
(a) If bp(g) contains a non-cyclic group B, then bp(g) contains a group X which has
В as a proper epimorphic image.
_ (b) If g Ф (1) and bp(g) Ф 0, then bp(g) is infinite; in particular, if g Ф (1), then
h(g) is either empty or infinite.
Proof, (a) If Zp 6 g, by IX, 1.9 the cyclic group Zp« belongs to g for all n e N.
Applying (4.8) with G, = В and G2 = Zp„, we can find a group X in bp(g) with
quotients isomorphic with В and Zp„, and so by taking n large enough, we can ensure
that |B| < |X|, in other words that В e (q — 1)(X).
Now suppose that Zp ф g. By A, 14.17 we can find a single-headed group X with
4. Fitting class boundaries I
811
X t g. But if M = Cosoc(A'), then M/N, . Co^TgVoTTT IT
quasi-Ro lemma MeR. Hence Xeb ,(g) if n. =N = i then Xt~m У
^nttaty to hypothesis. Hence В is a proper epimorpHc irmge of X, al dliTed^
To justify Statement (b), suppose that bp(g) is non-empty.
Case 1: bp(g) contains a non-cyclic group B. In this case repeated application of the
foregoing result yields an infinite sequence, starting with B, of groups in b (g) each
term of which is a proper quotient of the next. ’
Case 2: Zp e bp($). Let S be a simple group in g. Either (i) S S Z, for some prime
q^tp, or (n) S is non-abelian. If (i) holds, the class bp(g) contains the non-cyclic group
E(plq\ and if (ii) holds, then it contains S*1jZp. In either eventuality we can return
to Case 1 to complete the proof. q
The next result suggests that g‘ is a ‘large’ subclass of the Fitting class g.
(4.10) Lemma. Let % be a Fitting class and p a prime such that bp(g) 0. Let G be
a group in g.
(a) If G/Cosoc(G) = Zp, then Cosoc(G) e gp.
(b) If G is perfect, then G e gp.
Proof, (a) In Lemma 4.8 take Gj 6 bp(g) and G2 = G, the given single-headed group,
and let M denote the cosocle of the bp(g)-group X so obtained. Then gp contains M
and Cosoc(Gj) S M/N,. Since M/N,N2 e <ZP, by the quasi-R0 lemma gp therefore
contains M/N2 Cosoc(G), as asserted.
(b) Let W = GlbrceZp. Then IT is single-headed by A, 18.8(d) and Cosoc(W')
coincides with the base group G“. If We g, then G* e gp by Part (a). If W$ g, then
We bp(g), and by definition gp contains Cosoc(lL) = Gf In either case we have
G es^C) £ s„g£ = g£. D
Let g and (5 be Fitting classes with b(g) = b(ffi). Theorem 4.4 states that if bp(g) is
empty for all primes p, then g = gS = ®S = ©. The following is a corresponding
result for the case when some of the classes bp(g) are non-empty.
(4.11) Theorem (Pense [1]). Let g and 6 be Fitting classes and n anon-empty set of
primes. Assume that b„(g) = b„(6) Z 0 for all pen. Then gG„- - ®®«-
Proof. Choose a group G of minimal order in 6S„.\gS„-, re a
non-empty. Then G is single-headed with Cosoc(G) 6 gS„_- / °
n'-group we should have G e g^G.. = T±’“ haVe®T
O” (G) = G and consequently G 6 ®. Let p e n. Since bp(g) p( r
L g. Therefore if G = G‘, by (4.10) (b) we have Ge^.
of G t ge„.. It follows that G/Cosoc(G) s Zr for some rex , )
hypothesis:we can conclude from (4.10) (a) that Cosoc(G) e ®, - 5, £ g.
812
XI. Fitting classes—their behaviour as classes of groups
that G e br(g) since G ф g. But then G e br(&) by hypothesis, against Ge®. This
contradiction shows that our initial supposition was wrong and hence that ©S„. £
gS„.. Since the hypotheses are symmetrical in g and ffi, the reverse inclusion also
holds. □
We can now deduce the result mentioned earlier as our objective, namely that a
Fitting class is determined by its imperfect boundary when this is non-empty.
(4.12) Corollary. If g and © are Fitting classes such that b(g) = b(ffi) 0, then
5 = ©.
Proof. Let л denote the (non-empty) set of primes p for which bp(g) 0. Then
gS„. = ©<S„- by (4.11). But bp(g) = 0 if and only if g — gGp, and so g = gSp for
all p 6 n. Consequently g = gG>„.. Similarly ©£„ = © and the result now follows.
□
(4.13) Corollary. Let g be a Fitting class, and let X be a class of single-headed perfect
groups such that SB = b(g) о X is subnormal!}’ independent. Then SB is the boundary of
a Fitting class.
Proof. By (4.3) the class h(X) is a Fitting class, and therefore so is /i(SB) =
h(b(g)) h(X) = g n h(X). Since h[SB) and g have the same imperfect boundary,
namely h(g), by (4.12) the two classes coincide when b(g) 0. On the other hand,
ifb(g) = 0, then SB is a preboundary of perfect groups and is therefore a Fitting class
boundary by (4.4)(b). □
To end this section we investigate an equivalence relation on Fitting classes defined
by
g is equivalent to ffi if and only if gS = ffiS.
The Fitting classes of soluble groups form a single equivalence class under this
relation, and we shall see that the other equivalence classes share some of its properties;
for example, each equivalence class contains a unique smallest element, which we
now proceed to characterize.
(4.14) Lemma. Let be a class of groups, and let G be a perfect single-headed group
in Fit(R). Then G e s„R.
Proof. If G f s„R, then R s h(G). Since h(G) is a Fitting class by (4.3), we have
Fit(R) £ h(G), and therefore G ф Fit(R), against the choice of G. Hence G e s„R. □
(4.15) Definitions, (a) If g is a Fitting class, a class X, or a set -T of groups with
X = (S'), is called a generating system for g if g = Fit(X).
(b) If g is a Fitting class, we shall denote by s(g) the class
813
4. Fitting class boundaries I
s(8) = Fit(G e g: G = G'),
Clearly s(s(g)) - s(g), and s(g) = 1 when geg. Since the smallest memberg of
Locksec(g) is generated by the class (&: G e g), it follows that *
(4./?)
s(8)<=g«.
We now characterize s(g) as the smallest Fitting class in the set
{©: ® is a Fitting class and ffi® = gg}.
(4.16) Proposition. Let g and ® be Fitting classes. Then gg = ®g if and only if
s(g) £ 6 c; gS.
Proof. First suppose that gg = ®g. Then obviously (5 c fig c gg. If G is a
perfect group in g, then G 6 ffig, and so G = Gs e ®. Hence s(g) e 6 by definition
of s, and therefore the condition is necessary.
To prove the condition sufficient, one need only observe that gg = s(g)g and
then conclude that
gg = s(g)S <= ®S c (gg)g = gg.
(4.17) Example. According to (4.16), for each Fitting class g we obtain a map
/:3E-s(g)0 3E
from the family of soluble Fitting classes to the equivalence class
S' = {ffi: ffi = Fit(ffi) and ffiS = gS}.
We wish to show that f need be neither surjective nor injective,
(a) Let g be a Fitting class satisfying
(4.y)
g = g2 g.
We claim that g = s(g). We have already observed that s(g) S gt. By X, 4.20 the
Fitting class g is generated by the class (G': G 6 g), and so it will suffice to show
ST& ttenG'e S(g). Let X e g\S,andlet У = ^-evidently У isanon-tnvial
perfect group in s„g — g. Set
H = GOj^K
let В = G “ e Dog = 8 (B is the ba”*5 6roup^ and note that w e 82 = 8- By 18 4
(b) and (d) we have
B' < [fi, У] and H' = Y’
814
XI. Fitting classes—their behaviour as classes of groups
and it follows easily from A, 12.4 (b) that H' is perfect. Since G' < В' < H' < H e g,
we have H' g s(g) and hence G g s„s(g) s(g).
We remark that Fitting classes g satisfying (4.y) are not hard to come by. For
example, if 3 is a class of finite simple groups containing at least one non-abelian
group, then the class e 3 of all groups whose composition factors belong to 3 is clearly
such a class. Also the class g = h(3) is a Fitting class by (4.4) and satisfies g2 = g
by VI, 3.11 (c); if 3 £ 3\'2I, then additionally g = gS and g is a Lockett class.
(b) If g = 6, the map f defined above is not surjective. To see this, observe that
Locksec(g) £ by (4.16). Thus if / were surjective, every C e Locksec(g) would
have the form £' = О X by Part(a), and if Char(X) = n, then we should obtain
C.91. e j с
by X, 1.8 (a). But this would imply by X, 4.14 that the Lausch group A(G) is a direct
product of cyclic groups of prime order and, according to X, 4.21, this is not the case.
(c) To see that the map/defined above need not be injective, take for g any Fitting
class g = gS satisfying (4.y), Then by Part (a) we have
s(g) 0 91 = g/Л = g* = g,
and so s(g) о J = s(g) 0 ?) whenever 91 £ X n 91-
We now consider classes which, like s(g), have a generating system of perfect
groups.
(4.18) Lemma (Pense [1]). Let be a Fitting class which has a generating system X
of perfect single-headed groups. Then X is a minimal generating system if and only if
X is subnormally independent.
Proof. Suppose that G 61 is redundant as a generator of g. Then G 6 Fit(X\(G)),
and so by (4.14) we have G e s„X; thus X is not subnormally independent.
Conversely, if X is not subnormally independent, there exists a group G in
X n s„(X\(G)), and then G is obviously redundant as a generator of g. □
Although a Fitting class need not possess a minimal generating system, more can be
said when it does.
(4.19) Theorem (Pense [1]). Let % be a Fitting class generated by perfect groups, and
let Xbe a minimal generating system for g consisting of single-headed groups. Then
(a) X consists of perfect groups, and
(b) if 9) is a generating system for ^consisting of single-headed groups, then X £ 9).
Proof, (a) Since g is generated by perfect groups, by A, 14.16 (b) it is generated by
a class SB of single-headed perfect groups. Then SB £ s„X by (4.14). Let G be an
imperfect group in X and note that G/Cosoc(G) is abelian since by hypothesis G is
single-headed. Set
4. Fitting class boundaries I
815
Wi = Fit(Cosoc(H'): IFe'ffi).
Since the Fitting class ®I(d0(3\2I)) clearly contains 2B, it also contains g. Conse-
quently G 6 ЯЯ. Let H e Ж n s,(G). Then H is perfect and single-headed, and H e
s„(Cosoc(PV): IVe'lB) by (4.14). Thus H is redundant as a generator in 2B, and the
class SB n s„(G), since finite, is redundant as a whole from the generating system SB
Without loss of generality we can therefore suppose that SB ns (G) = 0 and hence
that SB £ s„(X\(G)). But then g = Fit(SB) £ Fit(X\(G)) e g, which contradicts the
minimality of X. We conclude that X contains no imperfect groups as asserted
(b) Let 2) be a generating system for g consisting of single-headed groups and, for
a contradiction, suppose that X 2). If G 6 X\S), then G is single-headed and perfect
by Part (a), and so G 6 s„S) by (4.14). Let G sn Ye S). If Y is perfect, then Ye s.X by
(4.14), in which case G is a proper subnormal subgroup of an X-group, contradicting
the minimality of X. Thus Y is imperfect, and by the argument of Part (a) we have
Y e Fit(Cosoc(X): X e X). But then, once more by (4.14), the perfect group G is a
subnormal subgroup of Cosoc(X) for some X e X, which again means that G is
redundant from X. This final contradiction proves that X £ V). □
(4.20) Examples, (a) Let R denote the class of single-headed perfect groups G such
that Cosoc(G) = Z(G). Then a group in the class 25 of generalized nilpotent groups
(see IX, 4.14) is the central product of its Fitting subgroup and groups in R. If p is a
prime, then SL(p, qr) is a R-group whose centre has order divisible by p for a suitable
choice of the prime power q'. Hence Sp, and therefore 91, is a subclass of Fit(R). It
follows that Fit(R) = SB and hence that s(25) = SB. In particular, SB has a trivial
Lockett section.
(b) If g is an E-closed Fitting class containing insoluble groups (for example the
class /1(3) described in (4.17)(a)), then s(g) has no minimal generating system. For
let X be such a system. If G e X, then G is perfect by (4.19) (a), and since G is generated
by perfect, single-headed subnormal subgroups by A, 14.16(b), without loss of
generality we can suppose that G itself is single-headed. Then G Qjree G is perfect and
single-headed by A, 18.8(d) and belongs to g2 = g. By (4.14) the group GffiG is
subnormal in an X-group H and G is a proper subnormal subgroup of H. Since this
contradicts the minimality of X, no such X exists.
Exercises
1.
2.
3.
4.
Let g be a Fitting class. Then gto £ h(fo(g)), and this inclusion can be proper.
(Pense [1]) Let g and ® be Fitting classes with ® = ®S. Show that the class
Rad(g, ®) = (G: GK e ®) is a Fitting class.
(Pense [1]) Let g denote the class of all groups G satisfying Cg(F(G)) _ F(G) and
let 23 be the class of generalized nilpotent groups. Show that g = Rad(23, to) in
the notation of the preceding question. Deduce that g is a Fitting class and tha
g2 = g Show that fc(g) = R, the class defined in Example 4.20(a).
(Pense Fl J) Let R, S and T be three different non-abelian simpe groups, let
25 = (T,(R^g S) 4.8 T). Show that the class g = /i(25) is a Fitting class satisfying
g 0 g = g and g2 # g. Is g2 a Fitting class?
g16 XJ Fitting classes—their behaviour as classes of groups
5. Fitting class boundaries II
In this section our universe is S, and we adopt the notation from Section 4 for this
universe.
We consider a range of problems in the following general ambit: Given a class ®
of single-headed groups, what can one say about Fitting classes ft when either
® £ b(g) or b(g) S ®? We will answer this question for some special classes ®. In
the following section we shall relate such questions to the behaviour of the so-called
“Frattini dual” subgroups.
Our first objective is to characterize the Fitting classes in a Lockett section by their
boundaries. For this purpose we define two classes of single-headed groups associated
with a given Lockett section.
(5.1) Definition. Let 5 be a Fitting class. Define
I(Locksec(g)) = Q {b(£>): & 6 Locksec(g)}, and
U(Locksec(g)) — (J {b(§): 5 6 Locksec(g)}.
The following striking theorem shows that each of the classes I( ) and U( ) deter-
mines, by reference to its boundary, whether a Fitting class belongs to a given Lockett
section distinct from Locksec(S).
(5.2) Theorem (Doerk and Hauck [2], Baldauf [1]). Let g and § be Fitting classes.
Any two of the following statements are equivalent:
(a) g* Ф S and ft 6 Locksec(g);
(b) g* Ф S and l(Locksec(g)) e b(§);
(c) 5* Ф S and b(fi) £ U(Locksec(g)).
It is clear from the definitions that Statements (b) and (c) are both consequences of
(a). We shall now present Baldauf’s proof that Statement (b) implies (a), but shall
postpone Doerk and Hauck’s proof that Statement (c) implies (a) until we have
developed the appropriate techniques.
Proof that (b) => (a): Assume that g* A G and I(Locksec(g)) £ b(§).
(1) We show first that § S g* by choosing a group G of minimal order in fi\g*
and deriving a contradiction. This choice means that GB. is the unique maximal
normal subgroup of G, and if |G: GB#| = peP, then G/GB. is a cyclic p-group; this is
because G' < GB< by X, 1.7 and because G is single-headed. If GB. = GB>, we obtain
G 61(Locksec(g)) e fc(jj)
by hypothesis, which contradicts the choice of G 6 £>. Hence Ga> < Gs.. By X, 1.5
we have (G x G)B> = (GB< x GB>)((<?, д'): g e GB.>, and by X, 1.9 we have
(G x G)B. = GB. x GB.. Now consider the subgroup
5. Fitting class boundaries II
L = Ws. * g8.)«9, g~'Y g e G>.
817
I < G x G and /с L u S' П L in 5*’ then We should have
71 tb , r d X )b* W°U d be subdirect in G x G. But this means by X
M that Ge g*, contrary to the choice of G; therefore Lt g*. On the other hand by
X, 1.5 we have 3
L^. - Lr^(Gs. x Gg.)
= (Gg. x GB.)<(0, g ‘): g 6 G> n(Gg. x Gg.)
= (G x G)g.,
and consequently Le £i\g* and Ls. = Let K/L^. be a composition factor of
L, and let S be a minimal subnormal supplement of Ls. in K. By A, 14.15 the group
S is a single-headed group whose unique maximal normal subgroup coincides with
Sn LB. = Sg. = S8.. Therefore S e I(Locksec(g)) e against the fact that S e
s„(L) e This contradiction shows that § c §*.
(2) We can suppose that g,\£>* # 0- For if g, £ §*, by (1) we have 8. £ fi* £
g* and hence § e Locksec(g) by X, 1.17, as required.
(3) Let G 6 and let p be a prime divisor of |G: Gg.|. We show that gt
contains G2 rbreKZ4 for all primes q Ф p.
Set H = G2 Qjreg Z9. We suppose that H fS* and derive a contradiction. Evidently
the base group В = (G2)fc of H coincides with Hs , and if H were in g*, then by X,
2.4 and X, 2.8 we should have H e gt, contrary to supposition. Therefore В = Hg. =
Hg«. Let S be a minimal subnormal supplement to В in H. Then, as in Step 1, we
obtain S 6 I(Locksec(g)) e /?(§); jn particular S belongs to £1*3,, which is a Lockett
class by X, 1.26(b). Since p||G: Gg.|, certainly G^§*S,, and consequently
G2 t 5*S,. Hence by X, 2.1(a) the £i*S,-radical of H is contained in B. But then
S < B, contrary to the choice of S. Therefore H e gt, as claimed.
(4) Let n denote the set of all primes p for which there exists a group G e g.\£i*
with p 11G : Gg.|. If p e n, we show next that gt contains H2 Tj„gZ, for any H e g.
and any prime q Ф p.
Let G 6 gt\£j* with p||G : Gg.| and H e gt; then H x Ge g„\&* and
p divides | W x G:(H x G)g.|.
From Step 3 we see that g. contains (H x G)2 and G2 Qi^Z,, and so by the
quasi-R0 lemma (IX, 1.13) it contains H2 rbreg Z„, as desired.
(5) We can suppose that |n| < 1.
If |n| > 2, then g. contains G2 for all q e P and all G e g. by Step 4. But
then g. is normal by X, 3.7. and in this case g* = 6, contrary to hypothesis.
(6) Conclusion of the proof that (b) => (a).
By Steps 2 and 5 we have |rr| = 1, say n = {p}. If Ge g., by Step 4 we have
G2 Tj Z e g* £ g‘ for all primes q + p, and hence g <£„ = g by X, 2.15. Now
by definition of л the Lockett class £>*tep contains g. and therefore contains (gj
gig XI. Fitting classes—their behaviour as classes of groups
5* by X, 1.8; thus g*Gp £ §*SPSP = But £>* £ g* by Step 1, whence
§*<Sp £ g*Gp, and so £i*Gp = g*Gp- Consequently
§*6P. <= g*6p- = g* £ 5*SP.
and it follows that &‘Sp. = £>*. Therefore §*9i = 5*Gp = g*Sp = g*91, and so
from X, 1.22 we can conclude that 5* = g*, which implies Statement (a) of the
Theorem. О
Using the notation introduced in (4.5), we have
(5.3) Lemma. Let g and fi be Fitting classes with g £ §. If p is a prime for which
bp(& # 0, then gp £ &p.
Proof Let G e bp(g). If G ф f>, then G e and Gs = G6e hj. On the other hand,
if G 6 fi, then GB 6 fip by (4.10) (a). Since gp is generated as a Fitting class by the
g-radicals of groups in fcp(g), we conclude that gp £ fip. □
(5.4) Lemma. (GJ11 = Gt.
Proof. By X, 5.32 we have hp(Gt) A 0 for all primes p. By (4.10) (a) the maximal
normal subgroup of each single-headed group in S* belongs to (G*)1’. Since every
group is generated by its single-headed subnormal subgroups (see A, 14.16(a)), it
follows that G* c (6*)*9L Therefore S = 2*91 = (G*)ll9i9l, and because 9191 =
s(919I) Ф S, we can conclude from X, 3.10(a) that (G*)b is a normal Fitting class. Thus
Gt £ (GJ11, and so equality holds. □
In fact, the following stronger result is true.
(5.5) Lemma. (G*)p = S* for all primes p.
Proof. First we show that, for a fixed prime p, we have
<5a> M®*) e (S*)‘PSP9l
for all primes q. If q = p, this is clear. Suppose that q Ф p, and let G 6 bt(<5t). Set
and, as usual, let Gfc denote the base group of W. Denote by .У the
set of minimal subnormal supplements to G11 in W, and recall from A, 14.15 that each
S 6 V is single headed, with S n G15 as its maximal normal subgroup; in particular,
S/(S r^Gb)^. Zp. Define
M = <S:Se.f/>, and
N= <SnGh;S6 У).
Evidently M, N < G, and M/N is a p-group by A, 8.6(a).
5. Fitting class boundaries П
819
Let S 6 У. If S 6 then S n G" e (6,£ by (4.10)(a). If S ф 6„ then S/Ss is a
eyehe p-group because the head of S is Zp. On the other hand, sin*ce G e b 6 j the
group G /(G )s. is a «-group, and consequently |S: Ss | = p. Thus S e b (6 ) and
tn this case too we have S n C* = Ss. e (6^. Hence N e (S,)*. jt is a property of
the regular wreath product W that (G")' < [G\ Zp] (see A, 18.4(b)). Therefore if
bd'O) denotes a direct component of G we have
(Go)' < (G")' < [G\Z„] < [G>, M] <G'nM,
and consequently AT (Go)' < G > n M. Since N e (SJbp and M/N is a p-group, it follows
that N(G0)' 6 (<S„)J;ep. Since (Go)' sn 7V(G0)', we conclude that C £ Goe (34)ь3 91,
and Assertion 5.a is justified. *₽ p >
By (5.4) we have (<S#)b = 6,, and by (5.a) it follows that (<3*)fl s (6 J <5 91. Hence
<5 = <S„9I = (<SJb919I = (St)‘9129I, and since 9129I is s-closed, we deduce from X,
3.10(a) that (G#)p is a normal Fitting class. Hence 3 £ (S )‘ £ Q and equality
holds. * n
The next result shows that inclusion between boundaries can only happen for
Fitting classes belonging to the same Lockett section.
(5.6) Theorem (Doerk [6]). Let % be a Fitting class different from S. Then
(a) (g‘)* = g*, and
(b) if ft is a Fitting class with b(g) £ b(ft), then 5* = ft*.
Proof, (a) By (4.7) (a) we have b(ft) £ bfft1’), and so I(Locksec(ft)) £ b(ftb). If ft is
not a normal Fitting class, then ft’’ 6 Locksec(ft) by (5.2), (b) => (a), and therefore
(ftb)* = ft*. On the other hand, if ft is normal, there exists a prime p for which
bp(g) ф since ft A S by hypothesis. From (5.3) it therefore follows that (3„)p £ ft’p,
and so by (5.5) we have £ ftp £ ft*. Consequently (g‘)* = S, and once again we
have (g‘)* = ft*.
(b) If b(g) £ b(ft), then ft* £ ft £ ft by (4.7). Hence by Part (a) we have ft„ £
ft11 £ ft £ ft, and therefore ft* — ft*. □
We can now finish off the proof of Theorem 5.2.
(5.7) Statement (cj implies Statement (a) in Theorem 5.2.
Proof. Let ft A S and b(ft) £ U(Locksec(g)).
Case 1: Assume that ft* = S. It will suffice to show that 6, £ ft. If this is not so,
then we can choose a group G e 6,\ft of minimal order, in which case G e b(ft)I £
U (Locksec(ft)). Consequently there is a Fitting class 9) in Locksec(S) with G e b( J),
and then Ge 6, £ ?, a contradiction. Therefore S„ £ ft, and ft* = S.
Case 2: Assume that ft* A 6, and suppose that ft t Locksec(g). Fr°m the equiva-
lence of Statements (a) and (b) in the Theorem 5.2 we can then conclude that there
820
XI, Fitting classes—their behaviour as classes of groups
is a group G in I(Locksec(g)) with G ф b(£>). If G £ £>, then Gs < M <G, and so there
exists a prime p and a subnormal subgroup N of G satisfying N < M and 17V/G*| = p.
Let S be a minimal subnormal subgroup of N with SG6 = N. Then S e b(§) since S
is single-headed by A, 14.15, and by hypothesis S e b(9)) for some Fitting class
'J e Locksec(g). But then Ssn M e ll) yields the contradiction Se4)o b(9)) = 0,
and we are forced to conclude that G e By (5.6) we have (§*’)* = §*, and since
b(§) £ U(Locksec(g)), we also have £ 5*. Thus f>* £ 5*. But then we have
G e § £ §* £ g*, contradicting the fact that G e I(Locksec(g)). Therefore our initial
supposition was false, and consequently § e Locksec(g). □
A natural question arising from this theorem is the following: Given a family SF
of Fitting classes and a Fitting class § with 6(§) £ (J {b(.F): X e J’"}, when can one
deduce that § e Я A comprehensive answer to a question of such generality is
hardly to be expected. If & is the family of all Lockett classes, for example, § certainly
need not itself be a Lockett class (see Doerk and Hauck [2]), although, if b(§) £ b(g)
for a Lockett class g, Theorem 5.6 (b) shows that § is then a Lockett class. In
fact, we do not know how to determine the Fitting classes § which satisfy b(§) £
(J (b(F): -F is a Lockett class}.Nevertheless, we do have a useful description of such
§ when we restrict the union to Lockett classes which satisfy an additional wreath
product property.
(5.8) Definitions, (a) We shall say that a Fitting class g satisfies the weak wreath
product property if for all primes p and for all single-headed groups Geg with
O₽(G) < G, the wreath product G 1iree Zp belongs to g.
(b) We shall call a closure operation c an amenable Lockett operation if every
c-closed class of groups is a Lockett class with the weak wreath product property. In
particular, C.F is always a Fitting class, and so <s„, n0> < c.
(5.9) Lemma. Let c be the closure operation <s, n„, q). Then c is an amenable Lockett
operation.
Proof. Let X = <s, n0, Q>I £ S. Then I is certainly a Lockett class by X, 1.25. Since
Ro sd0 < sn0, the subgroup-closed Fitting class I is a formation and hence by (1.7)
a primitive saturated formation. But by VII, 3.9 a primitive saturated formation of
bounded nilpotent length has the form
е„п(П i.e,en.)
я=Р
for suitable Fitting classes F„. In particular, if G e -F, then the subclass <s, n0, Q)(G)
of -F has this form, and if G is a p'-perfect soluble group, then it follows easily that
any extension of OP(G) by a p-group belongs to X. Consequently F has the weak
wreath product property and c is an amenable Lockett operation. □
(5.10) Lemma. Let cbe an amenable Lockett closure operation, and let g be a Fitting
class satisfying
5. Fitting class boundaries II
821
(5.P)
6(g)5=U{b(3£);3£ = cI}.
Then the following statement is true for all primes p- If G
5 with O"(G) < G, then В ф c(G) for all В e bp(g).
is a single-headed group in
Proof We begin with a simple observation. Let X e b(g). Then Hypotheses 5.8
imply the existence of a c-closed class I such that X e h(X). Evidently Xx = X
and X ф X э с(Хж), and therefore c(XR) is a proper subclass of c(X). *
Now let В e bp(g), and let G be a single-headed g-group with G/Cosoc(G) Z .
We write Л/ = Cosoc(G) and distinguish two cases: P
Case 1: c(M) - c(G). Let X e 6p(g) denote the group constructed in (4.8) with В and
G in the roles of G, and G2; it has normal subgroups N, and N2 satisfying X/N, =; B.
X/N2 S G with N, n N2 = 1 and X/N, N2 acyclic p-group. Furthermore, under these
isomorphisms, XB/N, corresponds to BR and XB/N2 corresponds to M.
Suppose, for a contradiction, that В e c(G). Since c-closed classes are by hypothesis
Lockett classes, we can apply the strong form of the quasi-R0 lemma (X, 1.24) to X:
since both X/Nj( = B) and X/N2(sG) are in c(G), we conclude that c(X) = c(G).
Applying it again, this time to XB, we then obtain M = XB/N2 e c(XB), and it follows
that c(G) = с(Л1) c c(X-) £ c(X) = c(G). Therefore c(X-) = c(X), which contra-
dicts the observation made at the outset of the proof, and hence В <£ c(G) in this case.
Case 2: c(M) is a proper subclass of c(G). In this case it is obvious that M = GC(M).
Let Z = Zp, and set IF = G rL„g Z. Then IF/Л/B s Z Z, which by A, 18.11 has a
cyclic subgroup C of order p2 whose intersection with the base group of ZliregZ
evidently has order p. Therefore G has a subgroup К containing M~ such that
KjM~ S C and (K n G’)/MB has order p. Since G 6 c(G) and c(G) has the weak
wreath product property, the Lockett class c(G) contains W and hence contains
К (sn IF). Because c(K) is also a Lockett class and К f GB, it follows from X, 21(a)
that G e c(K). Consequently c(G) = c(K) = c(lF).
Let S be a minimal subnormal supplement to M in K. Then IF e c(S) by X, 2.1(a),
and therefore c(S) = c(K) = c(G). Since S is single-headed by A, 14.15, the subgroup
A = S n GB is the unique maximal normal subgroup of S and S/(Sn MB) is cyclic of
order p2. We show next that c(A) = c(S), and consider first the possibility that
KcU) < GB. Since radicals for Lockett classes respect direct products by X, 1.9, there
is a bijection between the Lockett class radicals of G and those of GB, and con-
sequently either KC(A) = G" or KctA) < MB = (GB)C(M) < GB. If KcM) = GB, then
G e c(A), and so C(S) = c(G) S c(A) £ c(S), which yields c(A) = c(S), as desired.
Since the alternative that KC(A) < MB is ruled out by the fact that AfM.it only
remains to consider the possibility that KcU) f G\ But in this case we deduce once
more from X, 2.1(a) that W e c(A), whence S e c(A) and again we have c(A) c(S).
If S were in b(g), we should have A = SB and could conclude from our initial
observation that c(A) <z c(S), a contradiction. Since Ge g, we have Со^) -
A e g, and it follows that S e g. Now applying the argument of Case 1 to S instead
of G with A in place of M, we obtain В t c(S) = c(G) for all В e b„(g).
822
XI. Fitting classes—their behaviour as classes of groups
We can now state and prove the main result on Fitting classes with restrictions on
their boundaries.
(5.11) Theorem (Doerk and Hauck [2]). Let c be an amenable Lockett closure opera-
tion, and let g be a Fitting class which satisfies
h(S)sU{b(I):I = c3£}.
(a) g is a Lockett class which satisfies the weak wreath product property.
(b) If f is any one of the closure operations sF, s, Q or r0 and if every c-closed class
is r-closed, then g is also r-closed.
Proof, (a) First we show that g is a Lockett class. Suppose not, and let G be a group
of minimal order in g*\g- Then certainly G e b„(g), and since G e g*, the group
К = (G3 x Gs)<(p, g~l):geG) belongs to g by X, 1.2. Let S be a minimal sub-
normal supplement to Gs x ©э in K. Then S belongs to g, and by A, 14.15 it
is single-headed with 0p(S) < S. Therefore G f c(S) by (5.10), and in consequence
GC(S) < G3. But because c(S) is a Lockett class by hypothesis, by X, 1.9 we have
(G x G)C(S) = GC(S) x GC(S) < G3 x Gg,
whereas on the other hand S < (G x G)C(S) and S f Gs x Gg. This contradiction
shows that g = g*.
Next we prove that g satisfies the weak wreath product property. Let G be a
single-headed group in g with OP(G) < G, and set W = G rbree Zp. Suppose that
И7 # g, and let S be a minimal subnormal supplement to G" in W. Then evidently
Sebp(g), and therefore by (5.10) we have S^c(G). Since c(G) satisfies the weak
wreath product property by hypothesis, we have We c(G) and therefore 5 e s„c(G) =
c(G). This contradiction proves that W e g. Thus we have shown that Statement (a)
is true.
(b) We deal with each of the four closure operations in turn:
(1) f = sf . We assume that every c-closed class is a Fischer class, but that g is not
• a Fischer class, and derive a contradiction. In this case we can find a group G in g
with a subgroup U not in g such that If91 sn G. Among pairs (G, U) with these
properties, minimize first |G|, then |U|. Let К be a maximal normal subgroup of U.
Since К91 < К < U and K91 < I/91, we have K91 sn G, and therefore К e g by the
choice of U. Consequently, since U f g, it follows that U is single-headed and hence
I that U/U is a cyclic p-group for some prime p. Let S be a minimal subnormal
supplement to If91 in U. Then 5 is not in g, it is single-headed by A, 14.15, and its
maximal normal subgroup S c\K belongs to s„g = g; thus 5 e bp(g). Moreover, since
S sn V, we have S91 < S n U91 sn If91 sn G, and hence S = U by the choice of U; in
particular, V e bpffi). Let M < G. Since M e s„g = g, the choice of G means that
U £M, and therefore U n M = UR. If bf <• G and M, # M, it then follows that
• n M) is a maximal normal subgroup of G containing V, a possibility we
have already excluded. Therefore G is single-headed, and since UM = G, we have
|G. M\ = |L; l/g| = p, whence OP(G) < G. From (5.10) we conclude that U f c(G),
5. Fitting class boundaries II
823
against the fact that V ec(G) since c(G) is a Fischer class by hypothesis. Therefore
g is, after all, a Fischer class.
(2) f = s. The argument is closely parallel to the previous one. If ft is not s-closed,
we can find a group G in ft with a subgroup U not in ft, and among such pairs (G, U)
we first choose one with |G| minimal and then choose the non-ft-subgroup V of G
with 1171 as small as possible. This choice of V means that U e bp(ft) and, as before,
it follows that G is single-headed with 0₽(С) < G. But then by (5.10) we have V </ c(G),
contradicting the fact that V e s(G) £ sc(G) = c(G) by hypothesis. Hence ft = sft, as
desired.
(3) f = q. Here we choose, if possible, a group G of minimal order in ft with
a normal subgroup N such that G/N f ft. Then G has a subnormal subgroup
T/Ne b(ft), and Г = G by choice of G. If S is a minimal subnormal supplement to N
in G, then S is single-headed, S e ft, and S/(S nN) = G/N t ft. Again the choice of G
forces S = G, and then by (5.10) we have G/N </ c(G). But G/N e q(G) g qc(G) = c(G)
by hypothesis, we have a contradiction, and so no such G exists. Hence ft = Qft.
(4) f = r0. As usual suppose, for a contradiction, that Roft # ft. Then we can find
a group G f ft with the following properties: G has normal subgroups Nt and N2
such that G/Ni e ft, G/N2 e ft, and n N2 = 1. Assume, that G, among such groups,
has minimal order, and let M/Nt < G/Nj. Then M/N2 e ft and M/(M n N2) s
MN2/N2 e s„g = g. The choice of G forces M eg, and since G ft, the group
G/Ni must therefore be single-headed; similarly G/N2 is single-headed. Since s
N/Nj/Nj sn G/Nj e ft for {i, j} = {1,2}, we have If e ft. Hence N\ N2 e Dog = ft, and
therefore NtN2 < G because G ft. Moreover, if S is a subnormal supplement to
Nt N2 in G, then S f ft, and it follows easily that G = Se bp(ft) by the choice of G.
By, A, 19.1 and A, 14.17 there exists a single-headed group К with the following
properties: К has normal subgroups К1 andK2 with К, nK2 = 1 such that K/KtK2
is a non-trivial p-group and K/K, = G/Nt for i = 1,2. Since K/K, e ft in this case, by
the quasi-R0 lemma (see IX, 1.13) we have К eft. Since c(K) is a Lockett class by
hypothesis, for i = 1, 2 we also have G/Nt s K/Kf e c(K) by the strong form of the
quasi-R0 lemma (see X, 1.24). (We observe in passing that the assumed R„-closure of
c(K) already ensures that c(K) is a Lockett class.) Thus G e R«c(K) = c(K). But by
(5.10) we have G </ c(K), and this is the desired contradiction. □
We end this section with the following theorem, which is now an easy consequence of
(5.9) and (5.11).
(5.12) Theorem (Doerk and Hauck[2]). Let P be a non-empty set of subgroup-closed
Fitting formations, and let ft be a Fitting class satisfying
b(ft)<=\J{b(X):*e^}-
Then ft is a subgroup-closed Fitting formation.
824
XI. Fitting classes—their behaviour as classes of groups
Exercises
1. (Doerk [6]) For a soluble Fitting class let b(g) be as defined in (4.2) for the universe
G. Show that for all Fitting classes g, ft £ S, the inclusion b(ft) £ b(g) implies
that ft* = g*.
2. (Doerk [6]) Let g be a soluble Fitting class.
(a) IfRSp?6 g for some prime p and if G eg, then G/GBb is an elementary abelian
p-group.
(b) If g(Sp Ф g for all primes p and if G e g, then G/GRb is a direct product of
elementary abelian p-groups for various primes.
3. (Doerk [6]) (a) Let 111 > 1, and, for i e I, let g, be a Fitting class different from S.
Assume that b(g,)n b(gj) = 0 for i ф j and that g is a Fitting class with b(g) =
U,e/b(g,). Then g is a normal Fitting class, and there exist sets л; £ P with
rq n 7iji = 0 for i # J such that g, = gG.-, for all i e I.
(b) If g is a normal Fitting class and {л;: i e 7} a partition of P, then b(g) =
Uie/ b(gS„;)with b(Ss«;) =
4. (Doerk [6]) (a) If л Ф 0, then (<3„)fc = S„r,St = (S„)#.
(b) (9l2), <= (9l2)" с 9l2.
5. (Doerk and Hauck [2]) Let g be a Fitting class. Then the following statements
are equivalent:
(a) b(g) £ J W'): >: = 0, 1,...};
(b) There exist (possibly empty) sets of primes with nf 2 nj+1 for i e N such that
8 = U<e N 91.,... 91.,.
6. (Doerk and Hauck [2]) Let g be a Fitting class, and let i < k. Then the following
statements are equivalent:
(a) b(g)Sb(9l')ob(9l‘);
(b) If i = к or i < к — 2, then either g = S or g = 91‘ or g = 91‘, and if i = к — 1,
then either g = S or g = 91'91. for some л c p.
7. (Doerk and Hauck [2]) There exists a Fitting class g such that
b(g) £ U {^(£): I is a Lockett class}.
8. (Baldauf) Let g and ft be Fitting classes and ne N. Let b"(g) denote the class of
all single-headed groups G such that | G/GB| is the n,h power of some prime (whence
b’(8) = /’(g)), and set 9)„ = Q {b”(3£): I e Locksec(g)}.
(a) If g* = G # ft, then ft* = G if and only if 9). £ b"(ft).
(b) If g* # S, then g, = <s„, N„>(GB:Ge 9)„).
6. Frattini duals and Fitting classes
The Frattini subgroup Ф(С) of a finite group G is the intersection of all maximal
subgroups of G. The Frattini dual *F(G) of G, first introduced by Ito in [1], is the
subgroup generated by all the minimal subgroups of G, in other words, by its
subgroups of prime order. Whereas the Frattini subgroup plays an important part
in the theory of Schunck classes, in the dual theory of Fitting classes no role has yet
6. Frattini duals and Fitting classes
825
been discovered for the Frattini dual. One reason for this is Gaschutz’s Theorem В
11.13, which tells us that, in contrast to <U(G), which is nilpotent, the quotient group
G/H'fG) has no structural restrictions. An obvious way to remedy this deficiency is
to look for a new definition for a Frattini dual, one based perhaps on a different
characterization of the Frattini subgroup, especially one leading to a fruitful dualiza-
tion of the closure operation еф. Thus, if Л( ) denotes this ‘new* Frattini dual, we
would define a corresponding closure operation EA(dual to Ej thus:
ел(1) = (G: A(G) is contained in a subnormal T-subgroup of G)
and hope to find a large “interesting” family of Fitting classes which are Enclosed.
With Л = T, this is certainly not possible (see Exercise 3 below).
The starting point for the following investigations is the simple description of the
Frattini subgroup in terms of e4 given in Exercise 1 below. Its dualization leads to a
whole family of Frattini duals T,, one corresponding to each closure operation c.
This approach is developed in two papers of Doerk and Hauck ([1] and [2]) and
our treatment closely follows theirs. We will work, at first, in the universe G of
arbitrary finite groups, and will later restrict attention to soluble groups.
(6.1) Definitions. Let c be a closure operation and G a finite group. We say that a
subgroup N has the property vc in G, and write N vc G, if N < G and с(Л1) £ c(N n M)
for all subnormal subgroups M of G. Clearly GvcG, and it follows easily that if
Nj vc G and N2 vc G, then Nt n N2 vc G. Thus the subgroup
Tc(G)=n{N:NvcG}
is a characteristic subgroup of G satisfying 'PC(G) vc G; in particular, c(G) £ cfT^G)).
We call TJG) the c-Frattini dual of G.
(6.2) Lemma. Let c be a closure operation and G a group.
(a) If К sn G, then T((K) < К n TC(G).
(b) If TC(G) < К sn G, then Tr(K) = Tt(G).
Proof, (a) It will suffice to show that К n »FC(G) vc К whenever К sn G. Let M sn K.
Then M sn G, and since TC(G) vc G, we have
c(M) £ c(M n TC(G)) = c(M n (K n Tr(G)),
,ь.. ч-дк>;,с. ut и»с » ^""2
Tc we obtain c(M) £ c(M n Tr(G)) £ c((M n 4-c(G)) n 4-c(K)) - C(M n ^(K)^
desired.
reference.
826 XI. Fitting classes—their behaviour as classes of groups
(6.3) Hypotheses. Let c be a closure operation.
(1) For every group H the classes cs_(H) and c(H) coincide (or, equivalently, s. < c
or s.c = c).
(2) The class с(Я) is a Fitting class for all groups Я (or. equivalently, <s„, n„> < c).
(3) If I is a c-closed Fitting class and if'() = c'l), then c(J O4J) = Io '?).
(4) The classes (1) and S are c-closed.
(5) If X is a non-abelian simple group, then c(X) = <s„, n0>(A").
(6.4) Proposition. Let c be a closure operation and G a group.
(a) If c satisfies Property 1 of (6.3), then T, (G) is the join of all those subnormal
subgroups SofG which satisfy c(K) Ф c(S) for all proper subnormal subgroups К of S.
(b) If c satisfies Property 2 of (6.3), then Тс(С) is the join of all those single-headed
subnormal subgroups T of G with c(T) c(Cosoc(T)).
(We adopt the convention that 'PJG) = 1 when no subgroups with the stated
properties exist).
Proof, (a) Let S be a subnormal subgroup of G such that c(K) c(S) for all proper
subnormal subgroups К of S. Then evidently TC(S) = S, and if J denotes the join of
all such subgroups S, it is clear from (6.2) (a) that J < 4'C(G). Thus it will be enough
to show that J vc G.
Suppose, by way of contradiction, that J does not have the vc-property in G. Then
we can find a V sn G such that c(J n V) Ф c(U) and can suppose that U has minimal
order among such subgroups. Let IT be a proper subnormal subgroup of U. Then
c(J n W) = c(lT). Suppose that c(fF) = c(U). Then we have
c(lf) = c(J n W) £ cs„(J n 17) = c(J n U)
by hypothesis, and therefore c((7) = c(J n If), which contradicts the choice of 17. It
follows that c( W) Ф c(lf) for all proper subnormal subgroups W of If, and therefore
U < J. But then c((7) = c(J n If), against the choice of U. This contradiction proves
that J vc G, as desired.
(b) Let S be a subnormal subgroup of G which satisfies c(K) + c(S) for all proper
subnormal subgroups К of S. Let M <j- S, and let g = c(Af), which is a Fitting class
by the Hypothesis 6.3(2) that c-closed classes are Fitting classes. If S belonged to g,
we should have c(S) £ eg = g and this would imply that c(S) = c(M) because
M e s„c(S) = c(S). Since this contradicts the choice of S, we must have S ф g, and
therefore M = Ss.
Let T be a minimal subnormal supplement to M in S, and note that T is single-
headed by A, 14.15. If T were in g, then we should have S = MT eNog = g,
which is not the case. Hence T£g and TR = M n T( = Cosoc (T)), and since
c(Cosoc(T)) c eg = g, we conclude that
^•a) c(T) Ф c(Cosoc(T)).
Thus by A, 14.16 the subgroup S is generated by single-headed subnormal subgroups
6. Frattini duals and Fitting classes gj-y
crond^nlSXrui\ofd “ПСе Pr°Per,y 2 °f (63) C'early imP'ies РгоРег‘У >• ‘he
conclusion of Part (b) of our proposition now follows from Part (a). n
N®xt we ^ePlne a characteristic subgroup (associated with a closure operation c)
which will play the role of a dual of the Fitting subgroup. ’
(6.5) Definition Let c be a closure operation with Property 2 of (6.3). Let G be a
fim e group and let Z<(G) denote the set of all single-headed subnormal subgroups
S of G for which c(S) * c(Cosoc(S)). Then we define
Rc(G) = <Cosoc(S): S e .^(G)),
noting that RC(G) is obviously a characteristic subgroup of G.
(6.6) Lemma. Let cbea closure operation with Property 2 of (6.3), and let Gbea group
(a) If 4<C(G) <SsnG, then RC(S} = RC(G).
(b) The quotient TC(G)/RC(G) is a direct product of a nilpotent group with a direct
product of non-abelian simple groups.
(c) If Ч\(С) < S sn G, then
RC(G)< P|{M:M<i-S}.
In particular, R,(G) is a proper subgroup of G when G # 1.
Proof. Assertion (a) follows easily from (6.4) (b) and the definition of Rc( ), and
Assertion (b) from the fact that 'PC(G)/RC(G) is generated by subnormal simple groups.
(Apply the final statement of A, 14.16 and the fact that d0J is a Fitting class if J is a
non-abelian simple group.)
In order to prove Assertion (c) it will suffice, in view of (a), to prove it for S = G.
Let M <1- G, and let Г be a single-headed subnormal subgroup of G. Then either
T < M or Tr>M = Cosoc(T). In any case, we have RC(G) < M, as desired. By taking
S = TC(G), we obtain RC(G) < Cosoc(Tc(G)) < TC(G) < G when TC(G) # 1. □
Lemma 6.6(a) can be viewed as dual to the result: F(G/<b(G]) = F(G)/4>IG), and for
soluble groups, Lemma 6.6(c) may be regarded as the dual of the statement. If G # 1,
then <D(G) < F(G)” (see A, 10.6(c)).
In IV, 5.8 and V, 3.2(e) we proved the following generalization of A, 10.6(d): “If g
is a saturated formation and if F(G)/<b(G) is g-hypercentral in G (or, equivalently, if
F(G)/<b(G) is covered by an g-normalizer of G), then G e g.” The following theorem
may be seen as a dual of this generalization; it also lends further support to the idea
that, in the duality between Fitting classes and saturated formations, the radical
corresponds to the conjugacy class of normalizers.
(6.7) Theorem. Let cbea closure operation which satisfies Property 2 of (6.3), and let
g = cg If G is a group for which TC(G) < RC(G)G8, then GeS-
828
XI. Fitting classes—their behaviour as classes of groups
Proof. We argue by induction on |G|. If Тс(С) < G, we have
TC(TC(G)) = TC(G) = Rc(G)Gg n TC(G)
= Rc(G)(GgnTr(G))
= Rc(T'c(G))('Pc(G))g
by (6.2) (b) and (6.6) (a), and therefore by induction TC(G) e g. But then G e c(G) £
cfF^G)) £ eg = g, as required. Hence we can suppose that 'f'c(G) = G. If G~ < G,
we can find a maximal normal subgroup M of G containing Gs, and since G # 1, by
(6.6)(c) we then obtain
»PC(G) < GgRc(G) < M < G = S-JG).
This contradiction proves that G = Gg e g. □
(6.8) Proposition. Let c be a closure operation which satisfies Properties 2 and 3 of
(6.3), let g = eg, and let G be a group. Then
TC(G/Gg) < Tc(G)Gg/Gg.
Proof. Let S/G% be a single-headed subnormal subgroup of G/Gg, let K/G^ =
Cosoc(S/Gg), and suppose that c(S/Gg) Ф c(K/Gs); by (6.4)(b) such subgroups
generate TC(G/Gg). Let L be a minimal subnormal supplement to Gg in S. Then by
A, 14.15 the subgroup L is single-headed and L/Ls S S/G^, furthermore, if M =
Cosoc(L), then M/L^ corresponds to K/G-^ under this isomorphism. We aim to show
that c(L) Ф c(M\ for then the conclusion of the proposition will follow easily by
(6.4) (b) once again.
By way of contradiction, suppose that c(L) = c(Af). Since the class g О c(Af/Lg) is
c-closed by hypothesis and contains M, it contains c(M) = c(L) and hence also
contains L. But then L/Lg e c(M/Lg), and consequently c(L/Lg) = c(M/Lg). Since
L/Lg s S/Gg and M/L^ = K/Gg, we conclude that c(S/Gg) = c(K/Gg), contrary to
supposition. □
We now prove a structural restriction on G/TJG) for certain closure operations c.
(6.9) Theorem (Doerk and Hauck, [1]). Let c be a closure operation which satisfies
Properties 2, 3,4 and 5 of (6.3). Then G/T'JG) is soluble for all finite groups G.
Proof. Suppose that the theorem is false, and let G be a counterexample of minimal
order. Let N be a minimal normal subgroup of G, and let g = c(/V). Then Gg # 1,
and so G/Gs is not a counterexample; thus (G/Gg)/TC(G/Gg) e S. But then by (6.8)
we have G/4'c(G)Gge S, and if g £ <S, the desired conclusion follows. Because c
satisfies Property 4 of (6.5), we conclude that N is non-abelian and hence, because of
Property 5, that g = <s„ N„>(X), where X is a composition factor of N. Therefore
6. Frattini duals and Fitting classes
829
213’ a.nd conse4uentb G„ < Soc(G). But Soc(G) < 4-c(G), for if not,
Property^ that"1 ПОГта1 Subgr°Up M with M * ^(G), and it follows by
rivjjciiy *+ 11 Id I J
(1) = c(l) = C(M n TC(G)) = c(M) 2 (M),
a contradiction. Thus GB < TC(G), and therefore G/TC(G) e S. П
Remarks, (a) The five properties of (6.3) are all satisfied by the closure operation c =
<s„, n0>, and so our results about the c-Frattini dual are applicable to T* =
No>* By (6.4) (b) the subgroup T*(G) of a finite group G is generated by all the
subnormal subgroups of G which are boundary groups for some Fitting class.
(b) The conclusion of (6.9) that G/T*(G) is always soluble can not be improved
upon, for Cossey and Ormerod have shown in [1] that for each n e there exists a
soluble group G of nilpotent length n + 1 such that G/4/*(G) has nilpotent length n.
(c) For a soluble group G the usual Frattini dual 'P(G) is always contained in
T*(G), but for insoluble G this need not be the case (see Exercise 4(a) below). Further-
more, the operator T* does not respect direct products (see Exercise 4(b) below).
We devote the rest of this section to a closure operation, associated with 'Pt, which
is dual to еф.
(6.10) Definition. Given a closure operation c and a class JE of groups, we define
e'Pc(JE) = (G: 3 К sn G with Tt (G) < К e JE).
For notational convenience we will write ec as an abbreviation for еТс.
(6.11) Lemma. The class map ec is a closure operation.
Proof. If JE and T) are classes of groups with JE £ ?), then it is obvious that -t £
ecJE £ ec9). It only remains to show that ec is idempotent. Let G e (ec)2£. By definition
G has a subnormal subgroup К with T'C(G) < Ke ec(£). By (6.2) (b) we have TJK) =
TC(G), and since К e ec(JE), there exists a subnormal subgroup L of К with T, (G) =
TC(K) < L e JE. Because LsnG, we conclude that G e ecJE, and therefore (ec)2£ £
ecJE. Since the reverse inclusion is clear, ec is idempotent. □
From now on we focus our attention on Enclosed Fitting classes and confine our-
selves to the universe S and to closure operations c for which s„ < c. In this context
we have the following natural question.
Question 1. Given a closure operation c such that s„ < c in the universe S, which
Fitting classes are En closed?
We begin by establishing that this is equivalent to a question which we have already
investigated in Section 5 of this chapter, namely.
830
XI. Fitting classes—their behaviour as classes of groups
Question 2. Which Fitting classes 8 have their boundaries inside a given class of
single-headed groups?
(6.12) Definition. Let c be a closure operation. A soluble group G is called a c-
boundary group if G is single-headed and c(G) / c(Cosoc(G)). We denote the class of
c-boundary groups by b(c). (Evidently, if G e b(c] and s„ < c, then G = >FC(G) by
(6.4(b).)
The next two lemmas show, in particular, just how Questions 1 and 2 are related.
(6.13) Lemma. Let c be a closure operation for which s„ < c, and let ft be a Fitting
class cf soluble groups.
(a) 5 is ec-closed if and only if b(8) £ b(c).
(b) If 8 = eg, then 8 = ec8-
Proof, (a) First suppose that 8 = ec8- Let G e b(8), and set M = Gs. If the classes
c(M) and c(G) were equal, we should have 'FC(G) < M e 8 and therefore G e ec8 =
8, against G 8- Therefore c(M) / c(G), and G e b(c).
Conversely, suppose that b(8) — b(c) and, by way of contradiction, that 8 # ec8-
Let G be a group of minimal order in ec8\8- Then G e b(8). and since G therefore
belongs to b(c), we have G = 4^(0) by (6.4) (b). But the fact that G e ec8 now implies
that G e 8 by definition of ec. This contradiction proves that 8 = Ec8-
(b) Assume that 8 = c8, and let G e b(8)- If the classes c(Gs) and c(G) were
equal, we could deduce that G e c(G) = c(Gs) £ c8 = 8, against G £ 8- Therefore
G e 6(c), and hy Part (a) it follows that 8 = Ec8- □
(6.14) Lemma. For each class ® of single-headed soluble groups there exists a closure
operation c such that s„ < c and b(c) = ®.
Proof. If 3£ is any class of groups, we set
cl£ = (G: s„(G) n ® £ s„3£).
It is straightforward to verify that c is a closure operation and that c-closed classes
are s„-closed. If G and H are groups, then c(G) = c(H) if and only if s„(G) n ® =
sn(ff)n ®, and it follows easily that 6(c) = ®. □
In the new setting suggested by the two preceding lemmas, Theorem 5.2, (a) <* (c),
can be reformulated as follows.
(6.15) Theorem. Let Sj(£ S) be a Fitting class. Define a class map c by: c(3£) = S if
3£ § and c(3£) is the smallest Fitting class containing 3t in the Lockett section of §
if E Then c is a closure operation, and the following statements about a Fitting
class 8(£ S) are equivalent:
(a) 8 is Enclosed:
(b) Either 3 = <5ог%е Locksec (§).
f>. Frattini duals and Fitting classes
831
Proof. It is obvious that c is a closure operation with s„ < c and that
b(c) = (J {b(X): I e Locksec(Sj)}.
(6 BHaf aXheIenCe °, St"ntS (a>and <b>of the theorem follows at once from
(6.13) (a) and the equivalence of Statements (a) and (c) in Theorem 5.2. D
Similarly Theorem 5.11 admits the following reformulation.
1ЛЛГ 7 LetC ~ ,<S1 N°’ Q>’an</ let %bea ctass °f finite soluble groups. Then
о — E g if ana only if § is a subgroup-closed Fitting formation.
Proof. Since 6(c) = (J {6(X): I = <s„, n0, q>X}, the conclusion of the theorem fol-
lows at once from (6.13) (a) and (5.11). □
Exercises
1. Let c be a closure operation and G a finite group. We say that a subgroup N of
G has the property nc in G (and write N nc G) if
(i) N S G and
(ii ) c(G/M) £ c(G/MN) for all M < G. Set Фс(С) = <N: NncG>.
Then show that:
(a) C(G) £ C(G/®c(G));
(b) If К < G, then ®C(G)K/K < Ф, (G/K);
(c) If с = Еф, then ®C(G) = ®(G).
2. (Fotheringham [1]). Let Фс be defined as in Exercise 1 with c = <ТФ, q, r0>. Show
that:
(a) For any G e CE, the subgroup ®C(G) is soluble;
(h) If G is soluble, then ®C(G) = ®(G);
(c) For each n e N there exists a finite group G„ such that ®C(G„) has nilpotent
length n.
3. (Doerk and Hauck [1]). Let A be a map with associates with each finite group G
a normal subgroup A(G) such that A(G)K/W < A(G/N)forall N <G. Assume that
if Z is a cyclic group of order p2 for some p e P, then A(Z) < Z. If g = <s„, n0 > g £
S with Char(g) = n, and if A(G) e g always implies that G e g, show that g = S„.
Deduce that the only soluble Enclosed Fitting classes (see Definition 6.10 with
'F(G) the usual Frattini dual) are the classes S,(?r £ P).
4. (Doerk and Hauck [1]). Let c = (s., No)-
(a) If G e to, then Ч'(С) < ^(G), but this fails to hold in general for G e G.
(b) For each GeG, there exists an H e G with ^(G x H) = G x H.
5 (Doerk and Hauck [2]). Define a closure operation c as follows: If X £ <= and
there exists an r e N with r = Min{i: X £ 91'}, set cX = 9F; otherwise set cX = 5.
Then a Fitting class g is Enclosed if and only if there exists a chain of sets of
primes Э л2 S’ • • such that g = (Ji=i
S32
XI. Fitting classes—their behaviour as classes of groups
6. Let G be a finite group, and let 4'°(G) denote the intersection of all normal
subgroups N of G with the property that <s„, n0>G < (s„, n0> (W n U) for all
V < G. Show that:
(a) For all Ge6, the residuals 4/0(G)’1 and G” coincide; in particular,
G V(G) e 91.
(b) If G is soluble and g a Fitting class such that *P°(G) < СЯСЙ. then Geg.
Appendix a
A theorem of Oates and Powell
Our goal is to prove Theorem a.19 below. This is an expanded version of a theorem
of Oates and Powell [1], which appears as Lemma 1.7 in Bryant, Bryce and Hartley
[ 1 ] *, and it is needed in Chapter VII to prove Theorem VII, 1.1, one of the key results
in the proof that the formation generated by a finite soluble group contains only
finitely many subformations. The proof we present avoids the use of the theory of
varieties and is based on ideas of Roger Bryant.
Definitions a.l. Let F^, denote a free group on the free generators
(a) For each к e N let zk denote the endomorphism of F„ uniquely determined by
setting
fizk — f+k
for all i e N. We call tk the k'* translation of F„.
(b) For each к e N let dk denote the unique endomorphism of F„ defined by the
equations
fA = t and
fA = f
for all i e N, i / k. We call 6k the k'* deletion of F„.
Direct calculation shows that
(a.a) ZA =
(a.j?) # = *’ and
(a.y) W =
for alii, jeN, provided that 4 is interpreted as the identity map on F„ when к eZ\M-
^AproofofLernma”!., is given in Hannah Neumann’s book [1] under Theorem 51.37.
«34
Appendix a. A theorem of Oates and Powell
Next we introduce the notion of special commutators in and subsequently use
it to extend the idea to a free product.
Definition a.2. Let Fa denote a free group on generators {f: i e N [. For n e M the
special commutators of length n are defined inductively as follows: fa and fa~l are the
special commutators of length 1. If и and i> are special commutators of length к and
I respectively, with k, I e M and к + I = n, then [u, ст*] is a special commutator of
length n. We will denote the length of a special commutator и by l(u). (Thus, for
example, the special commutators of length 2 are the four elements [/,, /2], [/,,
Ui-’./il, and [/Г1,/,’].)
Lemma a.3. Let ue F„,be a special commutator of length n. Then
(i) u3j = 1 for 1 < i <n, and
(ii) u6i = и for i > n.
Proof We argue by induction on n, noting that the lemma clearly holds when n = 1.
Let n > 1, assume inductively that the result is true for special commutators of length
at most n, and let и he a special commutator of length n + 1. By definition we can
write
« = [f, wrj,
where v is a special commutator of length к < n and w is a special commutator of
length n + 1 — к < n. If 1 < i < k, we have
= [*>, = [1, WTt6,J = 1,
and if 1 < i — к < n + 1 — k, from Equation (a.a) in (a.l) we have
U^f = Itj] = 1.
Finally, for i > n + 1, we have
= [d, wrt] = u,
and the induction step is complete. □
Definition a.4. Let A be a group with subgroups Л,(1 eIs N). The group A is called
a free product of the subgroups At if the following two conditions hold:
(i) A = <Af: i e I);
(ii) Given homomorphisms af from At to a group B, there exists a homomorphism
а: А -ь В such that ал_ = a,- for all i e I.
We will write A = Л,- to denote that A is a free product of its subgroups A,.
It is clear from this definition that a free group on generators {f: i e 1} is a free
product of |/| infinite cyclic subgroups
Appendix a. A theorem of Oates and Powell
835
Definitions «.5. Let A =
(a) Given к e /, we define the k"- deletion Дк to be the unique endomorphism of A
satisfying
= 1 for all ak e Ak, and
ai&k = a, for all at e A, and i g /\{fc}.
(b) A homomorphism a: F, A is called a specialization if there exists a natural
number k(o) and a function Л: {1,..., k(cr)} —»1 such that for free generators
fk,fz,... of F, we have
1 fo e AZ(j) for 1 < i < A(cr), and
fo = 1 for i > A-(cr).
We call the set {A(i): 1 < i < k(o)} the spread of о and denote it by Spr(o). A
specialization furnishes a connection between deletions 6k of the free group Fr, and
deletions Ak of the free product, as the following lemma shows.
Lemma a.6. Let o: F^ -»f]*6/ A, be a specialization, let к g I, and set
I(k) = {i e N : A(i) = k}.
Then
П г. )<7 = сгДх.
.iell» /
where the left-hand product is the identity map on F when l(k) — 0.
Proof. Since a homomorphism of a free group is fully determined by its action
on a set of free generators, it is sufficient to show that for all j G M and к e I we
have
XI П А )сг = ЛсгД*-
Case 1: Suppose that j e I(k), in other words that A(j)-k. Then
fbjo = 1 er = 1, and fo^k = 1 since fo e Aitfl = Ak.
Case 2: Suppose that) * I(k). ТЬепЛ(Птпand f
Л\{ l}. (It follows easily from the definition of a free product that A.riAj—jl
whenever i Fyi }
The concept of a specialization allows us to
commutator to free products.
extend the definition of a special
836 Appendix a. A theorem of Oates and Powell
Definitions a.7. Let A = Р|*6, and let c e A.
(a) The element c is called a special commutator of A if c A 1 and there exists a
special commutator и e and a specialization a: F^ -* A such that
(a.<5) ua = c.
The smallest value of l(u) among such pairs (u, o) is called the length 1(c) of c.
(b) Let c be a special commutator of A of length n. If к e I, we say that c depends
on к if we can choose и and a with к e {Л(1),..., Л(и)}. If J is a finite subset of I, we
say that c depends on J if we can choose и and a with J £ {1(1),..., Л(п)}.
Remark. If и is a special commutator of length n, it involves a subset of fk±l, f2±l,
..., ff1, and a homomorphism <p of such that ffcp = 1 for some i e {1,..., n}
evidently satisfies u<p = 1. Since a special commutator c of A satisfies c 1, for a
specialization a satisfying Equation a.6 of Definition a.7(a) we have k(a) > n, and so
the numbers Л( 1),..., Л(п) are all defined in Definition a.7(b).
Lemma a.8. Let c be a special commutator in the free product A„ and let к e I.
(a) If c depends on k, then cAk = I.
(b) If c does not depend on k, then cAk = c.
(c) Suppose that u* is a special commutator in F,j and that u*a* = c for some
specialization a*. If c depends on J, then J £ {Л*(1),..., Л*(/(и*))}.
(d) If c depends on J and J*, then c depends on J v J*.
Proof. We will use и and a to denote a special commutator and a specialization such
that ua = c.
(a) If c depends on k, by definition we can choose и and a such that к = Л(т) for
some me /(«)}; thus me I(k) in the notation of Lemma a.6. Since u6m = 1 by
Lemma a.3, we have
=
(by Lemma a.6)
(b) If c does not depend on k, then к ф Spr(a) for any specialization a such that
ua = c. Thus I(k) = 0, and once more by Lemma a.6 we have
cAt = uaAk = и
bj ) a = ua = c.
Jellkl J
(c) If c depends on J, by definition we can find a pair (u, <r) with ua = c such that
J £ {z(l),..., A(I(u))}. If the conclusion is false, we can find a к in {A( 1),..., A(l(u))}\
{Л*(1),..., A*(l(u*))}. But then cAk = 1 by Part (a), and applying Part (b) for the
Appendix a. A theorem of Oates and Powell
837
specialization a*, we obtain cA„ = c. But this contradicts the
that c 1, and therefore the conclusion is true.
(d) This follows at once from Part (c).
defining requirement
□
a’9' Let C Ье a Spedal commutator in a free product. In view of Lemma
a.e(d) there is a uniquely determined maximal subset К of I such that c depends on
K. We call this subset the spread of c and denote it by Spr(c).
Lemma a.10. Let c and c* be special commutators in the free product [3* A,. Let
Spr(c) — К and Spr(c*) = K*. Then either [c, c*J = 1 or [c, c*] is a special commuta-
tor with spread К и К*.
Proof. Let c — uo and c* — u*a*, where и and u* are special commutators in the free
group = </•: i e and a and a* are specializations. Let l(u) = к and Hu*) = k*,
so that we can suppose that fa = 1 for i > к and fa* = 1 for i > k*. We define a
new specialization of thus:
fat = fa for i = 1, ...,k,
fa1 = f-ka* for i = к + 1,..., к + к*, and
far =1 for i > к + к*.
Then
[с, C*] = [u<7, U*<7*] = [u<7t,U*Tj<7t] = [u, и*Тц]<г\
and so the element [c, c*], if not 1, is the image under a* of the special commutator
[u, u*rj of length к + к* in F„. Moreover, it is clear that
К и К* = Spr(cr) и Spr(cr*) = Spr(<7f) = Spr([c, c*]). □
Next we show that a special commutator of length n lies in the nth term of the lower
central series of A.
Proposition a.11. Let A = П.*е/ A- be lhe free of A- Then K”{A}
(see Definition A, 7.7) contains all special commutators of A of length n.
Proof. We argue by induction on n, noting that the conclusion obviously holds when
n = 1 ’ Let c be a special commutator of length n > 1, and assume that the proposition
holds for all special commutators of length less than n. Let и be a special commutator
in F, of length n and a a specialization such that ua - с. I hen
и = [t>. wqj
for special commutators t>> w e F„ of length/с and/respectively, where n = it+ /, and
838 Appendix a. A theorem of Oates and Powell
therefore c = ua = [г. нтк]а = [c,, c2], where Cj = va and c2 = м>тк<т. Now a* =
тка is evidently a specialization with k(a*) = k(a) — k, and therefore c, and c2 are
special commutators of lengths at most к and I respectively. By induction ct e К*(Л)
and c2 e К,(Л), whence from A, 7.8(b) we obtain
c = [Cl. C2] e [K*M), К|(Л)] < К(к+0(Л) = К„(Л),
and the induction step is complete. □
Corollary a.12. If c is a special commutator in a free product A and if c depends on J,
then с e К|2|(Л).
Proposition a.13. Let Ak denote the kth deletion for the free product A = J3*Gr At, and
let Dk denote the normal subgroup Ker(Ak) of A. For J S I. let Dj = P|teJ Dk with the
convention that Do = A. Then
(a) Dk = (Ak) for all к e I, and
(b) if 1 / a e Dj, then the element a is a product of special commutators of A which
all depend on J.
Proof, (a) Obviously Ak < Ker(Ak) < A, and therefore <Л^> < Dk. To prove the
reverse inclusion, let 1 / d e Ker(Ak). The element d can be expressed, usually in
many ways, in the form
(a.s) d = a,a2 ... a„,
with each a, in some ЛЛ1). Among all such expressions, let m(d) denote the smallest
value of the number of entries which lie in the subgroup Ak. If m(d) = 0, then dAk = d,
which contradicts our assumption that d 1 and dAk = 1; therefore m(d) is a positive
integer. We will argue that d e <Лк > by induction on m(d).
First suppose that m(d) = 1. Then there exists an expression for d of the form (a.s)
such that a, e Ak and a, ф Ak for all j distinct from i, and so
1 =rfAt = (aiAl[) - (a),Ak) = aI ...a^kal+I ...a„.
Thus, on setting b = ai+I ... a„, we obtain d = b^'afr e (_Ak). Now suppose induct-
ively that m(d) = r > 1 and that, for all d with m(d) < r, it is known that d e <Л(*>.
Let Equation a.s denote a representation of d in which exactly r terms belong to Ak,
and let af be the first of these terms. On setting b = aka2 ... at_iy we obtain b~'db =
a,ai+1 ... anal ... а(_], and then the element c = af'b^db is evidently an element of
Ker(Ak) with 1 < m(c) < m(d) = r. By induction с e <Лк >, and so d = Ь(а,с)Ь 1 also
belongs to <Ak >, as desired. This completes the induction step and hence proves (a).
(b) We argue by induction on r = | J|, and note that the conclusion is obvious when
r = 0. Let r > 1, and suppose that the statement has already been proved for sets J
with I J| < r. Let J = {j(l),..., j(r)}, and set J* = J\{ j(r)}. If 1 / a e Dj, then a
certainly belongs to Dj., and so by induction we can write a = ct ... c„ where for
i — 1,..., t each c, in this product is a special commutator that depends on J*. If all
Appendix a. A theorem of Oates and Powell
ST61118 C‘ a‘SO dEPend °n J(r’’ ‘ЬеП WE °b‘ain ‘he des,red inclusion by Lemma
Next suppose there exists an i e {I,.
not depend on j(r). Since
., t} such that q depends on j(r) but q+1 does
cici+l ~ ci+lCi[Cj, q+1],
and since [q q+1] is either 1 or a special commutator which depends on
J о Wl) - J by Lemma a. 10, without loss of generality we can suppose that
a = q ...qq^ ...e„
where q does not depend on j(r) for i = 1, ..., s but does depend on J(r) for
1 — s + 1. •> t. Then by Parts (a) and (b) of Lemma a.8 and the definition of D,
we have
1 (ci • • • q)Aj(r)(q+i •.. с,)ДЛи — q ... q,
and consequently a = q+I ... q, a product of special commutators which all depend
on the set J* и {j(r)} = J- This completes the induction step and, with it, the proof
of Part (b). □
It will simplify our approach below to use the alternative definition of the repeated
commutator [Я,for subgroups ., Hr of a group G, namely that defined
recursively hy
[H„ H2J = <, h2] : Л, e Я,> and [Я,,..., Яг] = [[//,,..., Hr_J, Я,].
This differs from the meaning of Definition A, 7.5(b), although by A, 7.11 the
two definitions will, in fact, agree in the situations where they arise in what follows.
Lemma a.14. Let r and s be positive integers, and let t = r + s. If Nt, ...., N, are
subgroups of a group normal in their join ..., N,), then
(a-0 [[Nj,..., Nr], [N,+i, . 4+,]1 rUsymto twi«> ’ N,<,].
Proof. We will prove (aX) by induction on s, denoting the product on the right-hand
side of the inequality by P. If s = 1, then [[A\,.... Nr], 4+11 = CM. ЬУ
definition, and the result is clear. Therefore suppose that s > 1, and assume that («X)
is true for smaller values of s. Let g e [[Я,,. -, Nrl> W+i> • • • wr+sll- en g is a
product of elements of the form [u. b] with
ое[Я1,...,ЛГг] and b e [Nr+1, ••• > Nri-sli
furthermore b is a product of elements of the form [c fl with, c e1Я+1>.„ , N.-J
and d e Nr±,. Since the subgroups Nt are normal in their jo n, у , -
840
Appendix a. A theorem of Oates and Powell
element [a, b] is a product of elements of the form
[«, fe, d]] = [[c. d], a]’*,
with a e [Nj,., Nr], c e [Nr+1,.... N, _,], and d e N„ and so it will suffice to show
that each element of the form [[c, d], a]-1 belongs to P. Set
и = [[d-*, a *], c]"1, and
t)= [[fl,C~1],d~1]t‘’,
and observe that [[c, d], fl]~* = uv by the Witt identity (A, 7.2(d)). Again because
the subgroups N,,..., Nt are normal in their join, we have
и e L = [[Nj,..., N„ NJ, [Nr+1,..., Nr+(,_n]],
and
veM = NJ, [Nr+1,..., Nr+,_J], NJ.
But then by induction we have L < P, and also [[Nb ..., NJ, [Nr+1,..., N^-!,]] <
ILf Sym«-1> • • •. ЛГ(,_1(„]. Consequently M < P by A, 7.4(f), whence uve P and
the desired conclusion follows. □
With a view to finding an alternative description of the subgroups Dj defined in
Proposition a. 13, we now take a closer look at the special commutators which depend
on a given index set J.
Lemma a.15. Let A denote the free product fj* Л, of the subgroups {A,: i e /}, and let
J = {7(1),...,](t)} be a non-empty finite subset of I. Then each special commutator
which depends on J lies in the product
p= П [<^1К,>,...,<^„„>].
<rESym(J)
Proof. Let c( / 1) be a special commutator in A. To prove that c lies in P, we argue
by induction on the length 1(c) of c, noting that by Proposition a. 13 (a) we can write
<Л^> = £>,( = Ker(AJ) for all i e I.
If 1(c) = 1, then | J | = 1 and so с e £>Л1) by Lemma a.8(a). Therefore suppose that
1(c) > 1, and that the conclusion of the lemma has already been proved for all special
commutators of smaller length in A. It follows easily from the definition that we can
find special commutators ct and c2 of smaller length than c such that c = [сх, c2], For
i = 1,2 suppose that J, is the largest subset of J on which c; depends; then Jl\jJ2 = J
by Lemma a. 10. If a special commutator depends on a set, then it depends on every
subset, and therefore we can suppose without loss of generality that J, n J2 = 0. If
J,- = J for some i = 1, 2, then c; e P by induction, and in this case c = [c,, c2] e
[P, Л] <P. Consequently we may suppose that f = {y(l),... ,j(r)} and J2 =
{j(r + 1),..., j(t)}, and then by induction we have
Appendix a. A theorem of Oates and Powell 84!
................................w]
П , Dj(,)<,], [Dj(r+1)p,..., DJ(I)p]]
by repeated application of A, 7.4(f). Finally we apply Lemma a.14 to deduce that
each term of the latter product is a subgroup of the product P. ц
Theorem a.16. Let A = fj* f A„ and let J = {j(l),..., j(t)} be a finite subset of I.
Then, in the notation of Proposition ct. 13, we have
DJ= П [w-,w
aeSymft)
Proof. Denote the right-hand product by P. Since Diw < A, for <7 e Sym(t) we have
[®i(io)> • • > Pi(ir)l f) Pitioi — ®j,
i=l
and therefore P < Dj.
On the other hand, by Proposition a. 13(b) each non-identity element a in Dj is a
product of special commutators, all of which depend on J; therefore a e P by Lemma
a. 15, whence Dj < P and equality holds. □
In order to show that elements of a free product A can be expressed in a special form,
we will need some elementary calculations with endomorphisms of A, which depend
on the fact that the deletion endomorphisms A( satisfy the relations Д; A, = A; A, when
i / j and A? = A;.
Lemma a.17. Let A,(i e I) denote the deletion endomorphisms for the free product
A = PJ*e, Ai, and for a e Adefine:a(\ — AJ = a(aAJ l.Let J = {j(l),.... ,y(t)} S I.
(a) For all aeA, the element a(l - AJ(1))... (1 - AJ(r|) belongs to the subgroup Dj
(defined in Proposition a. 13).
(b) Order the set J by: j(l) < j(2) < • • • < j(t), and, for SsJ, write S =
{s(l),..., s(m)} in increasing order. Set As = As(1)... As(m). Then, for all ae A, we
have
a(l - AJ(i>)... (1 - = «( П 7
where the right-hand product must be taken in a certain fixed order, which we will not
need to specify.
Proof, (a) Let b e A and i e I. Since
b(l - AJA,- = hA.fhA,2)1 = 1>
842
Appendix a. A theorem of Oates and Powell
we have b(l — A,) e Ker(A,) = D(. By induction on t, we may suppose that
a(l — AJ(1))...(1 — Ajd-n)e DJX|j(I)|, and can then conclude that
u(l — AJ(|])...(1 — Ajui) 6 Цг\|л<>1 n ®ло = Dj,
since the subgroups Dt are obviously Arinvariant for all i e I.
(b) We proceed again by induction on t, noting that the desired equation obviously
holds whenr = l.Then,withK = {7(1),.. -j(t— 1)}, our induction hypothesis yields:
°(1 — (1 — — Aj(<>)
а П (n'-1,mAT) (1 - AJ(1))
k 0#TsK /
= a П «'-'ЧЛ П
0*TsK \ 0/TsK
= «( П n(-1,'7’ATV П a< 1,171 'Arv{J(n|')fl ‘АДО
Xo/TcK /\0^T£K /
= « п «'“’Ч-
0#S£ J
Since the final product has the desired form, the induction step is complete and
Part(b) of the lemma is proved. □
We can now describe and justify the promised special form for elements of a free
product.
Proposition a. 18. Let A = Aj be the free product of its subgroups Ah and let
J = {7(1), • • •, 7(f)} be a finite subset of I with t > 2. If 1 / a e A, then there exists an
element и e Dj and elements V; = As, where S = Sf runs through the non-empty
subsets of J for i = 1,..., n = 2' — 1, such that
a = uvt
In particular, the element и belongs to the tth term K,(A) of the descending central series
of A.
Proof From Lemma a.l7(b) we obtain
fl = n(l -Aj(1))...(l-AJ(1)) П (e,’1,l’l''As)
0/Sc J
for all a e A. Set и = n(l — AJ(1))... (1 - AJ(„) and p( = 'As for non-empty
subsets S of J, ordered so that the above product is 14 v2... v„. Since и e Dj by Lemma
Appendix a. A theorem of Oates and Powell
843
“sdelr IndX^fma’loh “ V ’ "°n’emp,y subsets-the conclusion of the theorem
is clear, and the final observation that D, s K,(A) is obvious from Theorem a.16.
□
We have now prepared the ground for the proof of the following key theorem, which
is the promised objective of this appendix.
Theorem a.19. (R.M. Bryant-unpublished). Let Lbea subgroup of a finite group G,
let Nk(k > 2) be normal subgroups of G such that G = /V, ... NkL, and assume
that [Nj „,..., Л/ц„] = 1 for all a e Sym(k). For any subset S of К = {1,..., к],
define
Hs = П N,L
ieS
with the convention that H0 — Ly and let
E=\HS,
SsK
the external direct product of all the groups Hs. Then E has a subgroup R such that
(a) the projection map ns: R —> Hs is onto (in other words, R is subdirect in £), and
(b) RnHs=l for all SsK.
Proof. Let A = Nl* -*Nk*L denote the free product of the groups Nk, L
(regarded also as subgroups of A by abuse of notation). Let e;: Nf -» G and el: L -+ G
denote the embedding maps into the group G. By the defining properties of a free
product, there exists a homomorphism e: A -* G such that e|Ni = and elt = e£, and
since G = Nt... NkL, clearly e is an epimorphism.
Recalling from Lemma a. 17(b) the definition of the endomorphism Ar: A -> A for
7s K, we define a map
Д:Л->£ = X Hs
SsK
by taking the S-component (nA)s in the direct product £ to be (пДк«)£ for all a e A
and S £ K; for, with the natural convention that До is the identity map on A, it is
clear that (nAKxS)e 6 Hs- We assert that A is a group-homomorphism. To see this, let
ak, a2 e A and S £ K. Then evidently
= ((ai Oi)^x\s)£
= fol^K\s)£(a2AK\s)£
= (01A)s(a2^)s
= (fol A)(a2A))s.
844 Appendix a. A theorem of Oates and Powell
and since the elements (a,a2)& and (а, Д)(«2 A) agree on all components, they coin-
cide. Thus our assertion is justified.
Let R denote the image ЛД. Since A is a homomorphism, R is a subgroup of £,
and because
it follows that Rns = Hs. Hence R satisfies Condition (a) of the theorem.
Let S £ К and g e R n Hs. To complete the proof, we must show that g = 1. On
the one hand, the fact that g belongs to R means that g = a& for some a e A, and on
the other, its membership of Hs implies that the '/’-components of g are 1 for all T / S.
Thus («Дкхг)е = (аД)г = (д)т = 1, and we have
(a.t/) °Дк\т e Ker(r)
for all Те К with T S. Therefore we shall be done if we can show that
(а.в) a&K.se Ker(r).
Let Dj = < NjA >, the normal closure of Nj in A for i = 1,..., k. By Proposition a. 13 (a)
we have Dt = Кег(Д,), and if DK = Кег(ДД by Theorem a.16 we have
£>к = П [Dla,...,DtJ.
ас Sym(k)
Let Uj = e~l(Nj) < A for i = 1, ..., k. Since e is surjective and Nt < G, we have
Dj <Gj<F\ and therefore
П [Dlo,...,Dto]e< П = 1
aeSym(fc) <7eSym(k)
by hypothesis. Consequently DK < Ker(r).
By Proposition a.18 we can write a in the form a = a&0 = uv, ... v„ with ue DK
(whence uc = l)and
Vj = («Д7..),-1,1Т’1 1
for i = 1, ..., n( = 2k — 1) as T* runs through the non-empty subsets of K. Write
T = K\T*, so that
Vi = («Дк^)'-1^’1-’,
and let Vj denote the term corresponding to T — S. Then u,e = 1 for all 1 < i n with
i # j by (a.q). We distinguish two cases:
Appendix a. A theorem of Oates and Powell
845
Case 1: S K. In this case (оА1: 5)’ 1>IK 51 ' = Vj = Vj21vJl2 I't'1' lov„l ... t^i e
Ker(e) because the elements u. a( =aA0), and i;,(i j) are all in Ker(e).
Case 2: S = K. Here we have a&K S = аЛ0 = a = uvt ... t’„ e Ker(e), since t)j,..., t)„
and и all belong to Ker(c) in this case. Thus, in any case, Condition a.O is satisfied,
and the proof is complete. □
Appendix fi
Frattini extensions
Let G be a finite group whose order is divisible by a prime p. We proved in Theorem
В, 11.8 the existence of a non-zero Fp G-module A which has a Frattini extension by
G. Our aim in this appendix is to give a more detailed account of the theory of Frattini
extensions, first developed by Gaschiitz [4] in 1954. Our treatment follows
Gaschiitz’s original approach and includes a summary of more recent knowledge. We
shall cite some well-known elementary facts about how a group may be represented
as an epimorphic image of a free group, but no homological machinery will be needed.
In order to formulate our results with precision, we first need a few simple concepts
about group extensions.
Definition fl. 1. (a) An extension E of a group К by a group G is a short exact sequence
E: 1 —» K~^-*E~^»G —► 1.
(Because we will usually be interested in the class of such extensions, we shall
sometimes refer to E as a G-extension.) Thus £ is a group with a normal subgroup
p(K) = Ker(e) such that that E/p(K) = G. We sometimes identify К with its image
p(K), in which case p is simply the inclusion map.
If К is an elementary abelian p-group, we call E (and also the group E) a p-
elementary extension, and if p(K) < Ф(Е), we call it a Frattini extension. If E has a
generating set X such that the restriction map ex is injective, we call the extension E
efficient. (Observe that Frattini extensions are obviously efficient.)
(b) Let E (as above) and
E*: 1 —> K* E* X G —> 1
be two G-extensions. A group-homomorphism 6: E -> E* such that
EXI: £ = 06*, and
EX2: Kp6 = К*р*пЕв
Appendix p. Frattini extensions
847
is called a G-extension homomorphism from E to E*. We say that E is equivalent to
E* if there exists an isomorphism в: E -> E* which satisfies EXI and EX2.
The first stage of the development is to establish the existence of an efficient,
p-elementary G-extension which is universal in the sense that the class of its epi-
morphic images includes all efficient, p-elementary G-extensions. We begin by recal-
ling some well-known facts about free groups which we shall need, in particular, the
Schreier Subgroup Theorem. (For an account of this and related material, we refer
the reader to Huppert [5], Chapter I, Section 19; a proof of the Subgroup Theorem
can be found in Huppert and Blackburn [1], Theorem IX, 1.14.)
Let F be a free group with a set Si' of free generators. (Huppert and Blackburn call
SF a “group basis”) Let R be a subgroup of F of finite index n. Then Schreier’s theorem
may be formulated as follows:
"There exists a left transversal SF to RinF with the property that if se SF is written
as a reduced word thus:
s = a„am^i ... a2a1 with each a, e ЗГSF~l,
then Sf also contains the element am_l... a2a2; in particular 1 e Sf. Such a transversal
у is called a Schreier transversal, and it turns out that for each xe9C and se Sf ,if t
denotes the unique element in Sf such that xs e tR, then the element
[i(x. s) = t-1xs
of R equals 1 for exactly n - 1 pairs (x, s)eSEx SF. Furthermore, the
n\SE\ -(n- 1)(= n(|ar| - 1) + 1) non-identity elements [i(x, s) form a set of free
generators for the (free) subgroup R. ’
In particular, we have
Theorem #2 (Schreier [1]). If R is a subgroup of finite index nina free group of rank
r, then R is a free group of rank n(r — 1) + 1.
For our purposes, an important special case of this result is the following: Let G
be a finite group, and let F be a free group of rank |G| - 1 with
^ = {/9:9eG*=G\{l}}
as a set of free generators. Let ф' F - G be the epimorphism determined by the map
Ф-fe^g (9eG*)<
, , . . .... spt V = f 1) и 9 forms a Schreier transversal
and set R = Ker(d). It is clear that the set г/ t x
f f pf R bv definition of R, we have fh) Jgh
to R in F. Since fefr efa R by de of Thus R 1S freeiy
equals 1 if and only if h = 1 (since g f иУ
848
Appendix P- Frattini extensions
generated by the set
and has rank (n — I)2 if | G| = n.
Now let
E: 1 —» 1
be an efficient p-elementary G-extension. By definition of “efficient”, E can be gen-
erated by elements lying in distinct cosets of К in E; thus, without loss of generality,
we can suppose that E = <e9: g e G * >, where ege = g. Now let F be the free group
with free generating set 9- = {/9: g e G *}, and let ф: F -> E be the epimorphism
defined by the requirement that /дф = eg. Then clearly
(fe = Ф,
where ф: F -+ G is the epimorphism defined above.
Let T = Ker(0). Since R = Кф~1 and therefore R/T = K, which is an elementary
abelian p-group, it follows that R'RP < T Let v denote the natural homomorphism
from F to F/R'RP, and for H < F, let El* denote its image Hv ( = HR'Rp/R'Rp).
Since R* = R/R'RP is an elementary abelian p-group, we obtain a p-elementary
G-extension
E*: 1 —► R* F* X G — 1,
where e* is the map sending the element fR’Rp of F* to /ф and is easily seen to be
a well-defined epimorphism. Moreover, the map 0: F* -> E which sends f R'RP to /ф
is evidently also a well-defined epimorphism and satisfies the requirement: fe = e*
and R*0 = K, which ensure that the G-extension E is an epimorphic image of E*.
These results can be summarised as follows.
Proposition ДЗ. Let G be a finite group of order n, and let 9- = {fg: g e G *} be a
free generating set of a free group F. Let ф: F -»G be the unique epimorphism for
which ]дф = g for all g e G *, and let R = Кег(ф). Then R is a free group with free
generating set
Furthermore, if v: F -> F* = F/R'RP denotes the natural homomorphism, then
E*: 1 —> R* -=♦ F* X G — 1
(see above for notation) is an efficient p-elementary G-extension, and every such
extension is an epimorphic image of E*. The set 9)* = .3?v is a basis for R*, and so
|R*| = p’”-1»2.
Appendix p. Frattini extensions
849
Remark. The normal subgroup К of any p-elementary G-extension 1 - К E
an7s n№ fXC LT? Г [₽£-m°dule in the usual ^ce К acts trivially
and since E/K = G, it can also be viewed as an FpG-module by deflation.
Corollary 0.4. In the notation of (0.3), let Q* be a normal subgroup of F* contained
in R , and assume that Q* is complemented in F*, by H* say. Then Q*. regarded as an
FpG-modide, is projective.
Proof. Let P be the projective envelope of the F^G-module Q* (see Definition B, 2.7).
Then by Theorem B, 4.8 the module P has a submodule M < Rad(P) such that
P/M =Q*- Since Q*, and hence P, are FpH*-modules by inflation, we can form the
semidirect product S = [PJ H*, and then we obtain the obvious exact sequence
Eo: 1 —»P © (Я* n R*) —»S —► G —•1,
which is clearly a p-elementary G-extension. Moreover, since M < Rad(P) < <h(S) by
B, 3.14, the inverse image in S of a set of generators of F* under the composite map
S S/M F*
is a generating set for S, and so Eo, like E*, is efficient. Hence by Proposition 0.3
there is an epimorphism from E* onto Eo and, in particular, |S| < |F*|. However
|S| = |M||S/M| = |M||F*|, and therefore |M| = 1. Consequently Q* = P,and hence
Q* is projective. □
The next stage of our exposition is to establish the existence of a Frattini p-elementary
G-extension (necessarily efficient) which is universal among such extensions. This is
easily described.
Proposition 0L5. In the notation of (0.3) let P* be a maximal projective submodule of
the G-module R*. Denote the image of a subgroup X* of F* under the natural
homomorphism from F* to F*/P* by А'ф Then
Еф: ! _ r® _£♦ F® X G —• 1,
(where £ф is the lift of the map e*: F* - G to F*/P’ = F®) is a Frattini p-elementary
G-extension which includes all such G-extensions among its epimorphic images.
Proof. First we recall the observation that all Frattini extensions are efficient Since
the module P* is projective, by B, 2.14 it is complemented in F , by H say. Thus E
is isomorphic with the G-extension
E*: 1 Я* n R* —♦ H*G —» 1,
where e * is the composition
Let E: 1 -K^E-G-
li -> hP* (h e H*l with еф: F*/P* ->• G-
elementary G-extension with К < Ф!Е). By
of the map
1 be any p-
852
Appendix p. Frattini extensions
we have Im(a) <= Ker(/f). But
Dim(Ker(/0) = Dim((pc. FpG) - Dim(Im(/?))
= n(n — 1) — (Dim(FpG) — Dim(Im(y)))
= n(n — 1) — л + 1 = (л — I)2
= Dim(R*) = Dim(lm(a)),
and so finally Im(a) = Ker(/?). □
We are now ready to prove Gaschiitz's important description of the Frattini kernel
R®.
Theorem fi.7. (Gaschiitz [4]). Let Pt denote the projective envelope of the trivial
^„G-module. and let P2 denote the projective envelope of Rad(Pt). Then P2 has a sub-
module M such that P2/M s Rad(Pt), and this module M is isomorphic, as Fp G-module,
with the kernel Ra of the universal frattini, p-elementary G-extension E® described in
Proposition [i.5. (In the language of extension theory, Ra is isomorphic with the kernel
e2 in a minimal projective resolution ...—► P2 —Pt (Fp)c—>0 of the trivial
module.)
Proof. Throughout the proof we shall tacitly use the fact that the notions of “projec-
tive” and “injective” are equivalent for FpG-modules. The key to the proof is the
exact sequence (fl.a) of Lemma /?.6. Since FpG/Ker(y) s (Fp)c, we have FpG =
P, + Ker(y) and P2 n Ker(y) = Rad(P,). Since FpG/Pt is projective, so also is
Ker()j/(P1 n Ker(y)), and so we can write Ker(y) = (Pj n Ker(y)) ф To, where 70 is
projective; furthermore, FpG = Pt ф To. Let S denote the direct sum of |G| — 1
copies of FpG, namely the module that appears as the domain of fl in the sequence
(/(.a). Since /i(S) = Ker(y), we can define the submodule Q = RadfPJ/T1 of S.
Then S/Q s Ker(',j/Rad(P,) s To, which is projective. Therefore S has a submodule
TjsTo) such that S = Q@Tt, and since fl(Q) = Rad(Pj), we have Q/Ker(/J) s
RadfPJ. Therefore we can regard the module P2 in the statement of this
Theorem as the projective envelope of Q/Ker(/i), and from this viewpoint we have
P2 + Ker(P) = Q because Q, like S and T2, is projective. It follows that Q/P2 is also
projective, and consequently
Ker(/J) = (Ker(j9) nP2) ф Qo,
with Q(l projective. If Ker(/J) n P2 had a projective submodule U A 0, it would follow
that P2 = P3 ф U with P3 projective and that P3 + Ker(/J) = Q, a contradiction of
the definition of a projective envelope. Hence Ker(/J) n P2 has no non-zero projective
submodules, and, in consequence, Q<: is a maximal projective submodule of Ker(/J).
Since a: R* -» Ker(/J) is an isomorphism, the module Q* = Qoct ' is a maximal
projective submodule of R*. Hence by (/i.5) (with Q* in the role of P*) we have
Appendix fl. Fraitini extensions
853
R® = R*/Q* s Ker(/?)/Co.
и"о'Ь»,0/c’:"“Vs~
г2/м - у/Кег(Д) = Rad(Pj), we have the desired conclusion. д
fn7w*S' r’p 'n Z С2Ье “ after the proof ofапУ two submodules M
and M* of P2 with P2/M s P2/M* s RadfRj are isomorph.c,
2. It is not difficult to prove that the module R® is indecomposable but we shall
not need this fact.
To complete our second proof of Theorem B, 11.8 it will suffice to prove the
following.
Corollary Д8. If p is a prime dividing the order of a finite group G, then the kernel
R® of the universal Frattini, p-elementary G-extension is non-zero.
Proof Suppose that R® = 0. If P, is the projective envelope of the trivial F„G-
module, we conclude from (/7.7) that Rad(PJ (sP2) is projective. But then (Fj^ s
P1/Rad(P1) is projective, whence Rad(P,) = 0 and Pt coincides with the trivial
module. If G is p-soluble, we know from Theorem B, 6.16 that DimE (PJ = |G|r, the
order of a Sylow p-subgroup of G, and we obtain |G|r = 1, contrary to hypothesis.
Hence, in this case, R® 0. The proof of the Corollary for an arbitrary finite group
follows from a special case of a theorem of Willems [1], which states that
Ker(G on PJ = Op.(G) and hence implies that G = OP(G) when P, = (Fjc. A
proof of Willems’ theorem can be found in Huppert and Blackburn [1], Theorem
VII, 14.6(c). □
We end this appendix with a summary of the known facts about R®, the kernel of
the universal Frattini, p-elementary G-extension, which, from now on, we call the
p-Frattini module of G and denote by Ap(G). The composition factors of AP(G) form
a subset of those of the projective envelope P2 of Rad(P,), vhere P, is the projective
envelope of the trivial module. By block equivalence, the composition factors of AP(G)
belong to the first block and by a theorem of Brauer [1] (see Theorem B, 4.22 in the
p-soluble case) they are centralized by OP- P(G). It follows that Op.(G) <
Ker(G on Ap(G)). .
If м is a simple FpG-module, by (/7.5) there exists a non-splitting extension of M
by G if and only if M is a composition factor (equivalently, direct summand) of the
head of AP(G) (such M are precisely the simple modules with non-vanishing 2-
cohomology) The simple modules with non-vanishing 1-cohomology are precisely
the composition factors of Soc^JG)) (in group-theoretical terms these are the
modules M for which M has more than one conjugacy class of complements in the
semidirect product [M]G). A helpful discussion of these facts can be found mGness
and Schmid [1], where it is shown that
(/?-9)
Ker(G on Soc(?4p(G))) = 0p. P(G).
854
Appendix p. Frattini extensions
There is no analogous result for the head of Ap(G); for example, Griess and Schmid
(loc. cit.) show that when p = 2 and G = Alt(5), then the head of A2(G) is just the
trivial module (F2)c. However, if G is p-soluble, more can be said. Gaschiitz’s
Theorem B, 6.18 tells us in this case that the head of Rad(F\) is isomorphic to the
direct sum of the complemented p-chief factors (including multiplicities) that occur
in a fixed (but arbitrary) chief series of G. Now the head of a projective module for a
group algebra is isomorphic with its socle (this follows from B. 4.9(b)). Therefore,
since Rad(R,), when non-zero, has a unique minimal submodule, namely (Fp)c, it is
straightforward to deduce that the head of Rad(Rj) is isomorphic with the socle of
the kernel of the canonical homomorphism P2 -» Rad(Rj) of its projective envelope
P2. Thus Soc(/4p(G)) is isomorphic with the direct sum of the complemented factors
of a chief series of G.
Again when G is p-soluble, Griess and Schmid have shown (loc. cit.) that Soc(^p(G))
is isomorphic with a summand of the head of Ap(G). From this we have:
(/(.10) If G is p-soluble, then CG(>lp(G)/Rad(>lp(G))) = Op. P(G).
and also
(/J. 11) Every complemented p-chief factor of a p-soluble group G admits a non-splitting
extension by G.
Of course, the example of F2Alt(5), mentioned above, shows that (/(.10) can fail
without the hypothesis that G is p-soluble.
Finally, in this context, we should draw the reader’s attention to work of Cossey,
Kegel, and Kovacs [1]. Here the authors are concerned with the more general
question of which groups К can occur in an extension
1 -» К-» E-» G-»1
with К < <I>(G). Of course, К has to be nilpotent, but the authors show more: namely,
that for a given variety of groups ® £ 91, the requirement Ke Я forces К to be an
epimorphic image of a certain group L whose Sylow p-subgroup is the free group of
rank rp in the variety ®p of p-groups in ®, where p'” = | Ap(G)|. When ® is the variety
of abelian groups of exponent p, we obtain Proposition p.5 as a special case of this result.
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List of Symbols
N set of natural numbers
г ring of integers
Q field of rational numbers
R field of real numbers
C field of complex numbers
P set of all primes
P* set of all natural numbers which
are a power of some prime
Fr field with p elements
multiplicative group of the field К
Ж lattice of all Schunck classes
Ж set of all formations
& lattice of all Schunck D-classes
ЖР 461
S, Q, S„, Rq, No, Do, E, Ez, Еф, P 264
Ep 268
Ro 346
Sp 602
s„, 340
Si 516
U1V 2
и 1 V the module U is isomorphic to a
direct summand of the module V
a||b 84
V<G 1
U<G 8
U <G U is a maximal normal subgroup of the group G
XG 5
Ay В 75
[G]H <U',G> [G, Ш] <UG> 16 48 48 50
X'b.G.X %G,X4alG 63
U p-ne G. 250
LneG 251
U proG 241
L'sn G 7
VG~HIN 60
U f-sn G, U fy-sn G 380
91, 91, 9lc, U S, ф, ф", 3, e 262
'M(p) class of elementary
abelian p-groups
O' 263
i'll
G1
Gz
G,
?)S«
8°®
So®
Sf®
8»
8*
<&ЦеЛ>
8«®
Alt(n)
Autz(H/K)
Aut(G)
n(.t)
B(G)
BLF(f)
b(X)
M*)
CAP
Char(JE)
262
582
263
272
275
538
516
337
566
333
575
681
677
324
426, 569
706
alternating group on n letters
44
5
308,612
118
224
370
283, 612
427, 809
37
263
872 List of Symbols
Chiefp(G) 590 Locksec(X) 682
Corec((7) 19 Locksub(X) 726
Cosoc(G) 274 E„(3.) 567
Cosoc(A/) 134 A(X) 722
Covj(G) 289 .//()!, K) 102
Crits(l) 517 NembfG -> H) 584
P/L) 396 ^i(S) 253
Dih(2n) 70 *'(G) 376
A(4 394 NC(S) 236
Р'(У,л/) 634 575
P.'(X) 630 On(G), O*(G), On„(G), O”°(G) 28
E/G), E/G, A) 389 J2(G), n,(G), 15(G) 31
E(n/p‘) 193 O(V) 78
EJG) 389 43, 43,, 432, 43з 53
F(G) 29 ProjjlG) 288
f 361 230
f* 383 Psi(G) 349
f 407 P(P) 97
FAG) 519 PJKG) 118
Fitset(G) 548 630
Fit(JE) 274 Rad(M) 104
<D(G) 30 RK 108
®.(G) 519 ^) 229
form(X) 272 Set(3E) 584
HP") 195 ^(G) 2
Hall,(G) 216 a(G) 4
H(G) 224 Snemb(G -> H) 584
HR(/, R) 577 S^(G) 230
HS( f, S) 581 Soc(G), Soc„(G) 12
HS'(f,S) 582 Soc.;, (G) 173
197 Soc(M) 104
h(X) 283, 612 Supp(3E) 323
I<AM) 140 <t(G), a(X) 263
Injj(G) 564 Sylp(G) 21
InjHG) 538 Sylfi(G) 315
JG4) 111 Sym(n), Sym(n) 1
Й(Л, d) 585 TrHG) 563
Ker(G on M) 105 У(и, К) 1
KG 101 iF(G) 330
K(G) 224 Z(G) 7
RAG) 24 Zf(G), Zf(G, A) 389
MG) 630 ZAG) 25
K,(X) 573 Z„ cyclic group of order n
LC(f) 322 Zi(G) 389
LF(f) 356 Za(G) 25
LF(F) 361
______________ Index of Subjects
abelian group(s), I
class of, 262
component of socle, 173
hcmocyclic, 14, 39, 191 fl., 7991T.
representations of, 157ff.
simple modules for, 161 fl.
tensor product of, 90
abnormal subgroup, 247
/-abnormal, 3771T.
ft-abnormal, 378ft, 402
X-abnormal, 308
sub-ft-abnormal, 402
absolutely indecomposable module,
151
absolutely
faithful, 184
irreducible, 121,150
action,
coprime, 41-43
/-central, 38711., 577
/-eccentric, 387ff.
/-hypercentral, 3870., 577
/-hypereccentric, 387ff.
of group on chief factor, 44 46, 3340,
3560
scalar, 156
of group on group, 15
active
p-active, 7430.
(ft, p)-active, 7430.
agemo subgroup U(G), 31
algebra,
dimension of, 101
division algebra, 108
extending field of, 120
group algebra, 101
multiplication constants of, 101
over a field, 101
quasi-Frobenius, 99
algebraic closure of field, 1220., 152, 591
amalgamated
centre, 75
factor group, 73
amenable Lockett operation, 8200.
anchor (type), 674
ascending series (see upper series)
atomic lattice, 437,460, 729
automorphism,
central, 661
inner, 5
of group, 5
fl-automorphism, 7
regular, 16
automorphism group, 5
induced on chief factor, 440
of abelian group, 8311.
of cyclic group, 84
of verbal product, 788
regular, 16
avoid section of group, 4
avoidance class,
of Fitting class, 6120
of Schunck class, 308, 31211., 428
В
Baer function, 370, 576, 713
Baer-local formation, 3700
Baer’s theorem, 373
generalization of, 374
balanced map, 90
bar notation, 192
base group of wreath product, 63
basis,
for Schunck class, 306
free, 90
group basis, 847
Z-basis, 91
Berger class, 676, 7500
not Fischer class, 758
Berger’s theorem, 735,7370
bimodule, 91, 101,129
block,
equivalence, 1160
first, 118,136
ideal, 117
membership of, 117
principal (see first)
874
Index of Subjects
boundary,
c-boundary group. 830
imperfect, 810
of Fitting class, 6121T., 691, 806-824
of Schunck class, 281, 285, 427, 806
Q-boundary, 283
the map b, 283, 806
Burnside’s basis theorem, 31
Burnside’s p^-theorem
preparations for, 204-210
proof of, 210-215
CAP
Schunck class, 461, 471 478
subgroup, 37, 38, 256, 260, 385, 395,
411,541, 564, 601
Cartan,
matrix, 116, 118
subalgebra, 279
Carter subgroup, 247, 305, 539
Cartesian product of groups, 10
central factor,
projector of, 328ff.
central product, 74ff., 328ff, 63Iff., 66011.
projectors of, 328ff.
central (/-central etc.),
automorphism, 661
characteristically, 679,697-698
chief factor, 8, 238, 356, 370, 378ff,
40011., 41611.
Frattini length, 7901T.
central series,
lower (descending), 24, 76-77, 837
of central product, 76- 77
upper (ascending), 25, 76 77
central socle (group with), 582-583, 619,
784, 796
centralizer of subset, 19, 10911.
chain of subgroups, 7ff.
ff-chain, 415ff
maximal, 309, 405, 483
stabilizer of, 42, 206
subnormal, 7ff
character,
as class function, 126
of module, 126
of representation, 126, 598
л-special, 598
characteristic,
conjugacy class, 21, 236
of class of groups, 263, 367, 565
of Fitting class, 565
of saturated formation, 367
series, 9
simple, 9
subgroup, 8
subgroup function, 414
characteristically
central, 679, 697-698, 707
hypercentral, 679
chief factor,
central, 8, 238
complemented, 31,54,335.594
composition type of, 370
eccentric, 8
/-centra(//-eccentric, 356, 378ff, 577
$-central/$-eccentric, 400ff.
^-critical, 408
Frattini, 31, 335
induced automorphisms on, 4411., 106,
334ff., 356ff.
p-chief factor, 8, 106, 590, 594
T-central/r-eccentric, 416
rank, 14,483
chief series, 8
class,
conjugacy class, 21
of nilpotency, 25
class function, 126ff
class map, 263
expanding, 264
idempotent, 264
monotonic, 264
the class map p, 264, 285
class of groups,
Berger, 750ff.
c-closed, 264
direct product-closed, 265
extreme, 5181T.
Fischer, 535, 554, 601ff, 734, 751
Gaschiitz class, 289ff
Lockett class, 567, 681, 686, 702, 758,
824
local, 322
locally defined, 322
projective, 289ff.
quotient-closed, 265
saturated, 272
skeletal, 528
subgroup-closed, 265
class product, 263, 566, 618
associativity of, 266
Clifford
correspondence, 143
theorems, 141,145, 153ff., 157,
165
closure operation (s), 263ff.
amenable Lockett operation, 820ff.
behaviour like, 679
Index of Subjects
875
finitary, 265, 501
Fitting classes with extra ones, 783fT.
for local definition, 364
join of, 267
list of standard examples, 264
partial order on, 267
product of, 264
the operation ec , 829
the operation sw, 340
unary, 270,516
cohomology,
group, 56
facts about, 148
1- and 2-cohomology, 853
commutator(s), 22
special, 834
subgroups defined by, 23, 839
commutator subgroup, 22
higher, 22
complements)
basis, 2211T.
conjugacy class of, 56, 59, 853
to subgroup, 4
complemented
abelian residual, 383
chief factor, 31, 54, 335, 591
lattice, 440ff.
minimal normal subgroup, 5311., 100
subgroup, 4
complete
lattice, 426,728
set of irreducible modules, 123
set of Sylow complements, 234
X-complete class, 516
completely reducible
module, 102, 139
Q-module, 11
representation, 106
completion.
(5, p)-completion, 7431Г.
complexion,
of Sylow tower, 359
composite Schunck class, 426
composition factor, 8
Q-composition factor, 7
composition series, 8
Q-composition series, 7
composition type of chief factor, 370
conjugacy of special subgroups, 216, 279,
294ff.
conjugacy class,
notation. 746
of complements, 56, 59, 853
of elements, 19
of subsets/subgroups, 5
conjugate,
Galois conjugate, 1231Г.
module/representation, 140-141
constructive,
^-constructive, 761
coordinate subgroup, 10,712
coprime
action, 41-43
operator group, 41 fl.
core of subgroup, 19
correspondence,
Clifford, 143
coset.
decomposition, 3
double, 137
left/right, 2
cosocle,
of group, 274,457,8081Г.
of module (see radical)
cover,
projective, 98, 115, 152, 8521T.
section of a group, 4
cover-avoidance property, 37,401,461
528,544
characterization by, 401, 544
for saturated formation, 528-534
for Sch unck classes, 461,471-478
covering subgroup, 280ff., 289, 535
critical,
^-critical chief factor. 408
(^-)critical maximal subgroup, 59,
330ff., 402. 515
formation critical. 480
S-critical, 517,525,781
subgroup, 330
cyclic,
group, 2
module, 108
D
D-class, 4531Г-
D-property, 453
D ^-property, 459
Dark’s construction, 602, 630-647, 768
variations of, 647-676, 75811.
Dedekind identity (law), 2
defect.
subnormal, 48
deflation of module, 105
closure under, 589
degree,
of matrix representation, 104
of primitive group, 311
of projective representation, 150
deletion endomorphism, 8331T-
876
Index of Subjects
derived
length. 35
subgroup, 22,63
descending series (see lower series)
detectable,
normally, 638-639
subnormally, 638
dihedral group, 87, 70
dimension,
dimension set, 595ff,
of algebra, 101
direct factor (see coordinate subgroup)
direct product, 9,638, 661,829
external, 10
^-radical of, 680ff., 690
internal, 9
irreducible modules for, 128
natural embedding into, 721ff
restricted, 9, 721, 736
uniquely decomposable, 71, 658
unrestricted, 10,736
direct sum, 10
of orthogonal hyperplanes, 83
directed set, 256, 566, 597
directly (S2-) indecomposable, 11, 648
distinguished set of normal subgroups,
6331T.
division algebra/ring, 108
dominance,
of Hall subgroups, 216, 279
of injectors, 617
of Lockett sections, 694
of nilpotent injectors, 623ff.
of the product of Fitting classes, 619ff.
of projectors, 453ff,
dominant
Fischer class, 684
Fitting class, 612, 617ff, 704, 713
Schunck class (see D-class)
double coset, 137
representatives, 137
double cover (see representation group)
dual,
criterion, 177
of core, 328
of D-property, 617
of Fitting pair, 736
of Fitting product, 566
of formation construction, 575
of formation product, 566
of Lausch group, 343, 735-737
of local formation, 604
of Lockett section, 343, 735-737
of projector, 564
of Schunck class, 535
s- and Q-closure, 763
theory of projectors, 281, 302, 535, 564
dual module, 132,177
dually atomic lattice, 434, 460, 729. 732
E
E-closed class, 264, 456
eccentric (/-eccentric etc.),
chief factor, 356, 3781T., 416ff
efficient extension, 8461T.
elementary abelian p-group, 14, 100, 778
as FpG-module, 106
embedding,
^-embedded normal subgroup, 390
natural embedding, 7211T.
normal/subnormal, 5841Г., 720ff.
theorem for wreath products, 68
empty class, 262
conventions about, 271ff
endomorphism,
G-endomorphism, 8
normal Q-endomorphism, 11
of group, 5, 83311.
Si-endomorphism, 11
envelope,
injective, 100
projective, 98, 136, 152, 852ff
epimorphism,
natural, 6
of group, 5
equivalence relation,
block equivalence, 1161Г.
on Fitting classes, 81211.
equivalent
extensions, 847
matrix representation, 106
permutation representation, 20
Eroc, 328
Euler ф-function, 84,159
expanded form, 78Iff.
exponent of group, 4
exponential notation for conjugacy, 5
extended affine group, 196, 672
extendible module, 147
extension(s),
efficient, 84611.
equivalent, 847
Frattini, 188,846 855
homomorphism of, 874
non-splitting, 853
of field, 120
of group, 846if.
of Hall system, 225
p-elementary, 846ff.
separable, 123
Index of Subjects
877
external
central product, 74ff.
direct product, 10
extraspecial (p-)group, 78IT.
classification, 79
representations of, 1660
extreme class, 518
smallest, 519IT,, 528
F
Я-critical maximal subgroup, 402, 515
Я-normalizer, 236, 400ff., 417,515
factor,
chief, 8
central, 328
factor group, 6
amalgamated, 73
faithful
linear representation, 105, 172ff.
module, 17211
permutation representation, 17
field,
additive group of, 195
algebraic closure, 122IT.
extension, 120
multiplicative group of, 195
perfect, 123
splitting, 35,122, 128
finitary closure operation, 265, 501
finite groups, the class of, 262
Fischer class, 353, 554,601IT., 734, 751
dominant, 684
Fischer set, 554IT.
Fischer (ft-)subgroup, 535,5541T., 600f.
not injector, 631,643ff.
Fitting,
family, 589IT.
generalized Fitting subgroup, 580ff.
length (also height), 519
product, 263,566, 605IT., 618IT.
subgroup F(G), 29, 36, 580
Fitting class, 271,274, 281, 563ff.
Berger class, 676, 7501Г.
boundary, 612IT., 691, 806 824
‘central under’ property, 683
dominant, 612.617ff.
factor, 760
generation of, 274, 548. 659ff„ 799ff,
812IT.
injector-closed, 762IT.
Lausch group of, 72111
Lockett section of, 682IT.
maximal, 732-735
metanilpotent, 783-806
normal, 586ft,606, 617IT., 703IT., 735
normally embedded, 6011F., 764,774
not permutable, 631,643-646
permutable, 60111,621,762
preboundary, 8081T.
product, 263, 566. 6051T., 618ff
radical, 274,281,563ft
Ro-closed, 79611.
repellant (abstoBend), 704
smallest normal, 716IT.
strictly normal, 675,719
strong containment between, 596 614
626ft, 691, 718
strongly normal, 683
subgroup-closed, 775-783,823
Fitting formation, 276, 599, 775-783
not saturated, 5921Г.
subgroup-closed, 775-780,823
Fitting pair, 584IT., 673,720ff„ 737IT.
dual of, 736
extension of, 767-768
kernel of, 585,720ff.
outer, 67311
transfer, 676, 737-738,743IT.
universal, 720IT.
Fitting set, 536IT., 564,633ft, 673
fixed-point-free
action of group, 16
automorphism group, 16
focal subgroup, 61
focal subgroup theorem, 61
form,
equivalent, 80
non-degenerate, 77
quadratic, 77
symplectic, 77
formation(s)
connection' with Schunck classes, 344IT.
finitely generated, 479,672
generation, 272
local, 356ff„ 3861Г.
pair, 736
product, 263,337,362,369, 566
saturated, 353,357IT., 367-384,400-
413,482, 497-515, 524ft
solubly-saturated, 370-375
formation-critical, 480
formation function, 356,374, 577
associated normalizer, 396IT.
full, 324,360ft
integrated, 324,360ff.
Fraktur symbols, 262
Frattini,
argument, 20-21
central Frattini length, 790ff.
chief factor, 31, 335
878
Index of Subjects
Frattini (continued)
closure of a class, 264, 302
dual subgroup 'P(G), 188, 824ff.
extensions, 188,846-855
kernel, 852
length, 789ff.
p-Frattini module, 853
subclass, 720
submodule, 104
Frattini subgroup, 30ff, 42 Iff., 824
of semidirect product, 107
free
basis, 90
cyclic module, 108
generators, 834, 847
group, 833ff, 834, 847, 854
module, 90,115,135
product of groups, 786, 834-845
Frobenius
product, 2
reciprocity, 131
frugality of module, 176
full
formation function, 360ff, 603
local function. 324
functorial property of tensor products,
93
fundamental bijection for Schunck classes
with Schunck bases, 307
with Schunck boundaries, 286
G-set (s),
homomorphism of, 20
isomorphic, 20
regular, 18
right, 17,7381T.
transitive, 17
Galois
conjugate module, 124
conjugate representation, 123
field extension, 123ff.
groups, 35,123,126,195
Gaschiitz,
class, 289ff.
fl-subgroup, 31 Iff.
Gaschiitz-Lubeseder theorem, 368
generalization of, 374
general linear group, 1
stable, 673
generalized.
Fitting subgroup F*(G), 580
hypercentre, 38611., 389
nilpotent group, 580, 624ff, 815
nilpotent injector, 62411.
quaternion group, 87
Sylow fl-subgroup, 315
generating system,
of Fitting class, 81211.
minimal, 814
generation
of Fitting class, 274, 584, 659ff, 799ff,
812ff.
of Fitting set, 548
of group, 2
of Hall system, 221ff
of saturated formation, 479-482, 503
group(s), 1
abelian, 1,262
automorphism of, 5
class of, 262
constructions, 9-10, 15-16, 62-73,
190 203
cyclic, 2
dihedral, 70,87
empty class of, 262, 27Iff.
extended affine, 196, 672
extension, 846ff.
extraspecial, 78ff, 166ff.
factor group, 6
free group, 833ff, 834, 847, 854
general linear, 1
generalized nilpotent, 580, 624ff, 815
group basis, 847
hamiltonian, 261
Hartley group, 1971T.
Lausch group, 343, 721 ff, 735 737, 757
monolithic, 272,672
normally detectable, 638
of operators, 15
fl-group, 6
fl-simple, 7
order of, 1
orthogonal, 78
p-(n-)soluble, 34
perfect, 50
primitive, 52
quotient group, 6
relatively free, 278
single-headed, 50ff.
symmetric, 1
symplectic, 77
subnormally detectable, 638
torsion, 728
with central socle, 582-583. 619. 784,
796
with Sylow tower, 344, 359
group algebra, 94,101-102
Jacobson radical of, 121
Group Theory Symposium, 536
Index of Subjects
879
H
£>-avoided/Sj-covered module. 462
Hall (n-)closed class of groups, 573,774
Hall (n-)subgroup, 216
existence and conjugacy of, 216
for sets of odd primes, 219
insoluble, 220
Hall system (s), 22011.
examples of, 318
extending to overgroup, 225
generating bases for, 22Iff.
reasons for terminology, 233-234
reducing into subgroup, 225
transitive action on, 222
Hall’s characterization
of soluble group, 204, 216-218
Hall-Higman,
version of Theorem B, 169
hamiltonian group, 261
Hardy (see Laurel)
Hartley group, 19711.
head,
of group, 274
of module, 104, 114, 134, 854
higher commutator subgroup, 22
homocyclic abelian (p-)group, 14, 39,
19Iff., 799ff.
homogeneous
component, 103, 141
module, 103, 124, 153ff.
homomorph, 272,290
saturated, 272
homomorphism,
natural, 6
of G-set, 20
of group, 5
of lattice, 376
CLhomomorphism, 7
hyperbolic plane, 77,171
hypercentral,
action, 387ff..577ff
characteristic section, 679
radical section, 5771К, 603, 623
socle section, 581ff, 713-714
hypercentre,
generalized, 38611., 389
f- and ft-hypercentre, 389ff„ 397,
417
of group, 25, 238
1
idempotent class of groups, 456, 705,
716,815
identical relations (see laws)
imperfect boundary, 810ff
indecomposable (fl-)group,
directly (fl-)indecomposable, 11
indecomposable module 11 39-40 114
118,136,151 ’ ’ ’
principal indecomposable, 118
projective indecomposable, 114
independent,
Fitting classes, 761
subnormally, 807
index of subgroup, 3
induced module, 129ff.
connection with tensor product, 135
induction,
of modules, I29ff.
of representations, 130ff
transitivity of, 130
inertia subgroup, 140
inflation of module(s), 105
closure under. 589
injective
envelope, 100
hull, 100
module, 98
injector,
characterization of, 6261T.
dominant, 6171T.
for Fitting class, 281,564
for Fitting set, 538ff, 646
for generalized nilpotent groups. 624ff
for nilpotent groups, 623-624
normally embedded, 601 ff.
of group, 548
p-normally embedded, 607,621
system permutable, 60Iff.
injector-closed Fitting class, 762ff
inner automorphism, 6
induced bj element, 5-6
integrated
formation function, 360ff., 603
local function, 324
internal
central product, 74
direct product, 9
interior,
Sje -interior of class, 516
intertwining matrix, 145
interval lattice, 424
invariant module, 144
irreducible,
complete set of modules, 123
module, 39,102, 121
modules for direct product, 128
Q-module, 11
representation, 106
subgroup, 673
Index of Subjects
880
isometry, 77
isomorphism
class, 262
of G-sets, 20
of groups, 5
of Ь-groups (modules), 7
J
Jacobson radical, 111, 121
join
of Schunck classes, 324-325, 426, 441
of subgroups, 2
Jordan-Holder theorem, 8, 9
strengthened form, 33
К
kernel,
Frattini, 852
of Fitting pair, 585ft, 720ff.
of homomorphism, 6
of module, 105
of principal block, 119
of representation, 105
key section, 630ff, 647ff.
Kronecker product of matrices, 93
Krull-Remak-Schmidt theorem, 12, 125,
638
L
largest local definition, 362,407ff.
lattice,
atomic, 437, 460, 729
complete, 426,728
dually atomic, 434, 460, 729, 732
homomorphism, 376
interval lattice, 424
lattice anti-isomorphism, 737
lattice homomorphism, 376-377
lattice isomorphism, 721,726fT.
modular, 455,728
of Lockett (sub)section, 72611.
of normal subgroups, 376
of Schunck classes, 426, 440-461, 509
subgroup lattice, 2, 232, 252, 301, 303
Laurel (see Hardy)
Lausch group,
dual of, 343, 735 737
of Fitting class, 721ff., 757
of formation, 737
laws in a group, 277, 786
left,
coset, 2
transversal, 3
Lie algebra,
Cartan subalgebra, 279
i methods, 202
length,
central Frattini, 790ff.
derived, 35
Fitting, 358,519fT.
Frattini, 789ft
nilpotent (see Fitting)
of special commutators, 8341Т.
lie over/under, 142
lifting
direct decomposition, 114
hypercentral action, 388
linear representation,
by linear map, 105
of degree one, 158
local class, 322, 334
local definition,
canonical, 361, 603
full, 324, 360fT., 485, 603
integrated, 324, 360ft, 603
largest, 362,407ff.
non-uniqueness of, 359
of formation, 356
smallest, 359, 36Iff., 502
local formation, 188, 281, 356fT., 3861Г.
generation of, 365, 482
X-local formation, 375
local function, 322
locally
central subgroup, 211
conjugate subgroups, 245
defined class, 322
finite variety, 278
pronormal subgroup, 245, 601
Lockett class, 567, 681, 686, 702, 758, 824
Lockett conjecture, 631,675, 683, 687,
716, 761ff.
Lockett section(s)/subsection(s),
boundaries of, 691
lattice isomorphism between, 729ff.
maximal element in, 732
minimal element in, 729
of a Fitting class, 343, 682ft, 707, 712ff.,
726ff, 758-760, 774, 816ft
of a formation, 343,696, 735-737
trivial, 758
Lockett’s star operation, 67711.
for Fitting product, 686ff.
lower star operation, 681ft, 731
lower,
central series, 24, 660, 837ft
nilpotent series, 526fF.
Loewy,
layer, 56,112
second Loewy layer, 136
series, 112
Index of Subjects
881
M
Mackey’s theorem, 137
special cases of, 138ff.
Magnus module, 184, 778
Maschke’s theorem, 38 39
matrix,
Cartan, 116,118
intertwining, 145
unitriangular, 197
maximal,
chain of Schunck classes, 4331T.
p-maximal subgroup, 377
projective submodule, 849
saturated formation, 514
Schunck boundary, 440
Schunck class, 433ff
subgroup, 1
subspace, 157
^-maximal subgroup etc., 281, 288, 538,
564
maximal chains
describing projectors, 308ff.
of subgroups, 309,405, 483
maximal subgroup(s), 1, 57-60,483, 552
and Schunck classes, 306
critical/$j-critical, 59, 402, 416
/-abnormal, 377
/-normal, 377
^-abnormal, 308, 3781T., 402
^-normal, 308,378ff.
inconjugate, 57
partial order on, 58
permuting pair, 57
sub-0--abnormal, 402
Jf-maximal etc., 281,288,538,564
McCann class, 6531Г.
meta-3E groups, 263
metabelian group, 218, 263
metanilpotent group(s), 263, 784-806
Q-closed Fitting classes of, 784-785
minimal
local definition, 359,36Iff., 502
projective resolution, 852
rank function, 484
Schunck classes, 437-438
minimal normal subgroup. 8, 5311.
as simple FpG-module, 106
modular lattice, 455, 728
modular law, 2
module/fi-module, 9. 102
absolutely faithful, 184
absolutely indecomposable, 151
absolutely irreducible, 121,150
bi module, 91
completely reducible (see semisimple)
cyclic (free), 108
deflation of, 105
dual, 132, 177
extendible, 147
extending field of, 120ff.
faithful. 1721Г.
free basis of, 90
free Л-module, 90
head of, 104, 854
induced, 129fT.
indecomposable, 11, 39-40,114 118
136,151 ’ ’
inflation of, 105
injective, 98
invariant, 144
irreducible (see simple)
kernel of, 105
lie over/lie under, 142
Magnus, 184,778
p-Frattini, 853
principal indecomposable, 118, 136
projective, 95ff., 115-116, 849ft
projective indecomposable, 114,152
quotient module, 9
regular, 108,180,851
semisimple, 11,102ff.
simple, 11, 102, 161, 1721T.
socle of, 104
special conditions on, 182fF_
monolithic group (having unique
minimal normal subgroup), 272,672
monomorphism,
of group, 5
Q-monomorphism, 7
multiplicative constants of algebra, 101
multiplicity of composition factor, 134
N
n-maximal Schunck dass, 433,440
Nakayama reciprocity theorems, 131,134,
168
natural basis of group algebra, 108
Noether-Deuring theorem, 128, 170
nilpotency class of group, 25
nilpotent,
generalized nilpotent group, 580ft, 624ft,
815
group(s), 25,262
injectors, 623-624,629
length of group, 358,51911
subgroup of maximal order, 629
nilpotent series,
lower, 526П.
upper, 519ff.
поп-degenerate form. 7711.
882
Index of Subjects
non-generator of group, 30
normal,
embedding, 584ff., 720AF.
/-(5-)normal, 377ff.
Fitting class, 586ff., 606, 617ff., 735
lattice of normal subgroups, 376
product closure, 264, 269, 274ff.
product extensions, 589
projectors, 303
subgroup, 6
subgroup function. 394ff.
X-normal subgroup etc., 308, 377ff., 586
normal closure of subgroup, 22
normalizer,
associated to subgroup function, 394
/-normalizer, 396ff.
^-normalizer, 236, 400fT., 417, 515
^-normalizer, 404ff., 410
of Schunck class, 326
of subset of group, 19
weak normalizer, 496
normally detectable group, 638-639
normally embedded
Fitting class, 601 IT., 764, 774
Schunck class, 461-471
subgroup, 25Iff., 46Iff., 536, 548 556
Q-group etc. (see group etc.)
omega subgroup Q(G), 31
operator,
Kn( ), 573ft
L„( ), 567, 576, 605ft, 615, 620ff., 69IfT,
735
N„( ), 575ft
operator group, 15
coprime, 41fT.
orbit of G-set, 17
orbit-stabilizer theorem, 18, 236
order,
of field element, 156
of group, 1
of group element, 4
orthogonal,
group, 78
hyperplanes, 83
sum, 83, 171
outer Fitting pair, 6731T.
P
p-active etc. (see active etc.)
p-chief factor, 8, 106, 590, 594
p-cover-avoidance property, 4721T., 529
p-elementary extension, 846ft.
p-group, 4
elementary abelian, 14
Priifer, 734
p-Iength, 358
p-maximal subgroup, 377
p-nilpotent group, 44, 118 119, 358
p-normally embedded
injectors, 607, 62Iff.
Schunck class, 461 ff.
subgroup, 250ff., 461
p-prefrattini subgroup, 136, 422-424
p-soluble group, 34, 118, 152, 163, 185,
358, 425,854
p-stably embedded,
Schunck class, 462
subgroup, 462
p-subgroup,
Sylow 21
partial order
on closure operations, 267
on conjugacy classes, 58-59
on saturated formations, 405, 509 516
on Schunck classes, 426-440, 509
perfect (see also л-perfect),
boundary, 808-809
field, 123,151
group, 50, 712,807ft
subnormal subgroup, 51,674
the map h, 283
X-perfect groups, 283
permutable,
abelian subgroups, 218
Fitting class, 60Iff., 621
join of two subgroups, 2, 229
nilpotent subgroups, 218
Schunck class, 471ft
subgroup, 234
permutation representation, 17, 141
faithful, 17, 52
Fitting pairs from, 738ff.
primitive, 52
regular, 18
transitive, 17
persistence of canonical subgroups, 280,
541, 565
л-avoidance class, 428
л-boundary of Schunck class, 427
л-element, 4
л-('^’-)1ас1огаЬ1е module, 598ft
л-group, 4
л-perfect group, 263, 303ft., 429, 672
projectors for the class of, 305, 320
л-separable group, 598
л-separated Schunck class, 428ff.
Index of Subjects
883
tr-socle, 12
groups with central л-socle, 582IT
619, 784, 796
n-soluble group, 34
tr-subgroup, 4, 216
Hall, 216
polyprimitive group, 782
power,
tensor, 1801T.
preboundary of Fitting class, 8070.
precursive subgroup, 397,4140.
preformation, 34811.
prefrattini subgroup, 4220.
generalizations of, 423
p-prefrattini subgroup, 136,4220.
prime,
relevant, 526
primitive class, 306
primitive groups, 52-57,272,2850., 3060,
697
construction of, 106
of types 1, 2, and 3, 53,291
soluble groups, 55, 6490., 697
the class of, 262
primitive saturated formations, 438,
4970, 626, 766, 779, 784, 794
primitive split images of a group, 349
principal
block 118.136
indecomposable module, 118,136
product,
Cartesian, 10
central, 740.
direct, 9
Fitting (class) product, 263,5660, 605,
6860, 710
formation product, 263, 337, 362, 369,
566
free, 786,834-845
of classes of groups, 263
of Schunck classes, 326
semidirect, 16
subdirect, 73
twisted wreath, 71-72
varietal (see verbal)
verbal, 278,7870.
wreath, 62-73
product formula, 3
projection onto component of direct
product, 13, 6480, 843
projective class of groups, 2890.
projective cover (see projective envelope)
projective envelope, 98, 115,152, 8520.
of trivial module, 136
projective indecomposable, 114,152
dimension of, 151
projective module, 950. 115 116 135
8490. ’ ’
indecomposable, 114, 152
smallest, 97
projective representation, 1440.
degree of, 150
irreducible, 144
projective resolution,
minimal, 852
projectors, 281,288,405, 535
characterization of, 309
connections with normalizers, 4080.
construction of, 381
described by subgroup indices, 3100.
dualization of, 281, 302, 535, 564
examples of, 316-320
for local formations, 3750.. 391
generalization of. 300
normal, 303
of central products, 3280.
of well-placed subgroups, 3300.
well-embedded, 301
pronormal subgroup, 2410, 256, 280,541,
564
Priifer p-group, 732-734
pure tensors, 91
Q
quadratic form, 77
quasidihedral group, 87
quasi-Frobenius algebra, 99,115
quasinormal subgroup 234
quasi R0-lemma, 567, 7000., 742
sharpened firm of, 685, 703
quaternion group, 87
quotient,
group, 6
module, 9
of class of groups, 669,706
radical, 343,538
as Frattini subgroup, 107
associated with Fitting set/class, 274,
343, 538, 563, 673, 729, 805
Jacobson, 111
of direct power/product, 6800.
of module, 104
radical section of group 577
subgroup, 760
ramification number, 142
884
Index of Subjects
rank,
absolute, 484,495
of chief factor, 14, 483
of relatively free group, 278
rank function, 484
absolute, 495
minimal, 484
of full characteristic, 484
saturated, 484
reciprocity,
Frobenius, 131
Nakayama, 131. 134, 168
reducer of subgroup, 249
reducibility,
into ft-chain 415ff.
of Hall system, 225ff., 243, 415ff.
of complement basis/Sylow basis, 243
of module/representation. 106
regular
automorphism group, 16
G-set, 18
group of operators, 16
module/representation, 108, 180,851
permutation representation, 18
wreath product, 66ff.
relatively free group, 278
relevant prime, 526
repellent Fitting class, 704
representation(s),
completely reducible. 106
equivalent, 106
faithful, 105, 172ff.
irreducible, 106
linear (degree one), 158
linear (of algebra), 104
linear (of group), 105
matrix, 104-105
modular, 366
permutation, 17
regular, 108
representation group, 696, 737
residual,
associated with Ro-closed class, 272ff.,
337, 391
complemented, 383
of class of groups, 669, 706
residual (Ro-)closure,
of class of groups, 264ff.,
of Fitting class, 567, 7 84ff.
of class of modules, 103
resolution,
minimal projective, 852
restricted direct product, 9, 721, 736
right
coset, 2
G-set, 17
regular G-set/representation, 18
transversal, 3
ring,
division, 108
with a ‘one’, 90
roof map J, 7231T.
S
s-critical group. 517, 525
saturated,
class, 272, 302
formation (see separate entry)
homomorph, 272
rank function, 484
solubly saturated formation, 370-375
X-saturated, 374
saturated formation(s), 188, 280ff., 290,
326ff., 353, 35711, 367-384, 400 413,
482, 497-515, 524ff., 775ff.
characteristic of, 367
generation of, 479-482, 503, 672
maximal, 514
primitive, 438, 497ff.. 626, 766, 784
strong containment between, 509ff.
with (p-)cover-avoidance property,
529ff.
saturation of formation, 502-508
scalar action of element, 156
Schreier subgroup theorem, 4-5, 188,847
weak form of, 4-5
Schreier transversal, 847
Schunck
basis, 306
boundary, 285
Schunck class(es), 272
and maximal subgroups, 306ff.
composite of (see join of)
connections with formations, 344-356
D-class, 453ff.
join of, 324И-, 426ff.
lattice, 426ff.
local/locally defined, 322, 334, 477
normalizer of, 325 326
normally embedded, 461-471
partial order on, 426ff.
product of, 326
strong containment, 325, 426ff., 612
strong inclusion (see strong containment)
that are Fitting classes, 594
Ф-Schunck class, 285
well-embedded, 301
with normal projectors, 303-305
Schunck D-class, 453 460
Schur’s lemma, 11
Index of Subjects
885
Schur-Zassenhaus theorem, 38, 150
section of group, 4
cover/avoid, 4
semidirect product(s),
internal/external, 15-16
isomorphism of, 190
semiiinear transform ation(s), 164,195,485
group of. 196
semisimple/Q-semisimple,
algebra, 112
(Q-) group, 11
(Q-)module, 11,39,102,139
series,
chief, 8
composition, 8
factors, 8
Loewy, 112
lower central, 24, 76-77, 660, 837ff.
lower nilpotent, 52611.
of group, 7
Q-composition series, 7
Q-series, 7
upper central series, 25, 76-77
upper nilpotent series, 519ff.
short exact sequence, 846ff.
sign of permutation, 73911.
simple,
class of finite simple groups, 262, 370
group, 6
module, 102, 108, 161, 172ft
non-abelian simple group. 15. 276,633,
826ft.
Q- simple Q-group, 7
simplicial complex, 424
single-headed
group, 50, 612, 806ft, 816ft 830
subnormal subgroup, 51, 671, 82611.
skeletal class, 528
smallest
extreme class, 519, 528
local definition, 502
socle/n-socle,
abelian component of, 173
groups with central socle, 582 583, 619,
784. 796
of group, 12
of module, 104
of primitive group, 53, 291
socle section, 581,713
solubility,
necessary condition for, 216
sufficient condition for, 217—218
soluble group(s), 34
derived length of. 35
the class of, 262
solubly-saturated formation, 370ff.
special commutators, 834
specialization of free group, 8351T.
splitting extension (see semidireci product)
splitting field, 35,122, 128
spread, 835ft
stabilizer
of module, 140,144
of point, 18.235
of primitive group, 52, 427
of subgroup chain, 421Г., 206, 386ft
stable linear group, 673
standard form, 781ft.
star operation, 677ft
for Fitting product, 710ff.
for wreath product, 697-704
on Fitting class, 677ff.
Steinberg’s theorem, 180
strictly ascending chain
of Schunck class, 433,438
strictly normal Fitting class, 675. 719
strong containment
for Fitting classes, 569, 614,626ft, 691,
718, 735
for saturated formations, 509ft
for Schunck classes, 325,426ft., 612
in Lockett section, 689-690
strong inclusion (see strong containment)
strongly normal Fitting class, 683
strongly pronormal subgroup, 251
subdirect
product, 13,662
subgroup, 73,663
subformations, 479ff.
subgroup(s), 1
abnormal, 247
CAP. 37- 38, 256, 260, 385, 395, 541,
564, 601
Carter, 247.305
characteristic, 8
commutator. 22,63
coordinate, 10
covering, 289ft
defined by commutators, 23,839
derived, 22,63
^-subgroup etc., 262. 537
fr-Untergruppe, 280
Fischer (ft-)subgroup, 535,554ft, 6000-,
Fitting 29,36, 580
Frattini, 30, 42Iff.
focal, 61, 743ft
generalized Fitting, 580ft
generalized Sylow, 315ft
generated by set, 2
Hall, 216ft
886
Index of Subjects
subgroup! s) (continued)
identity, 1
inertia, 140
join of, 2
locally central, 211
locally conjugate, 245
locally pronormal. 245, 256, 601
maximal. 1, 57-60,483, 552
minimal normal 8, 5311.
normal, 6
normally embedded, 251ff., 461 -471,
530, 536, 548-556
Q-admissible, 6
p-maximal, 377
permutable, 234
тг-subgroup 4,216
precursive, 397,414ff.
prefrattini, 422ff.
pronormal, 241ff.. 256, 280, 541, 553,
564
proper, 1
radical, 760
single-headed, 50, 612, 671, 80611.,
8161Г.» 826ff.
strongly pronormal, 251
subdirect, 13,663
subgroup function, 394, 414, 418
subnormal, 7,47-52,541,5631L, 648.674
subnormally embedded, 261
Sylow, 21
system perm utable, 230, 235, 256,401,
471, 483, 601ff.
Thompson, 214
verbal, m
T-maximal etc., 281, 288, 538, 564
subgroup lattice ^(G), 2, 232, 252, 301,
303
sublattice
of Lockett section, 726
of Schunck class lattice, 447,454, 458
of subgroup lattice, 232, 252, 301
submodule, 9
subnormal,
chain, 7
closure, 50
criterion for subnormality, 228
defect, 48,648
distinguished subnormal chain, 49
embedding 584ff., 720ft
/- and «^-subnormal, 380, 412
restriction, 589
subgroup, 7, 47 -52, 541, 5631Г, 648, 674
subnormalizer, 50
ft-subnormalizer. 380
subnormally delectable group, 638
subnormally embedded subgroup, 261
supersoluble group(s), 307, 483ft
class of, 262, 358
supplement to subgroup, 4
support of local function, 323, 396
Sylow
basis, 22211,
p-complement, 216,220
structure, 235
system, 233
Sylow П-subgroup, 3151Г.
Sylow (/’-|subgroup(s), 21
of linear groups, 84-89
well-embedded, 301
Sylow’s theorem, 21, 279
Sylow tower group, 344, 359
complexion of, 359
symmetric
algebra, I IS
group, 1
restricted group, 673
sympletic
form, 77
group, 77fT.
space, 77, 171
system,
Hall, 220ff.
Sylow, 233
system normalizer, 2361T., 6091К
system permutable subgroup, 230, 235,
256, 401, 471, 483,60HT.
T
tensor power, 18011.
tensor product,
associativity of, 93
functorial property of, 93
of abelian groups, 90
of matrices, 93
of modules, 94, 172
Theorem В of Hall and Higman,
non-modular version of, 169
Thompson subgroup, 214
three subgroups lemma, 23
torsion group. 728
trace,
of Fitting class, 537
of linear map/matrix, 93-94, 126
transfer, 60-62
Fitting pair, 676,737-738,743-750
map, 60-62, 746
transitive
action on Hall systems, 222
permutation group, 17
permutation representation, 17
Index of Subjects
887
transitivity of induction. 130
translation in free group, 833ff.
transversal (lefi/right), 3
Schreier, 847
twisted wreath product, 71 72,139
2-step maximal Schunck class, 440
type of primitive group, 53
U
unary closure operation, 270, 216
underlying set, 584ff., 7201T.
unitriangular matrices, 197
universal
class of modules, 462ff., 776
extension, 849ff.
^-Fitting pair, 720ff.
Fp-module property, 462
formation pair, 736
property for tensor products, 91
universe, 282
primitive groups in, 262
unrestricted direct product, 10, 736
upper
central series, 25, 76-77
nilpotent series, 5191T.
V
variety of groups, 277, 786, 854
locally finite, 278, 787
verbal
product, 278, 786fE, 787
subgroup, 277,786
Verlagerung, 60
W
weak
N0-closure, 631,632
normalizer, 496
enclosure, 631,632
wreath product property, 820fT.
Wedderburn’s theorem,
on finite division rings, 108
on semisimple algebras, 112,257
(also Wedderburn-Artin theorem)
well-disposed subgroup function, 418
well-embedded subgroup, 301
well-placed subgroup, 330ff., 340ff., 351ff.
word in free group, 277
wreath product(s), 62-73,69711.
base group of, 63
derived subgroup of, 63
fundamental embedding theorem for,
68
twisted, 71-72,139
weak property, 820ff.
X
X-complete, 516ff.
Index of Names
Akizuki, Y. 177
Alperin, J. L. 50, 240,413,414
Anderson, W. 536, 537, 546, 547, 552 553
554
Arad, Z. 218,629
Baer, R. 50, 53, 209, 280, 281, 370, 373
386,391,392,495
Baldauf, C. 816,824
Barlotti, M. 761
Barnes, D.W. 335
Becker, H.E. 150
Beidleman, J.C. 385, 675, 691, 710,719,
761, 762, 763, 764, 766
Bender, H. 108, 210, 580
Berger, T.R. 589, 599, 600, 675, 683, 738,
747, 755, 757, 761, 767, 768, 769, 784, 786
Bialostoki, A. 629
Blessenohl, D. 303, 305, 324, 325, 328,
344, 508, 572, 579, 584, 585, 619, 621, 623,
624, 625, 628, 629, 703, 705, 707, 708, 709,
714,715, 741, 760
Brandis, A. 136,423
Brandl, R. 278
Brauer, R. 119, 853
Brewster, B. 344,675,719
Brison, O.J. 572, 573, 574,669, 676, 692,
695, 738, 747, 755, 758, 759, 761, 764, 774,
805, 806
Bryant, R.M. 181, 203, 278, 340, 341,479,
481, 482, 672,757.833,843
Bryce, R.A. 139, 278, 340, 341,479,481,
482, 497, 499, 502, 573, 647, 652, 660, 672,
682, 683, 721, 726, 731, 732, 734, 736, 761.
762, 766, 775, 779, 780, 781, 782, 784, 786,
795, 796, 798, 833
Burnside, W. 210, 215
Camina, A.R. 740
Carter, R.W. 85, 89, 225, 235, 239, 240,
279, 305. 383, 400, 404, 405, 409, 411.413,
414,517,522, 523,527, 528
Chambers, G.A. 246, 252, 533, 546, 553
Chames, C. 761
Chillag, D. 629
Chouinard, L.G. 116
Clifford, A.H. 141,145
Cline, E. 426,509,511,513
Cossey, J. 118, 139, 152, 278,497, 499,
502, 503, 573, 589, 590, 594, 598, 599, 600,
647, 652, 660, 675, 682, 683,684.689,697
699, 710, 721, 726, 731, 732, 734, 736, 761,
762, 766, 769, 775, 779, 780, 781, 782, 784
786, 795, 796, 829, 854
Cusack, E. 574, 599
D’Arcy, P. 339, 344, 362, 381. 382,411,
412,509,512,514,515,516,604
Dark. R. 535, 540,556,630,643,647,670,
719
Doerk, K. 261,281,286.343,383,391,
407,414,425,429,430, 431,433,434,438,
440,454, 456,461, 478, 503, 505, 506, 507,
508, 528, 531, 532, 612, 614, 621, 689,693,
695, 696, 735, 810, 816, 819, 820, 822, 823,
824, 825, 828, 831
Erickson, R.P. 281, 300
Fischer, B. 244, 245, 249, 250,253, 254,
256, 260, 281,415,419,424, 517, 522, 523,
527, 528, 535, 536, 537, 539, 541, 554, 556,
564, 600, 601, 609, 623, 624, 630
Fitting, H. 274
Forster, P. 186, 281, 282, 293, 294, 297,
298, 300, 302, 305, 327, 328, 344, 348, 349,
350, 367, 374, 375, 403, 405,425,430,431,
434, 435, 436,437,438,441, 442, 445, 447,
448. 449, 455,457, 458, 459,461,465,467,
469,470,471, 474,477,478, 503
Fong, P. 85,89,118. 152
Fotheringham, G. 188, 190, 831
Frantz, W. 627
Gajendragadkhar, D. 589, 598
890
Index of Names
Gaschiitz, W. 59, 1 18, 136, 177, 188, 189,
242. 280. 281. 288, 290, 295, 299, 303, 305,
306. 310. 312. 313. 315, 316. 321, 327, 328,
337, 346. 356. 357, 368, 420, 422, 423, 535,
536. 537. 539. 540. 541, 564, 566, 577, 581,
585. 630. 703, 705, 707, 708, 709, 711, 712,
736, 825, 846, 852
Gillam. J.D. 401, 424
Green, J. A. 136, 151
Griess, R.L. 853, 854
Gross, F. 218,219.234
Haberl, K.L. 595, 597, 599
Hall, P. 216, 217, 219, 220, 233, 234, 235,
240, 241, 250, 262, 279, 386, 392
Harman, D. 484,488, 490,494,495
Hartley. В. 197, 254, 262, 278, 281, 340,
341, 479, 481, 482, 536, 537, 539, 541, 554,
564, 604, 630, 672, 833
Hauck, P. 424, 572, 573, 574, 575, 576,
638. 675, 685, 686, 687, 688, 691, 694, 695,
697, 699, 700, 701, 702, 703, 704, 708, 710,
719. 761, 762, 763, 764, 766, 783, 816, 820,
822, 823, 824, 825, 828, 831
Hawkes, T.O. 152, 290, 312, 313, 316, 343,
344, 346, 348. 383, 400, 404, 405, 409, 411,
412, 413. 414. 423, 424, 426, 431,436, 437,
440,441.442.461,497, 516, 517, 522, 523,
527, 528, 532. 535, 589, 592, 602, 664, 669,
675, 695, 696, 735, 779, 800
Hawthorn, I. 676, 760
Heineken, H. 484,488,490, 492, 496, 595,
597. 599
Herzfeld, U.C. 375
Higman. G. 383
Hill, R. 136
Huppert, B. 163, 222, 234, 280, 376, 385,
386, 390, 391, 483,484, 494,495
Iranzo, M.J. 624
Isaacs, I.M. 589, 598
Ito, N. 824
Iwasawa, K. 483
Johnson, K. 799, 805
Kanes, C.L. 589, 590, 598, 599
Kattwinkel. U. 344, 350, 352, 353, 355,
356
Kegel, O.H. 218, 228, 232, 335, 495, 854
Kleidman, P.B. 228
Klimovicz, A.A. 423
Kohler, J. 484,494
Kovacs, L.G. 181, 203, 599, 757, 854
Kurzweil, H. 423, 424
Lafuente, J. 303
Laue, H. 579, 584, 623, 624, 625, 629, 683,
705, 706, 714, 715, 717. 718, 719, 737, 740,
741, 747, 757,760,761,799.805
Lausch, H. 721, 722, 724, 726. 731, 737,
747, 757
Lennox, J.C. 234
Lockett, P. 228, 244, 246, 252, 253. 254,
567, 568, 569, 571, 574, 601, 605, 606, 607,
608, 609, 611, 615, 616, 617, 618, 626, 627,
629. 675, 677, 678, 679, 680, 682, 683, 686,’
690, 708, 709, 735, 761
Losey, G.O. 250
Lubeseder, U. 280, 281, 368
Lyons, R. 50
Magnus, W. 184
Makan, A.R. 533, 703, 708
Mann, A. 235, 243, 250,404,623, 629
Matsuyama, H. 212
McCann, B. 647, 652, 660, 672
McLain, D.H. 495
Meyer, K. 315
Michel, J. 536
Moran, S. 786
Nakayama, T. 173
Neumann. H. 277, 278, 786
Neumann, P.M. 342
Oates, S. 278,479, 502, 503, 833
Ore, O. 57, 234
Ormerod, E.A. 594, 647, 652, 660, 829
Pain, G.R. 737, 747, 757
Parker, D. 316
Peng, T.A. 250
Pense, J. 344,456, 673, 674, 675, 719, 783,
809,811,814,815
Perez Monasor, F. 624
Porta, M. 612,614,616,621,689,693
Pennington, E.A. 218
Powell, M.B. 278,479, 833
Prentice, M.J. 536
Rose. J.S. 184, 377
Salomon, E. 301
Scarselli, A. 761
Schacher, M. 494,496
Schaller, K.U. 254, 260, 350, 353, 355
Schmieden, K. 695, 696, 735, 736
Schmid, P. 281, 300, 366, 368, 386, 388,
389. 390, 392, 853, 854
Schnackenberg, F.R. 604
Index of Names
891
Schreier, О. 847
Schunck, J.H. 280, 290, 299, 535
Schur, 1. 696
Seitz, G.M. 494,496
Semetkov, L. 393
Shamash, J. 228,229, 239, 240
Shoda, K. 173,177
Shult, E. 383
Simoneit, V. 736
Steinberg, R. 180
Stonehewer, S.E. 234, 250
Suzuki, M. 209
Sylow, M.L. 279
Thompson, J.G. 207, 220,414
Torres, LM. 695,735
Tucci, S. 761
Vaughan-Lee, M.R. 342
Venske, P. 232, 234,483,495
Ward, M.B. 218
Weir, A. 85
Wielandt, H. 47,49, 217. 218, 219,235,
280, 536, 674
Willems, W. 152,853
Winter, D.L. 82
Wood, G J. 249,260,453,454,455
Wright, C.R.B. 536
Yen, Ti 253. 254. 376
Zappa, G. 588,760
Zmud, E.M. 177
Angewandte Lineare Algebra
Bertram Huppert, Universitat Mainz
1990. VIII, 646 Seiten. 17 x 24 cm. Gebunden ISBN 311012107 7
Inhalt:
Lineare Abbildungen. Vektorraume und lineare Abbildungen • Pblynome Die Jordansche
iNormaiiorm.
EmUichdlmensionale Hilbertraume.Normierte Vektorraume • Algebrennormen und Spektral-
radius • Der Ergodensatz • Endlichdimensionale Hilbertraume • Die adjungierte Abbildung •
Normale, hermitesche und unitare Abbildungen • Positive hermitesche Abbildungen • Eigen-
werte hermitescher und normaler Abbildungen • Konvexe Mengen • Der numerische Werte-
bereich • Zwei Eigenwertabschatzungen • Zum Helmholtzschen Raumproblem.
Lineare Differential-und Differenzengleichungen mii Anwendungen anf Schwingungsprobleme.
Beispiele von linearen Schwingungen • Die Exponentialfunktion von Matrizen • Systeme von
linearen Differentialgleichungen • Lineare Differenzengleichungen • Lineare Schwingungen
ohne Reibung • Lineare Schwingungen mit Reibung.
Nichtnegative Matrizen. Die Satze von Perron und Frobenius • Das Austauschmodell von
Leontieff- Bevolkerungsentwicklung und Leslie-Matrizen Elementare Behandlung stocha-
stischer Matrizen • Irreduzible stochastische Matrizen • Das Mischen von Spielkarten • Lager-
hal tung und Warteschlangen • Prozesse mit absorbierenden Zustanden Mittlere Ubergangs-
zeiten.
Geometrische Algebra und spezielle Relativitatstheorie. Skalarprodukte • Orthosymmetrische
Skalarprodukte - Orthogonale Zerlegungen Isotrope Unterraume und hyperbolische Ebe-
nen • Spiegelungen und Transvektionen Der Satz von Witt • Klassische Vektorraume uber
endlichen Korpem • Normalformen von Isometrien • Ahnlichkeiten • Minkowski-Raum und
Lorentz-Gruppe • Der Isomorphismus в+ SL(2,C)/(-£)- Spezielle Relativitatstheorie
Aus den Besprechungen:
“The essential motivation underlying the book is to treat and emphasize those topics of linear
algebra dealing with its applications, which are not directly related to its geometric interprets-
tion. The book deals also with further extensions of such applications. As a justification to
this perspective of presentation, it is stated that linear algebra had its origin in the develop-
ment of analytic geometry and many of its elementary results have direct geometric interpre-
tation. Notwithstanding, many of its deep-rooted statements such as linear mapping and its
eigenvalues, and normal form are scarcely geometric in nature ...
Topics are mostly directed to subjects in which the linear mapping and its eigenvalues, and
normal form play dominant role from the point of view of their application to various disci-
plines, specially in physics, probability theory, statistics, biometrics, econometrics, environ-
mental sciences, etc.... .
The material presented is quite well-organized and not much background is expected trom
the reader It is a pleasure to read the book. Students and workers, those who have occasion to
apply the principles of linear algebra will not only benefit from reading the book but will be
delighted to learn the underlined deeper mathematical structures of the tools they requen у
use.” Zeniralblatt fiir Mathematik
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de Gniyter • Berlin • New York
Group Theory
Proceedings of the Singapore Group Theory Conference
held at the National University of Singapore, June 8-19,1987
Edited by K. N. Cheng and Y. K. Leong
1989. XVIII, 586 pages. 17x24cm. Cloth ISBN 3 11011366X
Contents:
Workshop Lectures. О. H. Kegel .Four lectures on Sylow theory in locally finite groups D. J. S.
Robinson: Cohomology in infinite group theory.
Invited Lectures. S. I. Adian, A. A. Razborov, N. N. Repin: Upper and lower bounds for
nilpotency classes of Lie algebras with Engel conditions R. D. Blyth, D. J. S. Robinson:
Recent progress on rewritability in groups - W. Feit: Some finite groups with nontrivial centers
which are Galois groups • B. Hartley: Actions on lower central factors of free groups
G. Higman: Some countably free groups N. [to: Automorphism groups of DRADs
A. I. Kostrikin: Invariant lattices in Lie algebras and their automorphism groups В. H. Neu-
mann: Yet more on finite groups with few defining relations - M. Suzuki: Elementary proof of
the simplicity of sporadic groups - J. G. Thompson: Fricke, free groups and SL2
J. G. Thompson: Hecke operators and noncongruence subgroups.
Contributed Papers. B. Amberg, S. Franciosi, F. de Giovanni: Soluble groups which are the
product of a nilpotent and a polycyclic subgroup S'. Bachmuth, H. Y. Mochizuki: The tame
range of automorphism groups and GLn • A. J. Berrick: Universal groups, binate groups and
acyclicity A. K. Bhandari, I. B. S. Passi: Residual solvability of the unit groups of group
algebras - C. J. B. Brookes: Stabilisers of injective modules over nilpotent groups - R. A. Bryce,
J. Cossey, E. A. Ormerod: Fitting classes after Dark • С. M. Campbell, E. E Robertson,
R. M. Thomas: On groups related to Fibonacci groups H. Cardenas, E. Lluis: On the Chem
classes of representations of the symmetric groups • L. P. Comerford, Jr., С. C. Edmunds:
Solutions of equations in free groups Y. Fan: Block covers and module covers of finite groups
- A. M. Gaglione, H. V. Waldinger: A theorem in the commutator calculus - J. P. C. Greenlees:
Topological methods in equivariant cohomology W. Herfort, L. Ribes: Solvable subgroups of
free products of profinite groups - D. L. Johnson: Nomcancellation and nonabelian tensor
squares F Levin. G. Rosenberger: A class of SQ-universal groups J. Moori: Action tables for
the Fischer group F22 M. F. Newman, E. A. O’Brien: A Cayley library for the groups of order
dividing 128 - С. E. Praeger: On octic field extensions and a problem in group theory • B. Renz:
Geometric invariants and HNN-extensions • A. H. Rhemtulla. A. R. Weiss: Groups with
permutable subgroup products E. F. Robertson, С. M. Campbell: Symmetric presentations -
G. Schlichting: On the periodicity of group operations S. C. Shee, H. H. Teh: An application of
groups to the topology design of connection machines W. Shi: A new characterization of the
sporadic simple groups • E. Wang: Equicentralizer subgroups of sporadic simple groups
T Yoshida: Some transfer theorems for finite groups.
Appendices. List of participants List of lectures.
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de Gruyter • Berlin • New York