THE THEORY OF GROUPS

THE THEORY OF GROUPS
HOMER BECHTELL
University of New Hampshire
ADDISON-WESLEY PUBLISHING COMPANY
Reading, Massachusetts • Menlo Park, California • London • Don Mills, Ontario

This book is in the
ADDISON-WESLEY SERIES IN MATHEMATICS
Consulting Editor: lynn h. loom is
Copyright © 1971 by Addison-Wesley Publishing Company, Inc.
Philippines copyright 1971 by Addison-Wesley Publishing Company, Inc.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system,
transmitted, in any form or by any means, electronic, mechanical, photocopying, recording,
otherwise, without the prior written permission of the publisher. Printed in the United States
America. Published simultaneously in Canada. Library of Congress Catalog Card No. 71-1361

To my parents

PREFACE
This book is designed to be used as a one-semester text. It evolved from a set
of lectures that was to bridge the gap between the group theory presented in an
introductory graduate algebra course and a serious pursuit of the subject. I
feel that enough material is presented here for the student to make such a move
comfortably.
The style is moderate and an attempt has been made to develop as many
topics as possible in a "natural" way. Hence, the book should be suitable for
independent study. The overall approach and the methods of proof are varied
but standard, and no novelties are introduced. Students have had little
adjustment in the transition from this treatment to either special topics or
reference works. Exercises are included with the original intent of the book
in mind. For the most part, they are not difficult. In particular, this is true of
the proofs of theorems and corollaries that have been left to the reader to
complete. Most of the notation is standard. Since tensor products are not
introduced in this text, the symbol used for the direct product should not
cause confusion. I found that all of the topics and the association exercises
can be covered in a one-semester program with the possible exception of the
entirety of Chapter 8.
The opening chapter summarizes the rudiments of group theory that are
assumed known at this point. It is the material usually covered in an honors
section of an undergraduate modern algebra course and a few topics that may
have oeen delayed until the first year of graduate study. These possibly
delayed topics are redeveloped here, but of course in more detail. A glance
at the table of contents is enough to indicate the nature of the remaining
portion. The bibliography consists of books rather than articles since this
presentation wasn't intended to be a reference work.
A brief introduction to category theory is found in the Appendix, but not
as an afterthought. Undoubtedly many readers will find this useful as a re-
review. My experience has been that consistent use of the methods of category
theory at this level created unanticipated difficulties in the students' under-
understanding of the material. On the other hand, to apply the methods only to
selected portions formed an inconsistent theme. So the inclusion of this
introduction allows flexibility for an instructor who may prefer to begin with

Preface
it and then to give alternative proofs at his discretion. In particular, one
arrangement would be Appendix, Section 2.1, Chapter 9, and then Section
2.2 through Chapter 8.
An introduction to the cohomology of groups is another omission. The
reasons for this omission are much the same as those mentioned above for
category theory. However, a treatment of this topic following a course based
on the material in this text has been well received.
I thank our students Stephen Bacon, Jiann Jer Chen, Paul Estes, Marshall
Kotzen, Gail Lange, Paul Lepage, and Robert McDonald for their invalu-
invaluable comments and suggestions. Acknowledgement is given also to Pro-
Professors David Burton and Richard Johnson for their helpful suggestions in
the preparation of the manuscript.
Durham, New Hampshire H.B.
April 1971

CONTENTS
Chapter 1 Basic Concepts and Notation ....... 1
Chapter 2 Products, Direct Products, Direct Product with Amalgamated
Subgroup, and Subdirect Products
2.1 Products and direct products ....... 8
2.2 Direct product with amalgamated subgroup . . . .12
2.3 Subdirect products 16
Chapter 3 Splitting Extensions; Semidirect and Wreath Products
3.1 Products of subgroups .......
3.2 Extensions ........
3.3 Splitting extensions ......
3.4 Wreath products .......
20
23
26
29
Chapter 4 Theorems on Splitting; Hall Subgroups
4.1 On a theorem of Dixon .
4.2 Splitting theorems of Gaschiitz
4.3 On Hall Ti-subgroups
4.4 Additional comments .
35
39
43
46
Chapter 5 Nilpotent Groups; the Frattini Subgroup
5.1 Nilpotent groups .....
5.2 The Sylow structure of a nilpotent group .
5.3 The Frattini subgroup ....
5.4 Additional remarks on the Frattini subgroup
49
53
56
62
Chapter 6 The Fitting Subgroup; Supersolvable Groups
6.1 The Fitting subgroup ....
6.2 Supersolvable groups ....
67
69

Contents
Chapter 7 General Extension Theory
7.1 Extensions and factor sets ....... 76
7.2 Equivalent extensions ........ 80
7.3 Extensions of abelian groups ....... 84
7.4 Cyclic extensions ......... 87
7.5 Finite extensions over a cyclic maximal subgroup of prime power
order 90
Chapter 8 The Theory of the Transfer
8.1 The transfer .....
8.2 Burnside's theorem ....
8.3 The theorems of Griin ....
8.4 Some applications of the theorems of Griin
94
97
100
105
Chapter 9 Free Groups and Coproducts
9.1 Free groups ....
9.2 Free products and coproducts in Grp
9.3 Coproducts in Ab
110
115
120
Appendix Some Elements of Category Theory
A.I Categories and functors
A.2 Products and coproducts
Bibliography . . . .
Index of Special Symbols
Index ......
123
127
135
137
141

1
BASIC CONCEPTS AND NOTATION
The fundamentals of elementary set theory and mappings are to be assumed
known. However, the next remarks are for the purpose of unifying the use
of certain concepts and terminology.
A mapping (function), a, from a set A into a set B is a subset of the
cartesian product A x B consisting of couples (a, b) for all a e A and pre-
precisely one element beB for each aeA. It is denoted by a: A —> B and the
image will be denoted by Aa. A will be called the domain and B will be called
the codomain. A given couple (a, b)ea will be denoted by either a: a\-^b
or b = a*.
Two mappings a and /? are said to be equal iff a and fi have the same
domain, the same codomain, and x* = xp for all x in the domain.
If the mapping a is one-to-one it will be called injective, if it is onto it
will be called surjective, and if a satisfies both it will be called bijective.
The composite of two mappings a: A —> B and [5: B -* C will be denoted
by a/? and «a/? = {aaY for all a e A. Clearly, the composite of mappings (appro-
(appropriately described) is associative.
The concept of the commutativity of diagrams is a convenient way to
express the equality of mappings through the use of diagrams. For example,
each of the diagrams here is said to be commutative.
A-^> B
\/
C
y = ap ya = a/?
The collection of all mappings of a set B into a set A is denoted by AB.
The identity mapping on a nonempty set A is the mapping iA for which
^ = a for all <ze A If a: ,4 -» 5 and X is a subset of B, then the inverse
image of X, denoted by X**'1, is the set of all elements aeA for which
a*eX.

Basic Concepts and Notation
For mappings a: A -> B and /?: C -> D, the cartesian product of a and /?,
denoted by a x /?, is the mapping axfi'.AxC^BxD defined by
(a, c)«x ^ = (a", c>).
The projection pr(A) of i x B onto a nonempty set A is the mapping
pr(^4): A x 5 ->• ^4 defined by (a, b)\-^ a for all (a, 6) e ^4 x B. Similarly,
the projection pr(?): i x B-^5 exists for a nonempty set B.
Note that if pr(A), pr(B), pr(C), and pr(D) are the projections of A x B
and C x D and if oc: A -^ C, f}: B -> D9 then the diagram here is commutative.
C ^li?LC x D Pr(D) , D
The cardinality of a set ,4 will be denoted by \A\. The empty set is
denoted by 0.
The expression B \ A will denote the elements belonging to a nonempty
set B that are not contained in a subset A of ?. The notation A a B will be
used whenever B\A is not empty, that is, ,4 is properly contained in B.
A c ? denotes i as a subset of 5.
It is also assumed that the reader has had a background equivalent to
that normally presented in a first-year graduate algebra program or in an hon-
honors section of an undergraduate modern algebra course. The group theory
concepts usually covered, together with the notation to be used in this pre-
presentation, will be summarized in the remaining portion of this chapter.
Two elements x and y in a group G belong to the same left (or right)
coset of a subgroup A in G iff x"*y e A (or xy'1 eA). This defines an equi-
equivalence relation on G referred to as the left (or right) coset decomposition
of G with respect to A. G is the set-theoretical disjoint union of these equiva-
equivalence classes, that is, G = (J/ -V^ (or G = (Jj Axt) for the index set / having
cardinality |/| = (G:^4) of the collection of equivalence classes. If |/| = n
is finite, then this is expressed as G = xtA + ••• + xnA (or G = Axx + ••• +
Axn). No ambiguity will arise with the ( + ) sign since the group operation
will be taken to be multiplicative. (G: A) will be called the index of the sub-
subgroup A in the group G. For subgroups A and B in a group G with B c A,
(G:B) = (G:A)(A:B).
A subgroup of a group G is generated by a nonempty subset M in G if
each element of the subgroup is expressible as a finite product of the elements
and the inverses of the elements in M. It is denoted by <M>. Equivalently,
<M> is the intersection of the subgroups in G that contain M.
The order of a subgroup A in a group G, |v4|, is the cardinality of A
when considered as a set. The order of an element x e G is the cardinality of
<x>. A one-element group, as well as the identity element in a group, will be

Basic Concepts and Notation
denoted by 1. If G is finite and A is a subgroup in G, then \A\ divides \G\.
For subsets A, B in a group G, define AB to be
V6 e ?}, or simply {a6 | a e 4, 6 e B},
where V denotes "for all." If ^4, 5 are also subgroups in G, then AB is a
subgroup in G iff ^4B = BA.
The conjugate of a subset A in a group G by an element g e G is the set
{g ~ xag \aeA}. It is denoted either by g " ^g or ^ The element # " xa^ = ^
is called the conjugate of a by g. Since the operation of conjugation by a fixed
element is a function on the group, the notation coincides with that already
introduced. The meaning will be clear from the context in which it is used.
The set of all g e G for which A9 = A is called the normalizer of A in G and
it is denoted by JVG(A). JfG{A) is a subgroup of G. The set of distinct con-
conjugates of A in G is called the conjugate class of A and the number of elements
in a conjugate class is equal to (G:J^G(A)). The centralizer of A in G, ^G(A),
is the set of all elements # e G for which g~xag = a, ^ae A. ^G(A) is a sub-
subgroup of G. For an element geG, ^G«#» *s denoted by %>G(g). The cardi-
cardinality of the conjugate class of an element g e G equals (G:^G(g)). ^G(G)
is called the center of G and it is usually expressed as Z(G). Conjugation
on G by all the elements of G defines an equivalence relation on G in which
the equivalence classes are the conjugate classes of the elements in G.
Whenever G is finite, \G\ = ?(G: ^g(#))> f°r tne summation taken over
all the equivalence classes. This equation is called the class equation.
If ^VG(A) = G, then A is said to be normal in G (denoted by A <a G). A
subgroup A is normal in G iff each left coset of A is also a right coset. Con-
Consequently, if B is a subgroup and ^4 is a normal subgroup in G, then v4? = BA
is a subgroup in G. The collection of all left (or right) cosets in G with respect
to a normal subgroup A forms a group under coset multiplication. This
group is called the quotient group of G with respect to A. It is expressed as
GjA.
A mapping a: G -» H of groups G and H for which {ab)a = aaZ?a, for all
a, beG, is called a homomorphism of G into H. If a as a mapping is injective,
surjective, or bijective, then a as a homomorphism is called, respectively, a
rnonomorphism, an epimorphism, or an isomorphism. The notation G = H
designates that the groups G and H are isomorphic. A homomorphism of
G -> G is called an endomorphism of G and an isomorphism of G -^ G is
called an automorphism of G. The set of all homomorphisms of G into H is
denoted by Hom(G,H). If aeHom(G,if) and /? e Hom(#, K), then the
mapping given by #a/? = (ga/, V# e G, is a homomorphism of G into K. Let
a, /? e Hom(G, H) and assume that H has a multiplicative operation. The
mapping defined by ga+p = gagp, V^eG, exists, but it is not necessarily a
homomorphism of G -* H. For a homomorphism a: G -> H, the inverse

Basic Concepts and Notation
image of the identity element of H is called the kernel of a and it is denoted
by Ker(a). Ker(a) is the identity element iff a is a monomorphism.
The fundamental isomorphism properties are the following:
1. If A is a normal subgroup of a group G, then there is an epimorphism a
of G onto G/A such that Ker(a) = A. Conversely, if a is a homomorphism
of the group G into a group H, then the image of a, Im(a), is isomorphic to
G/Ker(a).
2. For a normal subgroup A in a group G, there exists a bijection between the
subgroups in GjA and the subgroups in G that contain A. A similar bijection
exists with respect to normal subgroups.
3. If A, B are normal subgroups in a group G and B c= A, then GjA =
(GIB)I(AIB).
4. If B is a subgroup contained in jVg{A) for a subgroup A of a group G,
then ^45 is a subgroup of G and AB/^4 ^ B\A n B.
The set of all automorphisms of a group G, Aut(G), is a group. A sub-
subgroup H in G satisfying Ha = H, for all elements a e Aut(G), is called a charac-
characteristic subgroup of G. A fully invariant subgroup H of G has H* = H for
all elements a e End(G), End(G) denoting the collection of all endomorphisms
of the group G. Each fully invariant subgroup of a group G is characteristic
in G and each characteristic subgroup is normal in G. A characteristic sub-
subgroup of a normal subgroup in G is also normal in G.
The center of a group is a characteristic subgroup.
If G is an abelian group, then End(G) is a ring under the operations
For an element g of a group G, the mapping G -> G9 defined by
xg, VxeG,
is an automorphism of G called an inner automorphism. An automorphism
that is not inner is called outer. The set Inn(G) of all inner automorphisms
of G is a normal subgroup of Aut(G). Inn(G) ^ G/Z(G). The normal (or
invariant) subgroups of a group G are those fixed under Inn(G).
Let A be a normal subgroup of a group G. The mapping ^4 -» ^ defined
by a a -> </, for # e G,,is an automorphism of A, but not necessarily an inner
automorphism of A. Hence conjugation on A by all elements of G induces a
homomorphism of G -> Aut(^4) having %G(A) as a kernel. In general, if B
is a subgroup of G, there is a homomorphism J^dB) -> Aut(B) with the image
isomorphic to JfG(B) / ^GE)-
The subgroup \_A, B~\ for subgroups A and J5 of a group G is the subgroup
generated by the elements
a~1b~1ab = [a, *], VaeA and VbeB.

Basic Concepts and Notation
If A <n G, then \A9 B~] ^ A. If A and B are both normal (characteristic)
subgroups, then [^4, B~\ is a normal (characteristic) subgroup of G. In
particular, the characteristic subgroup [G, G], or G', is called the commuta-
commutator subgroup of G. G / [G, G] is abelian and if G / A is abelian for a subgroup
A<i G, then G' c A
A subgroup ^ is said to be subnormal in G, ^^kj G, if there exists
a finite series
lc Ay c A2 c ... c ,4n = G
such that ^o ^4I + 1 for / =1, ...9n — 1. The series itself is called a normal
series and the ^4i+ x / ^ are called the factors of this series. A group that has
no proper normal subgroups other than the identity element is called simple.
If all the factors of a normal series are simple, then the series is called a
composition series. The Jordan-Holder theorem states that in a finite group,
any two composition series have the same length and there is a bijection
between the two sets of factors in isomorphic pairs.
A group is called solvable if it has a normal series in which each factor
is abelian.
A group G is the direct product of a finite collection of its subgroups
{Al9 ..., An) if Aj<a G forj=l, ...,«, G = AxA2...An9 and
Ai n Ax ... Ai-t Ai+1 ... An = 1 for /=1, ...,/?.
It is denoted by G = Ax ® ••• ® An. Aj is called a direct factor of G.
A periodic group has only elements of finite order and a torsion-free
group has no elements of finite order other than the identity element.
For a prime p, a />-group G is a periodic group in which each element
has order a power of p. A Sylow /^-subgroup in a group G is a subgroup
maximal with respect to the property of being a /?-group. Its fundamental
properties in finite groups are the following:
1. Each /7-subgroup in a group G is contained in a Sylow /?-subgroup of G.
2. All Sylow /^-subgroups are conjugate in the group for each prime p.
3. For each prime/? and Sylow ^-subgroup S in a group G, (G: J^G(S)) =1
mod /?.
Each finite /?-group G is solvable and has Z(G) ^ 1.
A group G has an invariant series if it has a set of normal subgroups
{Al9 ...,An} such that
\=AX c A2 c ... ^An = G.
A nilpotent group G has an invariant series such that ^4/+1 / At c Z(G / At)
for / =1,...,« — 1. Each finite /?-group is nilpotent. A finite group G is
nilpotent iff G is a direct product of its Sylow/^-subgroups for each/? dividing
G; say G = St ® ••• ® Sn, n the number of distinct primes in G. The Sy

Basic Concepts and Notation
are characteristic subgroups of G and Aut(G) ^ Aut(Sr1) ® ••• ® Aut(SJ.
Moreover, each subgroup and each epimorphic image of a nilpotent group
is nilpotent.
Abelian groups are nilpotent. If G is a finite abelian p-group, then G is
expressible as a direct product of cyclic factors G = (a^ ® ••• ® <aM> such
that \ax\ ^ \a2\ ^ ••• ^ |aj and this expression is unique with respect to the
number and the orders of the direct factors. If \at\ = pSi for / = 1, ..., n, then
G is said to be of type (sl9 ...9sn). An elementary abelian p-group is an
abelian /?-group of type A, ..., 1).
The automorphism group of a cyclic group of prime order p is cyclic of
order p — 1. But, in general, the automorphism group of a cyclic group is
abelian. The infinite cyclic group is isomorphic to the additive group of
integers and its automorphism group has order two. A finite cyclic group
of order n is isomorphic to the additive group of integers modulo n and it
will be denoted by Cn. Its automorphism group is isomorphic to the multi-
multiplicative group of units in the ring of integers modulo n.
An elementary abelian p-group can be thought of as an /7-dimensional
vector space over a field of characteristic p. Consequently, the auto-
automorphism group of an elementary abelian /?-group is isomorphic to the full
linear group, GL(n,p).
The collection of all bijections of a finite set of n elements into itself
forms a group called the symmetric group of degree n and it is denoted by Sn.
The basic terminology of permutations is assumed known. The set of all
even permutations forms a subgroup An of index two in Sn. An is referred to
as the alternating group of degree n. If/? ^ 5, then An is a simple group. For
n < 5, Sn is solvable. Cayley's theorem states that each finite group G is
isomorphic to a subgroup of Sn for n = \G\. This statement can be general-
generalized by replacing Sn by 5/? Sj the set of all bijections of a set / having |/| =
\G\.
The generating relations for S3 can be given by
53 = <a,Z> | a3 = b2 = 1, bab = a2}.
Other than S3, S4, and Cn, the following groups of order p3, for a prime /?,
are used frequently as examples of nonabelian groups:
1. For/? > 2,
a) G = (a,b\ap2 = bp =l,b~1ab = ap+1}, and
b) G = (a,b,c | ap = bp = cp = 1, ac = ca, be = cb, c = [a,b]>.
2. For p = 2,
a) G = <tf,6 | a4 = 1, b2 = a2, b~~1ab = a3}, the quaternion group, and
b) G = (a,b | a4 = b2 = 1, bab = a3}, the dihedral group.

Basic Concepts and Notation
The additive group of integers, Z, the additive group of rational numbers,
Q, and the multiplicative group of pnth roots of unity for all integers n and
a fixed prime p, Cp00, are examples of infinite abelian groups that are also
assumed to be familiar to the reader.
There may be some omissions in this review and hopefully they will be
few. However, they will be found in any standard advanced undergraduate
or first-year graduate algebra texts along with a more detailed description
of the topics already mentioned. A few of these topics will be redeveloped
in this text in order to examine them in greater detail.

2
PRODUCTS, DIRECT PRODUCTS,
DIRECT PRODUCT WITH AMALGAMATED
SUBGROUP, AND SUBDIRECT PRODUCTS
2.1 PRODUCTS AND DIRECT PRODUCTS
Consider the collection Grp of all groups together with the collection of all
sets of homomorphisms Hom(A, B) of the group A into the group B for each
pair of groups ,4,2? e Grp. Hom(A, B) is never empty since the homomor-
phism A -> \B always exists for 1B, the identity element of B.
Let {Gi}j denote a subset of Grp having index set / equal to the cardi-
cardinality of the subset. Gt is then just the image of / in a mapping of / -» Grp.
Denote by G = X7 Gt the cartesian product of {Gjj, that is,
G = {{*;}/ | xt e Gh ie I}.
Define an equivalence relation ( = ) on G by {xjj = {>'J7 iff xt = yt for each
/ g /. Furthermore, define a binary operation (•) on G by
Then set 1G = {1/}/, for lt the identity element of Gh and {x^J1 = {xf1}^
It is easily verified that the set G together with the relation ( = ) and the opera-
operation (•) is a group. This group will be denoted by
The projection mappings pr(G7): G -> Gj defined by {x^jt-^Xj are epi-
morphisms, that is, pr(G7) e Hom(G, Gy).
As noted above, Hom(H, Gt) is never empty and hence X7 Hom(H, Gt)
exists for each H e Grp, perhaps trivially. Consider an element {aj7 e
Xj Hom(//, Gf) for an arbitrary H e Grp. The mapping r/, defined by
(hn)i = If1 for all /ieH and for each j'eI, is a homomorphism of H -> G.
It is unique with the property of making the accompanying diagram com-
commutative for each ieL
H

2.1 Products and Direct Products
The uniqueness of r\ stems from the fact that if there is some other
\j/ e Hom(H, G) that satisfies the commutativity of the diagram for each
ze/, then Qfi)t = ha\ for heH. This implies that the /-components of hn
and h* are equal for each iel. Hence h'1 = h*, VheH. Therefore rj = \j/.
The product for a collection of groups {G^j in Grp is defined to be the
couple [{pr(Gf)}j; G]. That it may have some type of universal character
is suggested by the previous remarks and this notion will be developed
further in the next paragraph.
Consider a collection {Gj7 in Grp. Each H e Grp can be associated with
the set X7Hom(//, Gt). Suppose KeGrp and </> e Hom(K, H). If
Xi e Hom(/f, Gt), then it follows that 0Af e Hom(iC, Gt). Consequently,
each cj) e Hom(K, H) induces a mapping of
XHom(H, Gt) -+ XHom(iC, Gf).
So we have a two-variable function F that associates a group with a set and a
homomorphism between groups with a mapping between sets, as indicated
in the diagram here.
H—F—+HF = XHom(H, Gt)
i
= XHom(X, Gt)
The question arises: Is it possible to find a group G for which each element
of X7Hom(K, Gt) can be obtained from some one particular element of
Xj Hom(G, Gt) in precisely one way for each K e Grp? The product
G = *Yli G; with its set of projections {pr(Gf)}7 e Xf Hom(G, Gf) provides
the answer. In order to distinguish their roles, G = *Yli Gt is referred to as
the product object and {pr(G/)}/ is called the product element.
Let ^ denote a class consisting of elements
[{aJ/;H], {aJ.eXHomCH, G,),
for which the following property is satisfied:
For each K e Grp and for each {/?J7 e Xj Hom(iC, Gf), there exists a unique
homomorphism 4>: K -> H such that j8f = 0af, fe/.
The class is not empty since [{pr(G;)}7; G] e 0> by our previous comments.
Now it is to be shown the respect in which [_{pv(Gi)}I; G] is unique in 0*.
If [{a J j; H] e 0>, then there exists a unique homomorphism 0 e Hom(G, H)
such that pr(G^) = 0af, V/ e /. On the other hand, there exists a unique

10 Products 2.1
r\ e Hom(H, G) for which af = r\ pr(Gt). Hence
pr(Gi) = 0 (iy pr(G,)) = (« pr(G,), V/ e /.
Moreover, c^rj e Hom(G, G). Consequently, xpr(G|) = (**y <G|), for each
xe G, implies that the /-component of x and x*n is the same for each iel.
So x^n = x, Vx g G, that is <^ = iG. Therefore rj is an isomorphism of
H -> G. Since af = ?y pr(Gt), Vz e /, then rj induces a mapping of {pr(GI)}/ ->
{ajj. Therefore, each [{aJ7;H]e^ is associated with [{pr(Gi)}/; G] by
a unique isomorphism ?| e Hom(H, G). In this sense, [{pr(Gj)}7; G] is
unique.
Whenever / is infinite, the product object in its present form may be too
general a group in order to use it in the study of group structures when apply-
applying flnitary methods. However, there is a subgroup of *!!/ Gf that is quite
useful in this respect. It will coincide with the product object whenever /
is finite. Before considering this subgroup, let us discuss a concept of group
structure defined as follows:
Definition 2.1.1. A group G is a direct product of a set {Gj7 of its subgroups
Gt for an index set / if the following conditions are satisfied:
2. G = ({G^}, and
3. Gt n(Gj \jel\i} =1, for each iel.
The direct product will be denoted by ® 7 Gt and the Gt will be called the
direct factors. If G is abelian, it is denoted by ©7 Gt and called the direct
sum; then the Gt are called direct summands.
Note B) in Definition 2.1.1. This condition implies that each element
of ®j Gt is a product of at most a finite number of elements from the Gt that
are different from the identity element. Now consider G = *Oj Gf and the
collection of subgroups {Gf}t defined by
Gf = {{xi}I\xj=ljJeI\i,xieGi}.
If/ is infinite, then {Gf}j satisfies A) and C) but not necessarily B) unless
all but a finite number of the Gt are one-element groups. Hence, in general,
*FI/ Gt is not necessarily a direct product of {G*}^
If / is finite of cardinality n, there exists a bijection of / with the ordered
set {1,..., n}. So just assume that / has this form and express ®7 Gt as
n
Gx® ••• ® Gn= ®Gt.
i
Theorem 2.1.2. The following two statements are equivalent:
1. G = ®/GI,

2.1 Products and Direct Products 11
2. a) G = <{GJ7> for a collection of subgroups {Gj7 in G.
b) QiQj = gj9h V^j-gGj-, V^eGy, for each pair of distinct elements ijel,
and
c) each element g e G is uniquely expressed, except for order of the factors,
as a product g = gh ••• #ln, for giseGis, and no two factors from the
same Gf.
Proo/. Let G = ®iGt and consider elements gt e Gt and ^ e Gj for / different
from/ Since Gf <i G, then \_gh g~\ e Gh V# e G. Hence [gh g^] e Gt. Similarly,
Ldh 9j] e Gj. Hence [gh gj] e Gt n Gj. By condition C) of the definition,
GJn0i?Gln<Gt|/6/\!>=l.
So [gh gj~] = 1 and this implies that gtgj = g^g^ The remaining portion of
this proof is left as an exercise.
The subset of *fli Gt that consists of all elements in Xz Gt having at most
a finite number of components different from the identity elements of the
G( is easily shown to be a subgroup. This subgroup will be denoted by Yli Gt.
Theorem 2.1.3. Let G be a direct product of its subgroups {GJ7. There is a
unique isomorphism , ,
f?eHom ®G;, \\g\
such that V/ l i
(®iGtyMGi) = Gi9 Vie/.
Moreover,
® ct s *n g,
i i
iff / is finite or all but a finite subset of the Gt have \Gt\ = 1.
Proo/. Exercise.
Consequently, whenever / is finite, the direct product can be considered
both as a structural concept and, isomorphically, as a product object. Since
®G/ = I1 Gi9
i i
we are able to use Yli Gt in order to prove statements about direct products
whenever the properties involved are preserved under isomorphism. In this
sense Yli Gt is a model for ®7 Gt.
Other properties derivable from Definition 2.1.1 and Theorem 2.1.2 are
the following:
Let G = ®f Gt.
1. If H is a normal subset of some G/? then H is a normal subset of G.
2.

12 Products 2.2
3. lG,G-] = ®/[Gl-, GJ.
4. <{//,-}/> = ®jH,-, where Ht is a subgroup of Gt for each /eJ.
5. If Gt = ®Iu Hu, for each iel, then G = ® j i/l7, for J = Uj **./> whenever
/ is finite.
Suppose that H is a subgroup of G = ®7 G, and that each element of
H has the unique expression given by Theorem 2.1.2(c). The set of Gr
components, Hf, is a subgroup of Gt for each / e /. If H o G, then iff <a Gf.
Therefore by A) above, Hf ^ G. Moreover, H = ®/HI* iff Hf c H. So
we have the additional properties:
Let H be a subgroup of G = Gx ® G2.
6. If H* = H n Gu then H2* = H n G2 and H = H? ® H2*.
7. If G! c //, then // = Gx ® (// n G2).
EXERCISES 2.1
1. Consider the group G = Gt IT G2 for Gx = (x\x4 = 1> and G2 = (y\y4 = 1>.
a) Determine the subgroups iff and //f for // = <(x, y2)}. Is it necessarily
valid that H= H*®H*21
b) Consider the subgroup K = <x2, y2)}. Note that ir<a G and so defines an
epimorphism 6: G-+ G/K = G*. Show that G* is not G9X® Ge2. Hence
the image of a direct product under a homomorphism does not necessarily
preserve the direct factors.
2. If G = Gx ® ••• ® Gm then G/ ® G, ^ Gt for / = 1, ..., n.
i*j
3. If iVo G = ®7 Gf, then either JV is abelian or A^ has a nontrivial intersection
with Gt for at least one / e /.
4. Complete the proof of Theorem 2.1.2.
5. Prove Theorem 2.1.3.
6. Prove the properties (l)-E).
7. Prove the conclusions concerning subgroups of direct products that are given in
the last paragraph of this section. Also, show that the conclusions F) and G)
are valid in *Y\i Gt.
8. For a group G, show that (GII G)/Z(G U G) = (G/Z(G)) II {GjZ{G)).
2.2 DIRECT PRODUCT WITH AMALGAMATED SUBGROUP
In this section a generalization of a direct product having two factors is
presented. It involves C) of Definition 2.1.1. Suppose that A and B are
normal subgroups of a group G such that G = AB and A n B = C, but

2.2 Direct Product with Amalgamated Subgroup 13
C^l. Two questions arise. Can we find an a priori condition that would
assure us that GjC = AjC <g> BjCI If we can, then in what respect is this
group GjC unique?
Definition 2.2.1. A group G is the direct product of subgroups A and B with
amalgamated subgroup C if G = AB, A n B = C, and A ? ^G{B). Denote
it by G = (AB)C.
Theorem 2.2.2. If G = (AB)C, then
1. C<= Z(G) and
2. GjC = AjC ® BjC.
Proof. Since C ^ ^G(B), then [C, ?] = 1. Moreover, C ^ ? implies that A
centralizes C. Therefore G = AB = ^G(C), that is Cc Z{G). Clearly,
C^G, GjC = (A/C) {BjC), [A/C, J5/C] = C, and ^/C n B/C = C. Hence
G/C = AjC ® BjC.
Theorem 2.2.2B) answers the first question and the next theorem answers
the second one.
Theorem 2.2.3. Let G = (AB)C and H = (,4*?*)c* for which there are iso-
isomorphisms a: A -> A* and p. B -* B* such that Ca = Cp. Then there exists
an isomorphism y: G -> H such that y\A = a and y|B = p. (y\A is the re-
restriction of y to A.)
Proof. For an operation (•) in G and ( + ) in H, define the mapping
y\ab\-*aa + Z^5 Vaev4, and VZ>e?. Since A c ^G(?) and A* c #H(fl*),
then y is a homomorphism. The definition of the mapping implies that
y\A = a and y\B = p. Moreover, y induces an isomorphism of GjC -> H/C*
by the mapping (ab) C i-» (aa + M) C*. Note first that this mapping is an
epimorphism. Then {<f + #*) C* = C* implies that aa e B*. Hence aa e C*.
Since a is an isomorphism, then aeC. Similarly, be C. Therefore, Ker(y) c C.
However, y|^4 = a implies that c e Ker(y) c yl iff ca = 1H. Since a is an iso-
isomorphism, then c = 1G. Hence Ker(y) = 1.
Before continuing with the discussion of direct products with amal-
amalgamated subgroup, an answer to another question is needed. If there exists
a monomorphism a of a group A onto a subgroup A* in a group G*, then
does there exist a group G that not only contains A, as a set, but also for which
a can be extended to an isomorphism of G onto G*?
Lemma 2.2.4. For an injection a of a set X into a group G, there exists a
group G containing X and an isomorphism p:G-^G such that /?|X = a.
Proof If G is finite, then we shall let S denote a countable set that contains
X. Otherwise, for G infinite, let 5 be a set containing X such that the

14 Products 2.2
cardinality of S is that of the set of subsets of G. Furthermore, assume
that (S \ X) n (G \ Xa) = 0. In either case, there exists an injection
y:G\X«->S\X.
Set G = X u (G\ Xa)y, where in this case u denotes the set-theoretical dis-
disjoint union. Define the mapping /?: G -> G by gp = ga if g eX and gp = gy~l
if g e (G \ Xa)y for all g e G. Clearly, j8 is a bijection and ?|X = a. On G
define the operation ( + ) by gx + g2 = (g/ gi^Y'1- It can be easily verified
that the operation is associative and that (lG)p~l is an identity element. The
inverse of g e G is (G7/*)~1)/*~1. Consequently, G is a group with respect to
this operation. Moreover (gt + g2)p = #/ g2p- Hence ^ is an isomorphism.
Theorem 2.2.5. For a monomorphism a of a group A into a group G*,
there is a group G containing A and an isomorphism /?: G ->¦ G* such that
?14 = a.
Exercise.
A consequence of the isomorphism between G* and G in Theorem 2.2.5
is that one can assume, without any loss of generality, that A is a subgroup
of the group G* whenever there exists an isomorphism of A into G*. This
assumption will be used frequently.
Theorem 2.2.6. Let A, B, and C be groups for which there are monomor-
phisms aA: C -> Z(A) and aB: C -> Z(B). There are a group G and mono-
morphisms fiA\ A -> G and ?B: 5 -^ G such that G = G4^ BPb)cpa.
Proof. By Theorem 2.2.5, there are groups 4 and B in which a^ and ocB
can be extended to isomorphisms aA: A -> ZD) and aB: B -+ Z{B) such that
a^|C = a^ and dB\C = ocB, and having A n B = C. Furthermore, dA and aB
can be extended to isomorphisms a/ and a/ for groups A* ^ ^ and B* ^ B
such that ocA*:A*-+A, a/:B*^5, a/|4 = a^, a/|S = aB, a/|C = a^,
aB*|C = aB, and w;ith ^4* n 5* = C Moreover, ZD) and ZE) are iso-
morphic respectively to Z(A*) and Z(B*). So it will be assumed that
C c Z(i4), C c Z(.B), 4 n B = C, the operations on A and 5 coincide on C,
and a^ = aB = tc. (See the diagram following the proof.)
Form the product object .4 II # an^ consider the set D = {(<?, c~x)\
ceC). Since C ^ Z(A) and C ^ Z(B), then DcZ(ifl4 Hence
D<i[]^ Next, consider G = (Afl B)jD. The mappings
PA:a\-+(a,l)D,VaeA, and pB:b\-*(l, b)D, VbeB,
are easily verified to be monomorphisms of A -> G and B -* G, respectively.
For each ceC, (c, c)/) = D implies that (c,l)D = A, c)D. Therefore

2.2
Direct Product with Amalgamated Subgroup 15
Pa\c = Pb\c- Using this relationship, one can verify that APa = (A fl C)/D,
BPb = (C n B)/D, and C*A = (C n Q/D. Furthermore, G = Afi* BPb,
aPa n BpA = c/u? and AfiA c <gG{B*B). Consequently, G = (APa BPb)cPa.
(The inclusion map is denoted by t.)
Corollary 2.2.6. Let A, B, and C be groups for which C c Z(A) and C is
monomorphic to a subgroup in Z(?). Then there exists a group G and mono-
morphisms a:,4-+Gand?:?-+G such that G = (A*Bp)Ca.
Theorem 2.2.7. Suppose that G = (AB)C, a e Aut(A), j8 e Aut(jB), and a|C =
j8|CeAut(C). There exists yeAut(G) such that y\A = a, y|B = ft, and
(o6)y = «a^ for a e A and b e B.
Proof. Exercise.
Theorem 2.2.8. If G = (AB)C and C =
Proof. Exercise.
= Z(B), then C = Z(G).
Please note: If A, B, and C are subgroups of a group H such that
C c Z(A) n ZE), then it is not necessarily valid that the subgroup AB c //,
if it exists, is in fact (AB)C. It may be that A <? %>H(B).
The proof of Theorem 2.2.6 outlines a procedure for the construction
of the direct product of two groups with amalgamated subgroup. In practice,
the monomorphisms can be simplified by a judicial use of notation and
symbols of operation. One should do the first several exercises for the feel
of it.
EXERCISES 2.2
1. Construct an example of (AB)C for A and B cyclic groups of order four with
A n B = C having order two.
2. Repeat Exercise 1 by taking A to be the quaternion group, i.e. A = <#, b | a4 = 1,
a2 = b2, b~x ab = a), Acyclic of order four, and A n B = C having order
two. Identify the monomorphic images of A, B, and C.

16 Products 2.3
3. Prove Theorem 2.2.5.
4. Prove Corollary 2.2.6.
5. Prove Theorem 2.2.7.
6. Prove Theorem 2.2.8.
2.3 SUBDIRECT PRODUCTS
Whereas the direct product with amalgamated subgroup was a way of
generalizing the direct product, the notion to be defined next deals with a
subgroup of the product object, *Yli Gt.
Definition 2.3.1. A subgroup H of G = *Yli Gt *s a subdirect product of G
if pr(G;)|// is an epimorphism of H onto Gt for each iel.
Note that if H is a subdirect product of G = *YIi Gh then the set of
/-components in the elements of H is all of Gf. This differs from the case of
an arbitrary subgroup A of G in that the collection of /-components may
form only a proper subgroup of Gf.
EXAMPLE. If A = (a\a4 =\}U(b\b4 = 1>, then <(*, 6)> is a subdirect
product of A But <(a2, Z?)> is not.
The subdirect product is not unique in the sense that there may exist
nonisomorphic subdirect products for the same set of subgroups.
If there is a monomorphism a of a group G into A = *Yli A{ such that
Ga is a subdirect product of A, then a is said to be a representation of G as a
subdirect product of A. If a is an inclusion mapping, that is, xa = x, Vx e G,
then of course G is a subdirect product of A. As a result of Theorem 2.2.5,
one can refer to a group G as being a subdirect product with the under-
understanding that the representation exists and without any loss of generality in
the context. In particular, this is convenient whenever / is finite. For then,
*O/ &i = Tli &i is a product of the subgroups
Gi = {{*J I Xj = lj, jel\ /, xt e Gj.
So just consider G = ®7 Gt and then pr(G,) will correspond to the epi-
epimorphism G/H = Gh where H = GXG2 ... G^j G/+1 ... Gn, for each iel.
(See Exercise 2.1.2.) Without any loss in generality, let pr(Gf) denote this
epimorphism in the direct product. Then rephrase Definition 2.3.1 with
respect to a direct product.
It is evident from the definition that Yli Gt is a subdirect product of
*I1/ Gt. More generally, the following can be said:
Theorem 2.3.2. Each subgroup H of *H/ Gt is a subdirect product of *ri/ At
for some subgroup A( c Gb and for all iel.

2.3 Subdirect Products 17
Proof. Exercise.
Clearly, *Y\i Gf is a subdirect product of itself. There is a form of tran-
transitivity in the representations of groups as subdirect products such as stated
in Exercise 2.3.4. But transitivity isn't always valid.
EXAMPLE. The subgroup <(a, b)} is a subdirect product of
H = (a\a2 =1>II<7#2 =1>.
Set G = <(a, 6)> II <A, b)}. Define the mapping a: H -> G by
A, I)h->(A,1),A, 1)),
(a, l)i-((fl,6), A,6),
(l,Z>)h->((l, 1),A,Z>)),
and
(a,6)i->((<i, 6), A,1)).
Clearly, a is an isomorphism and Ha = G. Hence, trivially, Ha is a subdirect
product of G. But
<(*, ^)>a = <(a, 6), A, 1)>
is not a subdirect product of G.
One useful application of subdirect products is the next result:
Theorem 2.3.3. If a collection {Gj7 of normal subgroups in a group G has
f]j Gt =1, then there exists a representation of G as a subdirect product of
Proof. This will be proved by using the universal property of the product
that is given in Section 2.1. Let at be the epimorphism of G -> GjGt. There
exists a unique homomorphism
such that l
0Lt = apr(G/G;), V/eJ.
Since Ker(a) c Ker(a?) = G,-, then
So a is a monomorphism and the required representation.
Theorems concerning subdirect products and three or more direct factors
are usually quite complicated and this seems to be the rule rather than the
exception. But, for two factors, this is not the case.

18 Products 2.3
Theorem 2.3.4. A subgroup H of a group G = A <g) B is a subdirect product
of G iff there are subgroups Ax <a ^4, E^B, and an isomorphism
such that for aeA and beB, abeH iff (a^O = 651#
Proo/. Suppose that ^41? Bl9 and y satisfy the conditions. Since y is surjective,
then for each aeA there is beB such that (tf^O = bBt and <z&eH. There-
Therefore pr(A)\H is an epimorphism of H -+ A. A similar proof and conclusion
is valid for pr(B)\H. Hence H is a subdirect product of G.
Conversely, let if be a subdirect product of G. Consider subgroups
A± = H n A and BY = H n B. For each aeA, there is a beB such that
a6 e H. So for a* e Al9
a-1a*a = (a'^-^a^iba) = (ab)~xa*(ab) e H n A = Ax.
Therefore At<3 A. In like manner, 1^ <a 5. Next, define an association y
of AjAt with B/^! by y: a^! h-^ Z?jBx iff ab e H. In order to show that y is a
function, consider a*b*eH such that (a*^O = 6*1^. Then
then (fl'Vje^. But this implies that (b~1b^)eH. There-
Therefore (b~1b*)eBl. Hence Z?!^ = b*Bt. An analogous argument is used to
prove that y is bijective. Since y is also a homomorphism, it follows that y
is an isomorphism of AjAx onto
The following properties are derived from the preceding theorem. The
notation is that used in Theorem 2.3.4 and the proofs are left as exercises.
Suppose that H is a subdirect product of the finite group G = A (x) B.
2. If At is the identity element, then H ^ B. If Bx is the identity element,
then H ^ A.
3. # <i yl ® B iff ^4/^! and ?/#! are both abelian.
It is clear that a subdirect product of abelian groups is also abelian. The
same can be said for nilpotent and solvable groups. If H is a subdirect
product of *ri/ Gi9 then Z(H) c *YlI Z(Gt). The fact that the containment
may be proper is brought out in Exercise 2.3.3.
EXERCISES 2.3
1. Determine all subdirect products in the product of a cyclic group of order three,
C3, and a cyclic group of order nine, C9.
2. Replace C3 in Exercise 1 by C9 and repeat.

2.3 Subdirect Products 19
3. In order to show that Z(H) is not in general a subdirect product of Z(A) IT Z(B)
whenever H is a subdirect product of AH B, set A — C2, B = S3 (the sym-
symmetric group of degree three), and At = 1.
4. Let if be a subdirect product of A = *Yli At and suppose that each At is
representable as a subdirect product of Ylit ^u* ^or eacn /6/. Show that H
has a representation as a subdirect product of *JX/ Atj for / = (J7 It.
5. Show that every finite group has a representation as a subdirect product of a
product object in which each nontrivial factor has precisely one minimal normal
subgroup.
6. Prove the properties (l)-C). Note that A) is valid even if H is infinite.
7. Prove Theorem 2.3.2.

3
SPLITTING EXTENSIONS;
SEMIDIRECT AND WREATH PRODUCTS
3.1 PRODUCTS OF SUBGROUPS
A first venture into the identification of a group structure with respect
to its decomposition by subgroups would be the use of direct products. The
study of abelian groups is done in this manner. However, this approach to
noncommutative groups is limited in that not every subgroup is normal in
the group.
EXAMPLE. Consider the symmetric group of degree three given by S3 =
<a, b | a3 = b2 = baba = 1>. Note first that S3 = «Z>>, {ab>}, with both
<Z?> and {ab} having order two. Also, S3 = <<#>, <&>>. Since <a><3 S3,
then <a> <&> = {by <«> = S3. So there exists a permutative property
between the subgroups in the second situation that doesn't exist between
{b} and {aby.
The permutative property between subgroups brought out in this example
is found to be desirable and we will rely on it heavily. Recall that for sub-
subgroups A and B of a group G, AB = {ab | aeA, beB}. In general, AB is
not a subgroup of G; for example, form <Z?> {ab} in the above example.
Theorem 3.1.1. For subgroups A and B of a group G, AB is a subgroup of
GiWAB = BA.
Proof. Exercise.
Corollary 3.1.1. For subgroups A and B of a group G, G = AB iff
1. G = (A, By and
2. AB = BA.
Proof. Exercise.
Two subgroups A and B of a group G are said to be permutable whenever
AB = BA.

3.1 Products of Subgroups 21
Definition 3.1.2. A group G is a, product of two subgroups A and B iff G = AB.
In this definition, it is not assumed that either A or B is normal in G.
The ambiguous use of the word "product" is regrettable. However, no
confusion will arise since the term "direct product" replaces its previous
use whenever the internal structure of a group is considered.
Theorem 3.1.3. Suppose that A and B are permutable subgroups of a group
G. If C is a subgroup of G for which i^c, then the modular identity,
AB n C = A(B n C), is satisfied.
Proof. Since AB is a subgroup of G, then AB n C is also a subgroup. More-
Moreover, A ^ C implies that v4(? nC)c ^4? n C. For each element ie.45nC,
there exist elements aeA and Z?e? such that x = ab. Therefore,
a~xx = beBn C
and so xeA(B n C). Hence AB n C c A(B n C). Equality then follows.
Corollary 3.1.3. If G = AB and C is a subgroup of G such that ^ c C,
then C = A(B n C).
Protf/. Note that C = GnC = iBnC = A E n C).
Verification of the next three theorems should be a review.
Theorem 3.1.4. If A and B are subgroups of a group G, then
Theorem 3.1.5. If A and B are permutable subgroups of a group G, then
(AB:A) = (B:AnB).
Even though the conclusion for this theorem has an appearance of one
of the basic isomorphism theorems, it is not required that A be normal in
AB.
Corollary 3.1.5. If A and B are finite permutable subgroups of a group G,
then
\Ar\B\
Theorem 3.1.6. If ({A, B}:B) and «X, B}:A) are finite and relatively
prime, then
«v4, B}:B) = (A:AnB) and ({A, B}:A) = (B:AnB).

22 Splitting Extensions; Semidirect and Wreath Products 3.1
Theorem 3.1.7. Let A and B be two subgroups of finite index in a group G.
1. A and B are permutable iff ((A, B}:A) = (B:An B).
2. If (G: A) is relatively prime to (G: ?), then G = AB.
Proo/. Consider A). If A and ? are permutable subgroups, then {A, B} =
AB and by Theorem 3.1.5, (AB:A) = (B:AnB). So then assume that
equality holds and consider the coset decomposition
B = (A n B) + (A n B) b2 + -¦- + (A n B) bn for n = (B:An B).
Also consider the set A + Ab2 + ••• + Abn. Suppose that Abt n Abj is not
empty for i =? j. Then there exist elements ai9 ctj e A such that aibi = afiy
But aj1ai = bjb^1 eAnB implies that bt and bj belong to the same coset
in the above decomposition of B. This is a contradiction. Hence the cosets
Abt and Abj are disjoint whenever / ^ j. Since ((A, B}: A) = (B: A n B),
it follows that (A, ?> = A + ••• + Abn with b1=\. Hence {A, B} c AB.
Since AB c <A, ?>, it follows that <A, ?> = AB. Therefore AB is a sub-
subgroup of G. By Theorem 3.1.1, AB = BA.
For B), note that Theorem 3.1.6 together with part A) proves that AB
is a subgroup of G. Furthermore,
(G: A) = (G: AB) (AB: A) and (G: 5) = (G: 4?) (AB: B)
have a common factor (G: ^45). If (G: AB) ^ 1, then there is a contradiction
to (G: v4) being relatively prime to (G: B).
Whenever a group G can be expressed in the form AB for proper subgroups
A and B, then G is referred to as being per mutably decomposable or factoriz-
able. The reader is referred to the more recent texts in the bibliography for
further comments on these groups. As an example of such a group consider
the symmetric group, S4, of degree four. Note that for each Sylow 2-sub-
group P in S4 there is a Sylow 3-subgroup Q such that 54 = PQ = QP.
Neither P nor Q is normal in S4. (See Exercise 3.1.5.)
EXERCISES 3.1
Note: The proofs for Exercises A) through D) use only the basic notions about cosets.
1. Prove Theorem 3.1.1 and its corollary.
2. Prove Theorem 3.1.4.
3. Prove Theorem 3.1.5 and its corollary.
4. Prove Theorem 3.1.6.
5. As an example of a factorizable group having neither factor normal, show that
the subgroups P = <A234), A3)> and Q = <A24)> satisfy S4 = PQ = QP.

3.2 Extensions 23
6. Show that if (G: A) and (G: B) are finite and relatively prime, then (G: A n B) =
(G:A)(G:B).
7. Consider a subgroup N normal in a finite group G and a Sylow ^-subgroup P
of G. Using Theorem 3.1.5, show that TV n P is a Sylow ^-subgroup in TV. (That
this result is not always valid for nonnormal subgroups may be seen in the
symmetric group of degree three. Just note that the intersection of two distinct
Sylow 2-subgroups is the identity element.)
8. Consider an epimorphism 6 of a finite group G and a Sylow/^-subgroup P of G.
Prove that Pe is a Sylow ^-subgroup of G6. Also prove the converse: each Sylow
^-subgroup in Ge is the image of a Sylow p-subgroup in G.
9. Prove these properties of the group isomorphic to the multiplicative group of
the roots of unity for all powers of a prime/?, Cpoo. (This should be a review.)
a) Cpoo contains an ascending chain of subgroups 1 = Co c Cx c ••• c Cnc--
such that \Cn\ = p\ \Cn+1/Cn\ = p, and G = (J? C,.
b) Each proper subgroup of CpOo is finite cyclic of order pn for some integer n.
Hence each proper subgroup of Cpoo has finite order.
c) There is precisely one proper subgroup of order pn in Cpoo, for each integer
n. Hence the ascending chain in (a) is a characteristic chain.
3.2 EXTENSIONS
Suppose that a group G is factorizable, say G = AB, and either A or B is
normal in G. One has a greater control in this case. In particular, the normal
subgroup can be taken as the kernel of an epimorphism of G. Let A <3 G
and consider G\A ^ B\A n B = C. There exist a monomorphism a: \^A,
a monomorphism fi:A-> G, epimorphisms y: G -> C, and <5: C ->1, such that
Im(a) = Ker(jS), Im(j5) = Ker(y), and Im(y) = Ker(<5). This can be ex-
expressed as
1-2_>4-2->g-JUC-?->1.
On the other hand, suppose that ^4 and C are given. Is it possible to
determine a group G* that would satisfy the above conditions? If so, then
how would G and G* be related? The remainder of this section sets the frame-
framework for answering these questions. In the next section the case will be ex-
examined for which A n B = 1 in the first remarks, that is, B = C, and then in
Chapter 7 the general case will be examined.
Definition 3.2.1. A sequence
is an exact sequence if Ker(a?+1) = Im(a^) for /= 1, ...,«— 1. A short exact
sequence is the exact sequence 1-+A ->B -> C -> 1, 1 denoting a group of order
one. This short exact sequence is called an extension of A by C. (For brevity,

24 Splitting Extensions; Semidirect and Wreath Products 3.2
B is also called an extension of A by C with the understanding that the short
exact sequence exists.)
Several earlier definitions can be rephrased in terms of exact sequences.
The proof of these is left to the reader.
Let A, B, C, and 1 denote groups, 1 a group of one element.
1. An element a e Hom(y4, B) is an epimorphism iff A-^ B -»1 is exact.
2. An element a e Hom(A, B) is a monomorphism iff l->^[-> B is exact.
3. An element aeHom(v4,J3) is an isomorphism iff l-»l-> J3->1 is exact.
a P
Clearly, ifl-»;4->jB->C->lisa short exact sequence, then C ^ B/Im(a).
Moreover, for each pair of groups A and C, there always exists at least one
extension of A by C, formed with AH C.
Definition 3.2.2. Two extensions 1 ->4 A B^ C: -+1 and 1 -*^ -^ 5* -i C ->1
are said to be equivalent if there exists 7 e Hom(?, ?*) such that the diagram
here is commutative.
1 ,A-^ ?-?-> C—>!
This condition is stronger than it first appears. Consider bt e5*. Then
b\ = c and the commutativity of the diagram implies that there exists an
element b e B such that bp = c. Again by the diagram's commutativity,
byd = b\. Hence byb\^ e Ker(^). This implies the existence of an element
aeA such that aa = byb^K However, aay = cf = byb;x. So
Therefore y is an epimorphism.
Let b e Ker(y). Then bp = byd = 13B* = lc. Hence b e Ker(j8), that is, there
exists an element aeA such that a* = b. Consequently, aay =lB*= aa. So
a e Ker(cr) = \A. Therefore b = 1B and this implies that 7 is a monomorphism.
Combining the two results, one concludes that a is an isomorphism.
Summarizing,
Theorem 3.2.3. Two extensions 1 -> A^B^ C -»1 and l-+A^> ?*-i C -> I
are equivalent iff there exists an isomorphism y e HornE,5*) such that the
diagram here is commutative.
2 C

3.2 Extensions 25
It has already been noted that given two groups A and C, there exists
at least one extension of A by C. Hence the collection of all extensions of a
group A by a group C is not empty. The next theorem points out the signifi-
significance of the equivalence of short exact sequences with respect to this class.
Theorem 3.2.4. Consider the collection $ of short exact sequences arising
from the set of all extensions X of a group A by a group C. $ is not empty
and equivalence of extensions is an equivalence relation on S.
Proof. Verified by the definition and the preceding remarks.
Theorem 3.2.5. If l->^4-> B-+ C ->1 is an extension, then there exists an
equivalent extension of A by C, \->A-> B*-> C ->l, such that y = iB*\A.
Proof. Use Theorem 2.2.5.
EXAMPLE. Consider the groups <a|a3=l>, <6|62=1>, and S3 =
<x,y\x3 = y2 =1, yxy = x2}. Let a:ai->x and fi:y (x}\-^b. Clearly, a
defines a monomorphism of <a> -» S3 and ft defines an epimorphism
Therefore
is an extension. On the other hand, so is
i—> {ay ^u <a> n (by -U <z>> —> 1,
where y: a h-> (a,l) and <5 = pr«6». Since S3 is not isomorphic to <<z> II
then the two extensions are inequivalent.
EXAMPLE. It is possible to have two inequivalent extensions
1 >A-^-> B-?~* C >1
and
even when B and 5* are isomorphic. The requirement that the diagram be
commutative in Theorem 3.2.3 cannot be overlooked. Take B = B* =
(x\x9 =1>, A = C = <x3>, a = iB\A = y, (xj)p = x3J, and (xjf = x6j\ for
an integer j. Then the two resulting short exact sequences are extensions of
A by C, but they are not equivalent. The details are left as an exercise.
EXERCISES 3.2
1. Prove the statements (l)-C).
2. Complete the second example in detail.
3. Complete Theorem 3.2.4 and 3.2.5.

26 Splitting Extensions; Semidirect and Wreath Products 3.3
Consider the dihedral group of order eight, D8 = <#, b | a4 = b2 = 1, bab = a3}.
Express D8 as an extension with respect to each of its normal subgroups.
Show that the extension of a/?-group by a/?-group is also a/?-group. Then prove
that if a Sylow /?-subgroup P of a group G is normal in G, then P is a unique
characteristic p-subgroup of G. More generally, a Sylow ^-subgroup is a unique
characteristic /^-subgroup in its normalizer.
Use the preceding exercise and show that JfG(JfG(P)) = JfG{P) for a Sylow
^-subgroup of a group G (not necessarily finite).
3.3 SPLITTING EXTENSIONS
Definition 3.3.1. An extension B of a group A by a group C is called a
a 0
splitting extension if for the short exact sequence 1-+A-+ B-* C -*l there
is a monomorphism t : C -> B such that t/? = tc.
First note that the extensions of A by C in the second example in Section
3.2 are not splitting extensions. On the other hand, consider the two exten-
extensions given in the first example of that section. For the extension
define t : b -»y. Then t is a monomorphism of <Z?> —> S3 having the property
that t/? = t<b>. This example also points out that t is not unique since
p:b -> xy also defines a monomorphism of <Z?> -> S3 for which p/? = t<ft>.
In the second extension, the mapping p: b ->¦ A, Z?) defines a monomorphism
of <Z?> -> <a> II <&>. With respect to the short exact sequence
it follows that pc) = t<5>. Therefore <a> n <Z?> is a splitting extension of
<a> by <&>. It is evident that, in general, a direct product of two groups
gives rise to a splitting extension.
Theorem 3.3.2. Any extension of a group A by a group C equivalent to a
splitting extension of yl by C is also a splitting extension.
a ?
Proo/. Consider the splitting extension 1-+A-* J3-> C ->1 for which there
exists a monomorphism t: C -> B such that t/? = tc. If l-»ylA ?* -4 C ->1
is an equivalent extension for which y: 5 -> J3* is an isomorphism satisfying
the commutativity requirement, then u* = zy: C -» 5* is a monomorphism.
Moreover, t*E = (ry) ^ = T(y<5) = tj5 = tc.
a ^
For the splitting extension \^>A-+ B-> C-+l, it can be assumed that
a = iB\A, by Theorem 2.2.5. Denote the image of C under t by C*, t a mono-

3.3 Splitting Extensions 27
morphism of C -> B such that tj8 = tc. If x e ^4 n C*, then x e Ker(/?) and
x*-1 =lc. Therefore x =1B. For each 6e?, Z? = b{b~lfxbp\ bPxe C*9 and
Z?(Z7~YTe^l since
SoB = ^4C* = C* A Moreover, A^ B implies that each element c* e C*
corresponds to an automorphism of A under conjugation of the elements in
A by c*. Hence there exists a mapping 6 of C* -> Aut(^4) which is in fact a
homomorphism. One can anticipate that a group B having this type of
decomposition with respect to the subgroups A and C* would lend itself to a
representation as a splitting extension. It also suggests that, given any pair
of groups A and C for which there exists a homomorphism 6: C -> Aut(^4),
a splitting extension other than a product object could be formed whenever
Ker@) # C.
EXAMPLE. Consider the groups A = (a\a3 =l>andC = <c|c2 =1>. Clear-
Clearly, A IT C exists and so the class of splitting extensions of A by C is not empty.
Aut(^4) = (y\y2 = 1> for y: a i—> a2. Form the isomorphism 9 of C -> Aut(y4)
defined by 0:ch->y. Define an operation on C X^4 by (c1? ax) (c2, a2) =
(c1c2,a1C2 a2) where a° = ay. (This associative operation is suggested by
observing the product of two elements in a group G = CA for A <a G and
AnC=L) Then 5 = {A,1), (I, a), (l,a2), (c,l), (c, a), (c, a2)} forms a
group with respect to this operation. Moreover, there exists a monomorphism
a of A into jB defined by a: an A, a) and a monomorphism t of C into B
defined by t: ch-> (c,1). As can be easily verified,
4" = {A,1), (l,a), (l,a2)H?
a p
and B = ylaC. The extension l-*A-+ B-> C->1 is a splitting extension
whenever /? is defined by (c1,a1)i->clJ V(c1,«1)e5 since t^ = tc.
Theorem 3.3.3. For a pair of groups A and C, there exists a splitting exten-
extension B of ^ by C for each 0 e Hom(C, Aut(^)). (Hom(C, Aut(^)) is never
empty.)
Proof. Consider 6eHom(C,Aut(A)) and the set C x A = B. As in the
example, aCi is defined by cA. So (acl)C2 = aclC2. Define a binary operation
on B by the relation (c, a) (cl9 a^ = (ccu acia^). B forms a group with respect
to this binary operation. This group is dependent upon the 6 chosen. More-
Moreover, the set C* = {(c, lA) | ce C} is a subgroup of B for which there exists
an isomorphism y e Hom(C, C*) defined by -y: c \-> (c,l^), Vc e C. Also, the
set A* = {(lc,a)|ae^4} is a subgroup of J3 for which there exists an

28 Splitting Extensions; Semidirect and Wreath Products 3.3
isomorphism a e Hom(A, A*) defined by a: ai—> (lc, ?). Then A* <i B, B =
C* A*, and i*nC* = AOU), (WJ the identity element in B. The map-
mapping fi\B-*C given by [1: (c,a)\->c is an epimorphism of 5 onto C with
Ker(/?) = yl*. Consequently, one has a splitting extension l->^4-^ ? -4- C ->1
with respect to the monomorphism y that satisfies yfi = lc.
Definition 3.3.4. Denote the group B in Theorem 3.3.3 by B = {A\eC. B
is called the semidirect product of A by C with respect to 9 e Hom(C, Aut(v4)).
a 0
Theorem 3.3.5. Each splitting extension 1 -* A-> B-+ C -»1 is equivalent
to a splitting extension 1-*A -> [^4]0C -> C -> 1 for some 0 e Hom(C, Aut(^4)).
Theorem 3.3.6. A group B having subgroups A and C such that A <i ?,
inC=l, and B = ^4C is isomorphic to a semidirect product [v4]0C for
some 6 e Hom(C, Aut(A)).
One could say that the group B in Theorem 3.3.6 is representable as a
semidirect product. However, it usually is said that B splits over A by C.
Since 6 is implicitly defined by the internal structure of B, B is expressed as
B = \A~\C.
By the way, a direct product may be represented as a semidirect product
in a form other than as a product object. Consider S3II S3 for S3 the sym-
symmetric group of degree three. Then S3HS3 = \_A~]C where
A = {(x,x)\xeS3} and C = {A,y) \yeS3}.
EXERCISES 3.3
1. Express D8 as a splitting extension. Compare with Exercise 3.2.4 and note that
not every extension given in that exercise is a splitting extension.
2. Show that each short exact sequence determined by the normal subgroups of
the group G = (a,b,c \.a3 = b3 = c3 = 1, a ba = be, ca = ac, cb = be} of
order 27 is a splitting extension except the one determined by <c> o G.
3. Prove Theorem 3.3.4.
4. Prove Theorem 3.3.5.
5. For a group A, let 6 be the identity mapping of AutO4) -> AutO4). Then
[A]e AutO4) is called the holomorph of A and it is denoted by Hol(^). Consider
A = Cp, Cp a cyclic group of order a prime p, and C = Aut(Cp). Then
|Hol04)| = \Cp\\C\ = p(p-l).

3.4 Wreath Products 29
6. Determine all splitting extensions, up to isomorphisms, of a group A by a. group
C whenever A = C5 and C = C4.
7. If G = \_A~\C, then the normalizer of C in G is C ® B and the centralizer of C
in G is Z(C) ® 5, for ? the set of elements in A that centralize C.
3.4 WREATH PRODUCTS
A special application of the semidirect product will be made in this section.
It has been found to be quite useful in the construction of examples and it also
provides a description of the Sylow /7-subgroups of the finite symmetric
groups. We begin by introducing it in a primitive but natural form.
Consider a group G, an index set /, and a subgroup F of the permutation
group S/? Sj the set of all bijective mappings / -> /. Form Gf = *Yli &i
such that Gt is a copy of G indexed by / for all iel. Each element of Gf is
identified with a mapping /of/ -> G. Hence *I~Ii &i *s bijective with Gz, the
set of all mappings of / into G. Let/, g e G1. Since if ig = ifg is defined for
each iel, then G1 together with this operation is a group. Moreover, it is
isomorphic with Gf. So without loss of generality, this group of functions
will be denoted by Gf.
Next we relate F to a subgroup of Aut(G/). Let ip denote the image of
/g/ under peT. For yeF, define the mapping y*: G/ -> G/ by
y*: /' h-> (/y- Y = //y*, V/ e / and V/e Gf,
that is, y*:/n->/y*. Clearly/y* e Gf. Moreover, for each/e G*,/^ exists.
Hence the mapping is surjective. For/ g e G*.
Therefore y* e Aut(G/).
For p, 7 g F,
= .(/^p* = (ip-iyy = (jp-iy-y = (/(yp)/ = //GP)*? Vie/
Therefore y*p* = (yp)*. So the mapping 6:T -> Aut(G*) defined by
6:y H^y*, VyGF, is a homomorphism. Suppose that pGKer@). Then p*
is the identity automorphism and this implies that/ = /p*, V/eG*. Hence
/•f = (ip~xY9 V/e/ and V/e G*. There will always exist some element of
G* such that the functional values will differ at the / and the (ip'1) com-
component, except when Gf is a one-element group. Thus a contradiction is
reached unless p is the identity permutation. Therefore 6 is a monomorphism.
A similar result holds if attention had been given to the product object
n, g, = g,.
The definitions are stated with respect to the above notation.

30 Splitting Extensions; Semidirect and Wreath Products 3.4
Definition 3.4.1. The unrestricted wreath product of a group G with respect
to a set / and a subgroup F of S1 is [_Gf]dT, 0 the monomorphism of F into
Aut(Gj) that is given above. It is denoted by G Wr F.
The restricted wreath product, G wr F, is [G7]0 F.
For a group f/ and RH the right Cay ley group on H, the wreath products
formed by taking I = H and T = RH are called the standard unrestricted
and the standard restricted wreath products. They are denoted respectively
by G Wr if and G\H.
If I or H is finite, then G Wr F = G wr F or G Wr if = G \ if.
Much of our attention will be on the standard restricted wreath product.
However, keep in mind that G\ H = \_GH~]Q RH. Since the permutation cor-
corresponding to an element reRH is induced by right multiplication of some
element keH, then identify r with k and note that (h)fk = (M/, VA e H,
\f e GH. In the notation of the semidirect product, G \ if is the set H x GH
having a product defined by (a,f) (b, g) = (ab,fbg) for a,beH and/, g e GH
such that hp9 = (hb'^h9, heH. Then as before, if is isomorphic to
{(/z, e) | h eH and e the identity element of GH} and GH is isomorphic to
{Oh,/) 1/ e G#}- Hence l-+GHA>G\ffAif->l is a splitting extension
with respect to the monomorphism a:/i-> A#,/), the epimorphism
/l:(g,h) -* A, and the monomorphism t: /z i-> (/z, e), such that t/? = t#.
Some of the immediate properties of the standard restricted wreath
product are the following:
1. If \G\ and \H\ are finite, then |G \ if | = \G\ m\H\.
2. If G and H are solvable, then G \ if is solvable.
3. If G and H are ^-groups, then G \ if is a />-group.
4. If the diagonal D of G Wr if is defined by the set of all constant functions
of if into G, then D ^ G.
5. If in G Wr ii, |if | is finite, then D is contained in the centralizer of the sub-
subgroup H1 = {(h,e)\heH and e the identity element of GH}. In fact,
the centralizer of Hx in G\H is D® Z{HX).
6. Whenever \H\ is finite, the normalizer of H1 (defined in E)) in G \ if is
D®H1.
7. If |if | is finite, then Z(G \ H) = Zip).
8. If \H\ is infinite, then Z(G \H)=l.
The wreath product is not in general associative. (See Exercise 9.) How-
However, G1\G2\-'-\Gt will be used to denote (Gx \ ••• \ Gt-{) \ Gt.
From the definition of G \ if one can readily see that G \ ?, E a one-
element group, is isomorphic to G and also that ? \ if is isomorphic to RH.

3.4 Wreath Products 31
A generalization can be made of E \ H with respect to any normal subgroup
N of a group G.
Theorem 3.4.2. For each normal subgroup N of a group G, there is a mono-
morphism of G into N Wr (G/N).
Proof. For N <a G and H = G/N form the coset decomposition G = [)H Nh\
such that hx is a coset representative corresponding to heH. Let /^ denote
the right coset in H that contains the element g e G. Then JV7zT = h =
hQi^hg) = (hhj^)hg = NQih^yNg^ = Nihh;1)" g implies the existence
of an element (h, g)eN such that (h, g) = (h h~ 1)t gitfy1. Define the map-
mapping y:G -+ NWrH by y:gy-^(hg,co), where /20J = (/?,#) for each /ieH.
Let /y = (hf9 ii), such that A" = (hj). Then ^ = (hghf, p), where
However,
'yy1 (h,gf) =
Since this implies that
(A, a/") = (AA/ S ff) (*./) = *". VA e H,
then #y/y = (g/)y, that is, y is a homomorphism. Suppose that gy = (hg, co) =
(lH, e), AH, e) the identity element of N Wr H. Then (/^) = JfgQfy1 = 1
implies that g = 1. Hence y is a monomorphism.
Corollary 3.4.2. For each normal subgroup N of a group G, there exists a
group G* = [iV*] H containing G such that N* nG = N, H ^ G/JV.
Proo/. By Theorem 3.4.2, there is a monomorphism y of
G -> JV Wr (G/JV) = [N5]a (G/iV) = Gl5
^ arising from the definition of the wreath product. G1 contains a normal
subgroup
N1 = {(N,co)\coeN*}^N*
and a subgroup
Hx = {(bN, e)\bNe G/N, e the identity of N*} ^ G/N
such that Gj = N1Hl, Nx n Ht = (N, e). From the definitions of y and the
elements (/?, g), it follows that Ny = Gy n N^. By Theorem 2.2.5, there exists

32 Splitting Extensions; Semidirect and Wreath Products 3.4
a group G* ^ G and an isomorphism a: G* -> Gx such that a\ G = y. Set
N* = N*1~1 and if = HI. Then G* = iV*H, N*n/f = 1, and N = GnN*.
Since each finite /7-group is contained in a Sylow /^-subgroup of a sym-
symmetric group of a sufficient degree, it seems worthwhile to give some struc-
structure concerning these Sylow/^-subgroups. The wreath product helps to supply
part of the answer.
In brief, the procedure will be the following. Consider a collection of r
distinct symbols and determine an expression for the highest power of p
dividing r!, say pH. Then determine a partitioning of the r symbols into cells
containing pk elements for certain values of k. Because of the partitioning,
a direct product of the Sylow ^-subgroups, one from each cell, exists, and we
find it has order pH. This reduces to the study of the Sylow /^-subgroups of
the symmetric groups of degree pk and fortunately an inductive procedure
exists for handling these.
If [/?] denotes the largest integer in the number n, then p, 2p,..., [rjp~]p
are the factors of r\ that are multiples of p. So pH\ {d^pd\ dt = [rjp~\.
Repeat the argument on dx!, obtain d2 = [d1/p'] = [r//?2], and one has
PH\ (d2l)pdl+d2- Inductively, there exists an integer m such that H = Zf<4
d. = [rlpi~\. In order to partition the r symbols into cells containing a pth.
power of elements, use the fact that r = rQ + rxp + ••• + rtp\ 0 < r{ < p,
to obtain rt cells having pi elements for / = 0,..., t. Since r = r0 + [rjp~]p
and dt = {rjp~] = rx + ••• + rtpl~x, then it can be shown by induction that
for / =1, ...,w, di = rt + ••• + rtp%~\ So m = t and
H = (rt + ••• + r,/) + (r2 + ••• + rf/) + ••• + {rt_1 + rtp) + rt
= rx + r2(l + p) + r3(l + p + p2) + ••• +r,_1(l+ ••• +/~2)
Hence, for a Sylow /^-subgroup of a symmetric group on pl symbols, H = 1 +
p + ••• +/'1.
Consider the r symbols to be 1,..., r. For r = p, the greatest divisor is
of course p, the order of a Sylow/^-subgroup PA) is/7, and we can assume that
PA) is generated by the cycle A, ...,/?). For r = p2, partition the set into the
cells
{1, ...,p), {p+l, ...,2p}, ..., {0,-1)^+1, ...,p2}
and consider the cycles
xx = (\,...,p),x2 = (p +l,...,2p),...,xp= (p2 -p +l,...,/72).
Then X = ®px <xf> exists and \X\ = pp. Since H = 1 + p, it is enough to find
an element 9 e AutpQ and form [X] <#> in order to have a Sylow/?-subgroup

3.4 Wreath Products 33
PB). Since X is elementary abelian and hasp direct factors then 6:xi\-^xi+u
the subscripts modulo p, is such an automorphism. It can be effected by
relating 6 to the element y = (l,p +1,2/7 +1, ...,p2 ~- p +1) B,p + 2,...,
p2 —p + 2) •••(/?, 2p, ...,p2). Then >'~1^j = xi+1, again the subscripts
modulo p. So PB) = [X] O> and PB) ^ PA) \ <j>.
Next partition the set {1, ...,/?s+1} into cells
{1,...,/}, {/ +1, ...,2/}, ..., {(/> -1)/ +1, ...,/>s+1}.
If Po = P(s) is a Sylow /7-subgroup on the set {1, ...,ps}, then the element
z = (i,Ps +1,..., o -i);?s +i) ••• o,/?s + «,..., (P -\)PS + «)...
(ps,2ps, ...,/?s+1), « =1, ...?jps,
has the property that z'TqZ1 = Pt is a group of order A+ ¦•• +/?5) on
the set {ips +1,...,(/ +l)/?s}, for / = 0, ...,/> — 1. Furthermore, for i,j =
0. ...,/?-l, {z>s +1,..., (/ +l)^s}, and {j>s +1,..., (J + l)ps} are disjoint
whenever / ^ j. Hence P* = ®o~x Pt exists. Moreover, [P*] <z) also exists
and has order pk where A: = p\P(s)\ +1 = ^A + p + ... + p8'1) +1 = 1 +
p + ••• +ps. Therefore P(s+1) = [P*] <z> ^ P(s) \ <z> is a Sylow /?-sub-
group of a symmetric group on/?s+1 symbols. Inductively, PE+1) ^ ^41\^42\
•••\^4s + i for Aj a cyclic group of order p.
Theorem 3.4.3
1. A Sylow /7-subgroup P(s) of a symmetric group of degree ps is isomorphic
to A± \ A2 \ -" \ As for Aj a cyclic group of order p.
2. A Sylow /?-subgroup of a symmetric group of degree r is isomorphic to
err pA)] n err2
where r = r0 + r^ + ••• + rtpf and nri P(i) is the product object of rt copies
of a Sylow ^-subgroup P@ in a symmetric group of degree pl.
EXERCISES 3.4
1. Consider G = (g \ g2 = 1>, the index set / = {1, 2}, and T = S2. Form
GwrF.
2. Consider G = (g \ g2 = 1> and H = (h \ h2 = 1>. Form G\H.
3. Repeat Exercise 1 with G = (g \ g3 = 1>, / = {l, 2, 3}, and F the subgroup
of order three in S3.
4. Repeat Exercise 2 with G = (g \ g3 = 1> and H = (h\h3 = 1>.
5. Prove properties

34 Splitting Extensions; Semidirect and Wreath Products 3.4
6. Consider G = (g \ g4 = 1> and N = <#2>. Use G and TV to exemplify Theo-
Theorem 3.4.2 and its Corollary.
7. Express the Sylow 2-subgroup of S8 as a wreath product A 1\A2\A3 by giving
the generators of Al9 A2, and A3ina manner suggested in the proof of Theorem
3.4.3(a).
8. Express a Sylow ^-subgroup of S15 in the manner given in Theorem 3.4.3(b)
for each prime dividing 15!.
9. In general, the wreath product is not associative. Note that |(C2\C2)\C2| ?"
|C2\(C2\C2)|, for C2 a cyclic group of order two.
10. In general, G = A Wr B is not a ^-group whenever A and B are /7-groups.
Consider the group G= CWrCfor C= CpQ0. Then note that TV* contains
the element / defined by cf = c, for all c e C. Show that/does not have finite
order.
11. Show that G1 is isomorphic to G*.

4
THEOREMS ON SPLITTING;
HALL SUBGROUPS
A complement of a normal subgroup N in a group G is a subgroup A c G
such that G = JV4 and N n 4 = 1. Clearly, G is isomorphic to a semidirect
product of N by ^4 and a splitting extension can be formed. Consequently,
whenever a normal subgroup N has a complement A in G, G = [iV] A
This chapter will be primarily concerned with the determination of con-
conditions under which a group G splits over a normal subgroup N and the con-
conditions under which the complements of N are conjugate. Enough informa-
information is then available to say something about the structure of a finite solvable
group by using its Sylow subgroups.
Several basic properties of nilpotent groups have been assumed known
to the reader and they will be used occasionally in this chapter. (See Chapter
1. The reader may prefer to review these properties, as well as other basic
notions about nilpotent groups, that are found in Sections 5.1 and 5.2 before
continuing.)
4.1 ON A THEOREM OF DIXON
Definition 4.1.1. Let % denote a set of prime integers and %' the complement
of n in the set of all prime integers. An integer n is a n-number if each prime
divisor of n belongs to n. A group G is a n-group if each element of G has
finite order and the prime divisors of the order of each element different from
the identity element are in n.
It will be left as an exercise to verify that
1. subgroups of Ti-groups are 71-groups,
2. epimorphic images of 7r-groups are 7i-groups, and
3. the extension of a 7i-group by a jr-group is a 7i-group.
However, the product object for a collection of Ti-groups in general may not
be a 7i-group. (See Exercise 3.4.10.)
Lemma 4.1.2. Consider a group G having an abelian normal Ti-subgroup
N such that for subgroups A and B of G, G = [JV]5 = NA. If k = (B: A n B)
is a Ti'-number, then for some yeN, By c A.
Proof. Assume that the result is valid whenever N n A = 1 and consider

36 Theorems on Splitting; Hall Subgroups 4.1
then NjM is an abelian normal 7r-subgroup of G/M = [JV/M] (BMjM) =
(N/M) (A/M) and (BMjM : (BM/M n A/M)) = A: is a Tc'-number. Hence
there exists an element jeiV and yMeG/M such that (jM) (BMjM)
(yM) c= 4/M, that is, y^BMy\M <= 4/M. Therefore ?* <= (?M)y c 4.
In order to prove the case for N n A = 1 consider the mapping y: B -> A
defined by bN = byN for beB and byeA. One can readily verify that y is
an isomorphism. A complete set D of right coset representatives of A n B
in B has k elements. Moreover, since N is abelian and (by)~1beN, VbeB,
the element z = YId (by)~ib is well defined. If aeA n B, then ay = a and
((aZ?O) (a*) = (F) (flO ^ = (b7)'1 b. Therefore z is independent of
the set of coset representatives used. Also note that Dd is a complete set of
right coset representatives for each element deB. So
(dT1 zdy = Ud (d7)'1 (bT'bV = Ud ((bdy)-1 bdy
= z(d~1dy)\
Since N is a 7r-group and k is a rc'-number, then Nk = {nk \n e N} = N.
Moreover, the mapping a defined by n \~> nk is an automorphism of N. Hence
there exists an element xeN such that xk = z. Consequently, (d7)'1 xkdy =
[OFT1 x^]' = ^{d-1d7f. Applying a, one obtains (dyyxxdy = xd~xdy.
Therefore (dy)~x x = xd~\ or xdx~x = dy, Vc/eB. Hence By ^ A for
Lemma 4.1.3. Let NWrH = [N*] H* where JV* = {AH,/) |/ e NH } =
N* and H = H* = {(h,lN*H)\heH}. If a subgroup C contains JV* and
C = [JV*] ,4 = [iV*] B, then there exists an element x e iV* such that ,4* = B.
By the modular identity, C = Cn iV*H* = N*(C n if*) = [N*]//*
for Hx* = Cn H*. Let if x = {A | (A, 1) e Hf}. Note that X, B, and Hf are
each a complete set of coset representatives for JV* in C. Hence for each
a e A, there is a unique element (A, 1) e if x* such that aiV* = (A, 1) N*. Con-
Consequently, there is a unique elementyj, e iVH* such that a = (A, /,,). Similarly,
each beB is uniquely expressed as b = (A, gh) for heH1 and #,, e iVH*. More-
Moreover, since (h,fh) (k, fk) = (hk, fhkfk) e A, thmfhk = fffk (by the uniqueness).
Similarly, ghk = g? gk. Consider a set jf of left coset representatives of Ht
in if. Define the mapping f.H -> iV by f: xA h-> (xA)^ (xA)^ for xg jf
and A e ifx. For each /: eHu
(xh)tk (xh)9* = (xhk-1)' (xh)g*
= (xhk-iyf»*-1)'1(xhk-1)gH'c-1 (xh)9*
(^ -1>" * (xh)9HK ~l (xh)g*
(xh)f^ (xh)9n
= (xhY* (xhj.

4.1 On a Theorem of Dlxon 37
Therefore tkgk = fkt. So for all keHl9
(k9 A) A, 0 = (k, fkt) = (k, tkgk) = A, 0 (k, gk).
Hence
(I,ty1(k,fk)(l,t) = (k,gk).
It follows that A* = B for x = A, t).
Theorem 4.1.4. Consider a group G having a subgroup B such that B con-
contains an abelian rc-subgroup JV<i G, (G: JV) is finite, and (G:B) = k is a
Ti'-number.
1. If 5 splits over JV, then G splits over JV.
2. The number of distinct conjugate classes of complements of JV in G does
not exceed the number of distinct conjugate classes of complements of
N inB.
Proof. Form K = N Wr (G/N). From Corollary 3.4.2, it follows without
loss of generality that we can assume G to be contained in K = [JV*] H such
that JV* = NH, N = N* n G, and H ^ G/JV. Clearly, JV* is a Ti-group. Since
K = GJV* (see the proof of Theorem 3.4.2) and BnN* = N = GnN*,
then (K: BN*) = k. Let C denote a subgroup such that B = [JV] C. This
implies that ?JV* = [JV*] C. Then note that
?JV* = f/JV* n ?JV* = N*(Hn BJV*)
and
JV* n (J^ n J5JV*) = (JV* n H) n ?JV* = 1.
Therefore ?JV* = [JV*] (/f n ?JV*). By Lemma 4.1.3, there exists Je JV*
such that Cd = H n ?JV* c #. However,
A: = (X: BJV*) = (H: if n BN*) = (//: Cd) = (H: Gd nH)(Gd n H: Cd).
So m = (H:Gd nH) divides fc. Therefore m is a Tr'-number. Since K = GdJV*,
then, by Lemma 4.1.2, there exists an element yeN* such that Hy c Gd,
that is, Hr <= G for ? = yd'1 e JV*. Thus G = NH* and JV n H{c JV* n Hf = 1.
Hence G = [JV] H'. This proves A).
As for B), suppose that G = [JV] C = [JV] A. Then ? = [JV] (C n B) =
[JV] (,4 n 5). If there exists ^eJ5 such that C n B = (A n B)b, then C nB c
Cnyl5. Therefore
(C: CnB) = (C:Cn ,46) (C n ^L6: C n B) = (BC: B) = k
implies that m = (C:C n ^4fr) divides k. So mis a Ti'-number and, by Lemma

38 Theorems on Splitting; Hall Subgroups 4.1
4.1.2, C is conjugate to Ab in G. Consequently, if B n C and B n A are con-
conjugate in B, then C and A are conjugate in G. The result follows.
Corollary 4.1.4
1. (Gaschiitz) Consider a subgroup B in a finite group G such that 5 contains
an abelian subgroup JV <i G and having ((G: 2?), | JV|) = 1. If B splits over JV,
then G splits over JV.
2. (Schur) If for an abelian subgroup JV normal in a finite group G, ((G:N),
|JV|) =1, then G splits over JV and all the complements of JV are conjugate.
The first of the following examples shows that if in Theorem 4.1.4 the
normal subgroup JV is nilpotent but nonabelian, then conclusion A) does
not hold. The second example indicates that strict inequality may occur in
conclusion B).
EXAMPLE. (Baer). For A a quaternion group, B, C, and D cyclic groups
of orders four, two, and three, respectively, there exists a group G = [H] D
for H = (AB)C given by the following defining relations:
A = (a,a1\a4 = aj =1, a2 = a2u a^1aa1=a~1y, B = <b\b* =1>,
C = <Z>2>, D = (d\d3 = 1>, b2 = a2 = af, ab = ba, a^b = bau db = bd,
d corresponding to an element in Aut(A) such that ad = au a\ = aat, and
(a2)" = a\
Note that ab has order two and it is not contained in A. So H = [A~](ab}.
Since H o G, then H is the unique Sylow 2-subgroup of G. Moreover,
\G\ = 48 and |if| =16 imply that ((G:#), \A\) =1.
Suppose there exists a subgroup K cz G such that G = [^4] X. It can be
assumed that D c= K. Since G/^4 is cyclic of order six, then K contains an
element x of order two. Therefore xeH and has the form x = a*b* for a* e A
and Z>* e ?. Suppose that xeB. Then ^4 n K ^ 1 since the only element of
order two in B is in A n B. Therefore x$B. So a* $B and consequently
(a*)d =? a*. Hence xd = (a*b*)d = a*d b*d = a*d b* =? x. This contradicts
K being abelian. Consequently, G does not split over A.
EXAMPLE. (Dixon). Consider a group G = <a, b, c | a3 = b3 = c2 = I,
c~1ac = a'1, c~1bc = a~1ba = b} and subgroups A = <a>, H = (a,b}.
Then </?>, <«Z?>, and <a2Z?> are three nonconjugate complements of A in H.
K = <c, &> complements ^4 in G. Suppose that X* is a second complement
yl in G. Since <6> = ?<3 G, |JB| = 3, and \G\ =18, then |G/?| = 6 =
\ABjB\ \K*B/B\ = 3|iC*?/?|. So \K*B/B\ = 2. This implies that B c X*.
Therefore KnH = B = K*nH. By Lemma 4.1.2, X* is conjugate to X.
Summarizing: A has precisely three conjugacy classes of complements in a
Sylow 3-subgroup H but ^4 has only one conjugacy class of complements in G.

4.2 Splitting Theorems of Gaschiitz 39
EXERCISES 4.1
1. In Lemma 4.1.2, verify that the mapping y is an isomorphism.
2. Verify each index relation in the proof of Theorem 4.1.4.
3. Verify each part of Corollary 4.1.4.
4. Supply the details in the examples.
5. Let G = NA for an abelian subgroup TV normal in the group G. If A splits over
N n A, then prove that G splits over TV.
6. Let TV be an abelian subgroup of a finite group G such that TV = ®jNi9 Nt a,
Sylow j?rsubgroup of TV for each prime pt\ |TV|. Then G splits over TV iff G splits
over each subgroup N* = ®ji=i Nj for all i.
7. Consider a subgroup B in a group G such that B contains a periodic abelian
subgroup TV for which there exists an integer k divisible by the orders of the
elements of TV and such that ((G: B), k) = 1. If B splits over TV and TV<a G, then
prove that G splits over TV.
8. Prove properties (l)-C) for 7i-groups.
9. Properties (l)-C) for Ti-groups suggest a generalization of the Sylow ^-subgroup.
For a set of primes n, define a Sylow n-subgroup of a group G to be maximal
as a 7r-subgroup.
Prove the next statements.
a) Each group contains a Sylow 7i-subgroup.
b) Each 7r-subgroup of a group is contained in a Sylow 7r-subgroup.
c) The number of Sylow 7i-subgroups in a subgroup of a group G is equal to or
less than the number of Sylow 7i-subgroups in G.
d) J^G(S) = J^Qi^QiS)) for a Sylow 7r-subgroup S of a group G.
e) A normal Sylow 7i-subgroup of a group G is a unique characteristic sub-
subgroup of G.
4.2 SPLITTING THEOREMS OF GASCHUTZ
Only finite groups are to be considered in this section.
Theorem 4.2.1. (Ore, Zassenhaus). If N is a normal subgroup of a group G
and JfG(P) is the normalizer of a Sylow /?-subgroup P of N, then
Proof. If No G, then conjugation by the elements of G induces auto-
automorphisms in N. Since \N\ is finite, the Sylow /^-subgroups of TV are conjugate.
Hence for each g eG there is an element x e TV such that P9 = Px. Thus
gx~x e JTG(P). So g e J^G(P) TV, that is, G = JTG(P) TV = TV ^(P).
Lemma 4.2.2. The intersection of the maximal subgroups of a group G is
a characteristic nilpotent subgroup of G.

40 Theorems on Splitting; Hall Subgroups 4.2
Proof. Let <I> denote the intersection of the maximal subgroups in G. Clearly,
<I> is a characteristic subgroup in G. By Theorem 4.2.1, G = Q>JfG(P) for a
Sylow ^-subgroup P in <I>. If J^G(P) is a proper subgroup in G, then e/FG(P)
is contained in a maximal subgroup M in G. Hence G = Q)M = M leads
to a contradiction. Therefore P <a G and soP<$. Since P is arbitrary,
one concludes that each Sylow subgroup of Q> is normal in <3>. Consequently,
<I> is nilpotent. (See Chapter 1.)
Theorem 4.2.3. If N is a normal subgroup of a group G and \N\ is relatively
prime to its index in G, then G splits over N.
Proof. If iV is nilpotent, then the Sylow /7-subgroup P of N is the Sylow
/7-subgroup of G for a prime p dividing |JV|. Consider G/P', for the commu-
commutator subgroup P' of P. Then (G/FiP/F) is relatively prime to |P/P'|. By
Corollary 4.1.4B), GjP' splits over PjP'. Hence there exists a proper sub-
subgroup A of G such that G = NA. If JV is not nilpotent, then N ? €>, €> the
intersection of the maximal subgroups of G, by Lemma 4.2.2. Hence there
exists a maximal subgroup A of G such that G = NA In both cases
N nA<3 A and (^4: JVn^4) is relatively prime to |Nn^4|. By induction
on the group order, A = [N n A]C Hence G = NC and
Therefore G splits over N by C.
The remaining theorems in this section relate the splitting of a group
with the effect on its Sylow subgroups.
Theorem 4.2.5. (Gaschiitz). For an abelian normal subgroup N of a group
G, G splits over N iff for each prime/? dividing |N|, each Sylow /7-subgroup
P splits over P n N.
Proof. Consider the prime p dividing |N|, the corresponding Sylow
/7-subgroup Np of N, and the direct factor Np, such that N = Np® Np,.
Clearly, both Np and Np> are normal in G. So if G = [N]X, then
G = [Np](NP' A). If P* is a Sylow /7-subgroup of Np, A, then P = lNp]P*
is a Sylow /7-subgroup of G. Furthermore, Pff = [iVp]P*sr for each # e G.
Hence each Sylow /7-subgroup of G splits over N n P.
Whenever N is of prime-power order, the converse is valid by Corollary
4.1.4A). So assume otherwise and use induction. For a prime p dividing
|JV|, denote by P a corresponding Sylow /7-subgroup of G. Np = N n P is
a Sylow/7-subgroup of N since No G. (See Exercise 3.1.7.) There exists a
subgroup Np> of N such that N = Np® Np> and clearly Np, Np> <a G. In-
Inductively, GjNp, = IN/Nfl (C/Np,) for a subgroup C cz Np, So G = NC
and N nC = Np>. However, for primes q dividing |N|, q ?" p, the Sylow

4.2 Splitting Theorems of Gaschiitz 41
^-subgroups of C are the Sylow ^-subgroups of G. By induction, C =
[Np^D. Hence
G = NC = N(NP, D) = ND and JV nD c Np, n D = 1.
Therefore G splits over JV.
Theorem 4.2.5. (Gaschiitz). If the Sylow /^-subgroups of a group G are
elementary abelian for each prime p dividing |G|, then G splits over each
normal subgroup.
Proof. Consider a subgroup JV<a G. Theorem 4.2.4 resolves the case if JV
is nilpotent. Use induction on |G|. If JV is not nilpotent, then there is a Sylow
subgroup P of JV for which JTG(P) ± G. By Theorem 4.2.1, G = JV^TC(P).
Since JV n «/KG(P) <= JV properly and N n J^G(P) <a ^gCP)* tnere exists a
proper subgroup A ofjVG{P) such that, inductively, ^VG(P) = [N n ^VG(Py]A.
So G = NJTG(P) = N((N n JTG(P)) A) = NA. Then N nA=l implies
that G splits over JV.
The converse of this theorem is not valid. It is enough to consider the
symmetric group of degree four, S4, and show that it splits over each normal
subgroup. Clearly a Sylow 2-subgroup of S4 is not elementary abelian.
As another application of Theorem 4.2.4 we can develop a criterion for
the existence of a complement for a normal subgroup.
Definition 4.2.6. For a normal subgroup JV and a subgroup A of a group
G, G is a reduced product of JV by A if G = JV^4 and ^4 does not contain a
proper subgroup 5 such that G = JV?.
Theorem 4.2.7. (D. G. Higman). For a subgroup ^4 and a normal subgroup
JV of a group G, G = [JV] ^4 iff G is a reduced product of JV by A and there
exists for each prime p dividing |G|, a Sylow ^-subgroup P of G such that
P = [JV nP](An P).
Proof. Assume throughout that JV ^ 1 or G. If G = [JV] A, then G is a re-
reduced product of JV by A. Let P be a Sylow ^-subgroup of G containing a
Sylow /7-subgroup Px of A. Since JV n P is a Sylow ^-subgroup of JV and
NnA=l, then P = IN n P] Px = [N n P] (A r\ P).
For the converse, we first prove that if G is a reduced product of JV by A,
for JV <i G, then B = N n A must be nilpotent. Otherwise, by Theorem
4.2.1, B contains a Sylow subgroup Q such that A = B JfA{Q), ^a(Q) a
proper subgroup of A. Consequently G = NA = N (B NA(Q)) = JV JVA{Q).
This is a contradiction.
For each prime p dividing |G|, a Sylow ^-subgroup exists of the form
p = [JV n P] 04 n P). Let Px be a Sylow /^-subgroup of G that contains a

42 Theorems on Splitting; Hall Subgroups 4.2
Sylow ^-subgroup P% of A for which (inP)g Pf. Then there exists
g e G such that
px = ps = [JV n PJ {A n P)* = [JV n PJ (A n P)».
Since JV n D n P) = 1, then P1 = [JV n PJ (A n P). Consequently,
p* = p* n Pt = [iV n Pf] (inP) = [5n P?] D n P).
So for each prime/? dividing \A\, each Sylow/?-subgroup of A splits over the
/^-component of B.
If 1? is abelian, then A splits over B by Theorem 4.2.4. A contradiction
results. For B' = [?, B~] #1, consider the epimorphism A-> AjB'. The
image of P* is isomorphic to 5' P*/B' and it is a Sylow /^-subgroup of AjB'.
Moreover, B'(A n P) n B = B'((A n P) n B) = Bf. Hence B'(AnP)IBf
is the complement of BjB' in ^4/?'. Now apply Theorem 4.2.4 to AjB' and
observe that there exists a subgroup C of A such that yl/U' = [?/?'] (C/Bf).
Therefore ^4 = BC. B n C = B' cz B properly implies that Cci properly.
But then G = NA = NBC = NC yields a contradiction. So B must be the
identity. Hence G splits over JV.
It is not necessarily valid that if G splits over JV <i G, then each reduced
product of G over JV is a semidirect product. Consider G = [^4] C for
A = (a\a2 =1> ® (b\b2 =1>, C = <c|c2 =1>,
and the identities ac = b, bc = ab. Set D = <ca>. Then |?| = 4. Clearly,
G = AD is a reduced product, since D is cyclic and has {(caJ} = (ab)> cz A
as the only nontrivial subgroup of D.
EXERCISES 4.2
1. Prove that each finite group of square-free order splits over each normal sub-
subgroup.
2. Verify that the symmetric group of degree four splits over each normal subgroup.
Hence the converse of Theorem 4.2.5 is not valid in general.
3. Using the results of Exercise 2, show that the property 0> of splitting over each
normal subgroup is not usually subgroup inherited. That is, there exists at least
one subgroup that does not split over each of its normal subgroups. (See Exercise
3.3.1.) However, this same property & is subgroup inherited whenever each
Sylow ^-subgroup is elementary abelian.
4. If a finite group G has a minimal normal abelian subgroup JV that is not con-
contained in the intersection of the maximal subgroups of G, then show that G
splits over JV.

4.3 On Hall 71-Subgroups 43
5. Consider the group G = (a,b,c | a3 = b3 = c2 = 1, ab = ba, c lac = b}.
Show that B = <<z, b} is normal in G and that TV = <(#/>> <a G. Determine if N
has a complement in G. If it has one, then find all the complements.
4.3 ON HALL tt-SUBGROUPS
Except in the following definition, only finite groups will be considered in
this section.
Definition 4.3.1. A subgroup if of a group G is a Hall n-subgroup of G if
H is a 7r-group and (G: H) is a 7i'-number. (If there is no need to identify
7i, one refers to H as a Hall subgroup.)
In finite groups, H is a Hall subgroup iff ((G:/f), \H\) =1. Moreover,
Sylow /7-subgroups are just a special case of Hall 71-subgroups for n consist-
consisting of precisely the prime p.
The reader should verify that if H is a Hall /r-subgroup of a group G
(not necessarily finite), then
1. Hd is a Hall 7u-subgroup of G0 for an epimorphism 6, and
2. if JV is a normal subgroup of G, then N n H is & Hall 7r-subgroup of JV.
The next theorem is just a restatement of Corollary 4.1.4B).
Theorem 4.3.2. If H is an abelian normal Hall subgroup of a group G, then
G splits over H and any two complements of H are conjugate.
Theorem 4.3.3. (Zassenhaus). If H is a normal Hall subgroup of a group G
and either H or G/// is solvable, then any two complements of H are con-
conjugate.
Proof. Use induction on \G\ and assume that G = [//] A = [//] ?.
Consider H to be solvable. By Theorem 4.3.2, H'AjH' and //'JWJ' are
conjugate complements of HjH' in G\H'. This implies the existence of g e G
such that (H'A)g = H'B = H'A9. However, since [H'~\B = IH'~\A9 c G
properly, then by induction, B and A9 are conjugate in jPTI?. Therefore, B
and ^4 are conjugate in G.
Next consider G/H to be solvable and note that there exists an iso-
isomorphism a: A -» 5, via G/H, such that a* = b = ha, heH. Let M #1
denote a minimal normal subgroup of A. Then M is an elementary abelian
/7-subgroup since ^4 is a finite solvable group. M = A implies that A and B
are Sylow ^-subgroups. Hence A and B are conjugate in G. Then assume
that M a A properly and note that [H]M = \H]M* c G properly. So, by

44 Theorems on Splitting; Hall Subgroups 4.3
induction, there exists geHM such that M = (My. Since Ma<a B, then
M = (My <a #s implies that A and jBg are contained in
JfG(M) = AH n JTG(M) = A(H n JTG(M)).
Furthermore, //<G implies that /fn/G(M)< ^G(M)- So # n«/TG(M)
is a normal Hall 7i-subgroup of J^G(M). Moreover, (H n J^G(M))M/M is
a normal Hall subgroup of ^VG(M)/M and has complements ^4/M and B^/M.
By induction, AjM and J5ff/M are conjugate and hence A and B are conjugate
in G.
Corollary 4.3.3. If H is a normal Hall subgroup of a group G, then any two
complements of H are conjugate.
Proof. A result by Feit and Thompson states that a group of odd order is
solvable. So clearly either (G:H) or H must be of odd order and the result
follows from the Theorem.
(The proof of the quoted result is beyond the scope of our work. But the
importance of this corollary indicates that it is appropriate to state it here.)
With this theorem at hand, one is able to develop several results in solv-
solvable groups analogous to the theorems of Sylow.
Theorem 4.3.4. (P. Hall). For a solvable group G, a set of primes n dividing
|G|, and a 7i-subgroup S of G,
1. S is contained in a Hall 7r-subgroup of G, and
2. each pair of Hall 7c-subgroups of G are conjugate.
Proof. Use induction on |G|. Since G is solvable, then G contains a minimal
normal subgroup M that is an elementary abelian /?-group. Moreover,
MS/M is a 7c-subgroup of GjM. Hence there exists, by induction, a Hall
7i-subgroup B/M of GjM such that MS/M c BjM. So MS c B.
Ifp e 7T, then B is a Hall 7r-subgroup of G and A) results. If C is another
Hall 7r-subgroup of G, then MC is a 7r-group (by property C) for 7r-groups),
implying that M c C and that CjM is a Hall 7i-subgroup of GjM. Therefore
B\M and CjM are conjugate in GjM and so ? and C are conjugate in G.
Hence B) is valid.
Suppose p $ n and B <= G properly. By induction, there exists a Hall
Ti-subgroup H of B containing 5. Since (G: 22) is relatively prime to the ele-
elements in 7i, then H is a Hall 7i-subgroup of G. A) follows. If ?* is a Hall
7r-subgroup of G, then MB*/M is a Hall 7i-subgroup of GjM. Inductively,
it is conjugate to BjM. Therefore B being conjugate to MB* implies that ?*
is conjugate to a Hall 7r-subgroup D of B. By induction, /f and D are conju-
conjugate. Hence so are H and ?*. B) is valid.

4.3 On Hall 7r-Subgroups 45
If B = G, p $ n, then G = [M]# for H a Hall 7r-subgroup of G by
Theorem 4.2.3. By Theorem 4.3.2, any two Hall Ti-subgroups of G are conju-
conjugate. So B) is valid. Suppose that S is a Ti-subgroup of G, but not a Hall
7r-subgroup. Then MS cz G = IM]H and MS = MS n MH = M(H n MS)
= [M] (if n MS). Therefore S and H n MS are Hall Ti-subgroups of MS
and by induction they are conjugate, say S = (if n MS)g for some g e MS.
Hence S c H9, H9 is a Hall Ti-subgroup of G, and A) is satisfied.
Recall that each nilpotent group can be expressed as a direct product of
its Sylow /?-subgroups. These Sylow /^-subgroups are obviously pairwise
permutable since each is normal in the group. We can extend this result in a
weaker form to solvable groups, as the next theorem points out.
Theorem 4.3.5 (P. Hall). If pu ...,pn is the set of distinct primes dividing the
order of a solvable group G, then there exists a set {St}n of Sylow/?rsubgroups
St, i=l,...,/?, such that StSj = SjSt for ij = /,...,« and G = StS2 ... Sn.
Proof. Denote by {Ht}n a collection of Hall ^/-subgroups, p/ the comple-
complement of pt in the set of primes dividing |G|, one and only one Ht for each
i=1,...,ji. Set
Sj=C\Ht and Su= f) Hk-
i*j k*i,j
By Exercise 3.1.6, Sj is a Sylow ^-subgroup of G and Su is a Hall {pi9Pj}-
subgroup of G. Since St and S^ are contained in Sij9 St nSj =1, and \Stj\ =
ISJ ISjl, then Su = StSj. This holds iff StSj = SjSt by Corollary 3.1.1.
Assume that three or more primes divide |G|. Consider Sk for k ^ / ?" j i=- k.
Then StSk = SkSt and SjSk = SkSj implies that (StSj) Sk = S^S^j) is a
subgroup of G. The conclusion follows by induction on the number of primes
dividing |G|.
A Sylow basis of a finite solvable group is a set of Sylow /^-subgroups
having the properties of {St}n in Theorem 4.3.5.
Theorem 4.3.6. Any two Sylow bases of a solvable group are conjugate.
Proof. Suppose S? = {Sjn and 0* = {Pjn are two Sylow bases of a solvable
group G. Then Sf = <S,|j # /> and Pf = (Pj\j ^ /> are Hall 7r-subgroups
for n = {pj\j ^ /}. Let J = {Qt}n be a set formed by Qt = Sf, for some
g e G, / = 1,..., /?, that is, J is "conjugate" to ?f {?fg = J). J is also a Sylow
basis. Furthermore, suppose that J has the additional property that the
number of primes for which Q* = P* is a maximum among the set of all
conjugates to Sf. Assume that for some7, Pf 7^ Q*. Then G = Q*Qj and
there exists an heG such that Qfh = Pf by Theorem 4.3.4. However
h = qq, for q e Qf, q e Qp leads to P* = Qf = Qfqq = Q?q. But Q?q = Qf

46 Theorems on Splitting; Hall Subgroups 4.4
since q e Qj, i ^ j. Consequently, the set Qq = &?hq has one more prime such
that Tf = Pf, Tf = (Qfli^j}, than does Q. This contradicts the maxi-
mality conditions on Q. Therefore for all /,
Q? = P? and Qt = Q Q? = f). pf = Pf
So J = 0> and Sf is conjugate to 0>.
The converse of Theorem 4.3.5 is also valid. Its proof will be omitted
since the known proofs are dependent upon methods not developed in this
text.
One of the exercises will examine the symmetric group of degree four
for all Sylow bases. This is the group of least order in which the bases exist
having neither of the factors normal in the group and at least one nonabelian
Sylow subgroup.
EXERCISES 4.3
1. Determine all the Sylow bases in the symmetric of degree four.
2. Verify for primes p and q, p > q, that groups of orders andp2q have a proper
normal subgroup other than the identity element. Then classify all such groups.
3. Identify the Hall Ti-subgroups for each set of primes n arising in the order of
the holomorph H of a cyclic group of order seven. Determine all subgroups
that complement the normal subgroup of order seven in H.
4. Prove properties A) and B) on Hall Ti-subgroups.
5. Let M be a minimal normal subgroup of a finite solvable group. Prove that M
is an elementrary abelian /7-subgroup. (Note that Ap = [ap \ a e G}, for a given
prime/?, is a characteristic subgroup of an abelian group A.)
6. Prove that a "conjugate" to a Sylow basis is also a Sylow basis.
7. Let Sf = {Si}n be a Sylow basis for the finite solvable group G. Define JfG(9*} =
On ^G^d to be the system normalizer of Sf. Prove that JfG(Sf} is nilpotent.
(First show that each Sylow /^-subgroup in j\^G{Sf) is contained in St.)
8. Consider a normal subgroup TV in a finite solvable group G and a Hall 7r-sub-
group H c N. Show that G = NJ^G(H). (Compare with Theorem 4.2.1.)
9. Prove that a normal Hall 7i-subgroup of a group G, G not necessarily finite, is
the Sylow Ti-subgroup of G. (See Exercise 4.1.9 for the definition of Sylow
Ti-subgroup.)
4.4 ADDITIONAL COMMENTS
Consider a linear group G acting on a vector space V. Suppose that Vx
is a subspace of V such that FXG =Vl and V =V1 ® V2- It is not always
valid that F2G =V2. An analogous situation is suggested by Theorem 4.1.4.

4.4 Additional Comments 47
Suppose that the abelian subgroup B is normal in a group G and that B con-
contains a subgroup JV<=a G such that G = [iV]v4. Then of course B = [iV]
(/I n ?). Since under conjugation by elements of G there is an epimorphism
G -» G* c: Aut(jB) such that JVG* = JV, one may look for conditions under
which (A n B)G* = A n B. In general, not too much can be said. However,
whenever N is abelian and has unique kih roots we will arrive at some con-
conclusions that include a case for finite dimensional vector spaces over a finite
field. A group G has unique kth roots if for each element g e G there is
precisely one element xeG such that xk = g, k an integer. The proof of the
next theorem is suggested by the proof of a well-known theorem in linear
groups, called Maschke's Theorem.
Theorem 4.4.1. Consider a finite group T of automorphisms on an abelian
group N (N may be infinite). If N possesses unique kth roots for k = \T\,
then any direct factor A of N for which AT = A, has a complement B such
that BT = B.
Proof. Suppose that N = A ® C and AT= A. Denote by d the projection
of N -> A defined by x = ac t—> a and by fi the endomorphism of N defined
by x h-> x1/k, x1/k denoting the Mi root of x e N. Define the mapping
VxeiV,
r '
the product over all y e F. Of course, 6 e Hom(iV, A). Since
l\ayay'i = ak9 VaeA,
r
then 6\A = la. If xe An Ker@), then x0 = x = 1. Therefore A n Ker@) = I.
Moreover, for each xeN,xd = aeA implies that x°2 = a6 = a = xe. Hence
(x~ 1)exe Ker@). So xeA Ker@). Consequently, JV = A ® Ker@). It
remains to show that
Ker@)r c= Ker@).
For each a e F, aF = F. Consider j e Ker(#). Then
n) (rj) (n
For any element x e N9 xp = 1 iff x = 1. Since j e Ker(#), then
Therefore ^=1^=1. Hence ^ffeKer@). So Ker(O)r c Ker@). Set
B = Ker@) and the conclusion of the theorem is satisfied.
Corollary 4.4.1.1. Let iVbea normal abelian subgroup of a finite group G.

48 Theorems on Splitting; Hall Subgroups 4.4
If N is a Hall subgroup of G and N = A ® B for A o G, then there are sub-
subgroups D and D* of G such that
1. G = [A]D and
2. D = [JV n D]D*.
Proof. Use Theorems 4.2.3 and 4.4.1.
Corollary 4.4.1.2. If a normal elementary abelian /?-group P is a Hall sub-
subgroup of a finite group G, then
such that Py is a minimal normal subgroup of G.
Consider a finite dimensional vector space V over a field i7 of charac-
characteristic p # 0 and denote by G a finite group of nonsingular linear trans-
transformations acting on V. Maschke's theorem states that if (p, \G\) =1, then
V is completely reducible with respect to G, that is, V can be represented
as a direct sum of subspaces each of which is minimal with respect to being
invariant under G.
This chapter has given a selection of some of the basic theorems involv-
involving a group splitting over a normal subgroup. Restrictive as at first they may
seem, they are surprisingly quite adaptable to a variety of situations.
EXERCISES 4.4
1. Prove Maschke's Theorem.
2. Prove the following: If TV is elementary abelian /?-group, \N\ = pn, of a finite
group G = [AQ^, then A is epimorphic to a subgroup of the general linear
group of a vector space of dimension n over a field of characteristic p.

5
NILPOTENT GROUPS;
THE FRATTINI SUBGROUP
5.1 NILPOTENT GROUPS
A normal series of a group G is a finite collection of subgroups {Nt}n of
G such that
1= jv0<1 Nt^i ...<i Nn = G.
If, in addition, Nt<3 G for i = 1,...,«, then the series is called an invariant
series and if each iVf is a characteristic subgroup of G for z = 1,...,«, then
it is called a characteristic series.
One can replace the finiteness in the above by a linearly ordered index
set, or even to an index set with a weaker ordering, and generalize the material
to follow by using transfinite methods.
Definition 5.1.1
1. The upper central chain of a group G is the set of subgroups {Z J such that
1= Z0^Z1 c ... c Zfe c ...
and for each / ^1, ZijZi-1 = Z{GjZi^1). The hypercenter is
The w/?/?er central series is an upper central chain for which there exists a
least positive integer n such that ZtczZi+1 for / < n and Zn = G. The length
of the upper central series is n.
2. An ascending central series of a group G is an invariant series {Nt}n for
which
NJNt-! cZiG/Ni-J for i=l,...,«.
3. The lower central chain of a group G is the set of subgroups {Dt} such
that
G = Do ^ Dx 2 ••• 2 Dfe 3 •••
and for / ^1, ?>, = [D;_i, G]. The hyper commutator is
= H D,.

50 Nilpotent Groups; the Frattini Subgroup 5.1
The lower central series is a lower central chain for which there exists a least
positive integer n such that D^_x dD; for i < n and Dn =1. The length
of the lower central series is n.
4. A descending central series of a group G is an invariant series
such that [Lf, G] c L/+1 for / = 1, ...,«.
5. A group G is nilpotent if G has an upper central series. The class of the
nilpotent group is the length of the upper central series.
Theorem 5.1.2. For a group G
1. the elements of the upper and lower central series are characteristic sub-
subgroups of G,
2. the hypercenter and the hypercommutator are characteristic subgroups
of G,
3. if {Nt}n is an ascending central series, then Nt <= Zt for i = 0,...,«,
4. if {Ni}n is a descending central series, then Dt c Nt for i = 0,...,«,
5. if G is nilpotent of class n, then Dn =1 but Dn_1 #1, and
6. if G is nilpotent, then G is solvable.
A) follows by induction and B) is a consequence of A). As for C),
use induction. Clearly No c Zo and so assume that Nk c zfe for fc ^ 0.
This implies that [Nfc+1, G] c jVfe c Zfc. Hence ZkNk+1/Zk c Z(G\Zk).
Therefore iVfc+1 c Zfc+1. Again, use induction in D) noting that Do c iV0.
Assume for fc ^ 0, that Dfc c Nk. Then Dfc+1 = [Dfe, G] c [JVfc, G] c Nk+1.
For E), note that if G is nilpotent of class n, then
G = Zn z> Zn_x z> .-¦ z> Zo =1
is a descending central series. By D), Dj c Zn_j for j = 0,...,«. Hence
Dn = 1. If Dfc = 1 for k < n, then G = Do ^ Dt z> ... Dfe = 1 is an ascending
central series. Therefore by C), Zn_k = G. If k =? n, then a contradiction
is reached. F) is a result of the definition of a nilpotent group and the defini-
definition of a solvable group.
Corollary 5.1.2.1. For a finite group G having hypercenter H and hyper-
hypercommutator D, GjD and H are nilpotent. Each abelian group is nilpotent.
Corollary 5.1.2.2
1. If a group G has an ascending central series {Njn, then G is nilpotent
of class k ^ n.

5.1 Nilpotent Groups 51
2. If a group G has a descending central series {Lt}n, then G is nilpotent of
class k < n.
Corollary 5.1.2.3. Each finite j^-group is nilpotent.
Proof. A finite/7-group P has a nontrivial center Z(P) and the homomorphic
image of a p-group is a />-group. Consider PjZ{P). Inductively, an upper
central series exists and it terminates with the group.
Corollary 5.1.2.3 is not necessarily valid for infinites-groups. An example
is the /?-group Cp \ Cpoo. By properties C) and (8) for wreath products in
Section 3.4, this group is a />-group with trivial center.
Theorem 5.1.3
1. Each homomorphic image of a nilpotent group is nilpotent.
2. Each subgroup of a nilpotent group of class n is nilpotent of class k ^ n.
Proof. Let G denote a nilpotent group of length n and let 6 be a homomor-
phism of G. Then {Zf}w forms an ascending central series for Gd. So A) is
valid.
Let Di(H) denote the zth term in the lower central chain for a subgroup
H in G. Clearly, D0(H) c Do = G. Assume, for an integer / ^ 0, that
Dt(H) <= Dt. Then Di+1(H) = [Dt(H), H~] c [?>., G] = Di+1. Therefore,
if G is nilpotent of class n, then Dn(H) =1. Hence for an integer k < 77,
=1. This proves B).
As the symmetric group of degree three indicates, the extension of a nil-
potent group by a nilpotent group is not usually nilpotent.
Theorem 5.1.4. If N is a normal subgroup of a nilpotent group G and N ^ 1,
then iVnZ(G) #1.
Proof Consider the upper central series {Zt}n and suppose that N nZt = 1.
For some integer 7 > 1, N nZj ^l but N nZj_1 = 1. Since JV<i G, then
[JV n Z;, G] c jv n Z,-_ x = 1. This implies that JV n Zj c zx. So N n Zx # 1
whenever JV ^1.
Theorem 5.1.5. Each proper subgroup ^4 of a nilpotent group G is properly
contained in its normalizer.
Proof For the upper central series {Zt}n of a nilpotent group G, consider
the integer j such that Zj a A and Zj+1 ? A. Then
[Z^^JciEZ^^GlczZ,,
So there exists an element zeZj+1\A such that \_z,A~\ ci. Therefore
Az — A and this implies that A c JfQ(A), properly.

52 Nilpotent Groups; the Frattini Subgroup 5.1
Corollary 5.1.5. If a nilpotent group G has maximal subgroups, then each
maximal subgroup is normal in G.
Proof. Just apply Theorem 5.1.5 to a maximal subgroup.
Theorem 5.1.6. Each subgroup H of a nilpotent group G is subnormal in
the group. Moreover, if is a member of a normal series for G.
Proof. Consider the upper central series {Z JM of a nilpotent group G. Form
the set {HZt}n,
H = HZ0 c HZX c ... c HZn = G.
Then [Zj+^i/ZJ c [Zf+1,G] ? Zf, thatis,Zi+1 <^JfG(EZ^). Hence {HZJB
is a normal series and so //<a<a G. Since if is nilpotent, consider its upper
central series {Zjf. Then
1 'y >y #^ y y rj f-f 7 CH J~f7 d ••• J-f7 C*
is a normal series for G.
EXERCISES 5.1
1. Prove Corollaries 5.1.2.1 and 5.1.2.2.
2. Prove that a maximal normal abelian subgroup TV of a nilpotent group G is also
a maximal abelian subgroup of G.
3. Show that if M is a maximal subgroup of a /?-group G and |G| = pn, n > 0,
then |Af| = jp".
4. Prove that if G is a /?-group and G/G' is cyclic, then G is abelian. If also G is
finite, then G is cyclic.
5. If G is a finite nonabelian /?-group of order y, n > 0, then show that |G/G'| ^
6. Show that each nonabelian group of order p3 is nilpotent of class two.
a) For p = 2, show that the only nonabelian 2-groups of order eight are the
dihedral and the quaternion groups.
b) For p ^ 3, show that the only nonabelian p-groups of order p3 are those
given in Chapter 1.
7. Classify all i?-groups oi order at most p3. (See Exercise 6.)
8. If \G\ = p2, then G is abelian. Furthermore,
a) if G is cyclic, then |Aut(G)| = p2 - p, and
b) if G is noncyclic, then | Aut(G)| = p(p -1J (p +1).
9. If G is a finite /7-group and contains a normal subgroup iV of order p2, then N
is contained in the center of some maximal subgroup of G.
10. Let G = D PI C2, where D is the dihedral group of order eight and C2 is
cyclic of order two. Show that the lengths of the upper and lower central
series are equal, but that the terms of each series do not coincide.

5.2 The Sylow Structure of a Nilpotent Group 53
5.2 THE SYLOW STRUCTURE OF A NILPOTENT GROUP
The general theory of nilpotent groups has been developed around properties
found in the finite case. The finite case reduces to the study of /^-groups
because of the next theorem.
Theorem 5.2.1
1. Each Sylow ^-subgroup of a nilpotent group G is normal in G.
2. A nilpotent group G contains a subgroup S such that
a) S is a direct product of the Sylow ^-subgroups of G and
b) S contains each periodic element in G.
Proof. Consider a Sylow ^-subgroup P of G and the chain
Pc/G(P)c/G(/G(P)).
The containment is proper by Theorem 5.1.5. Since P<3 */TG(P), then P is
actually characteristic in JfG{P). (See Exercise 3.2.5.) So P</G(/C(P)),
The contradiction implies that JVG(P) = G. Hence A) is valid.
It is clear that if P and Q are two Sylow subgroups with respect to distinct
primes, then P n Q = 1. Consequently, the direct product S exists in G and
each element of S has finite order. This proves 2(a). Denote the set of all
elements in G having finite order by F and consider an element g e F. Then
\g\ = p\l -"pnn, {Pi}n the set of all distinct primes dividing \g\. Since <g>
is a cyclic group, then it is a direct product of Sylow /?rsubgroups, each cyclic
of order pfl, i =1, ...,n. Consequently, the generators of <(#> are contained
in S. This implies that g e S. Therefore F ?= S. Since S c= F, then S = F.
Before examining the proof of the next theorem, the reader should first
verify the commutator identity,
for elements a, b, and c in a group.
Theorem 5.2.2. If A and B are normal nilpotent subgroups of a group G,
then AB is a normal nilpotent subgroup of G.
Proof. Clearly AB<3 G. Adopt the notation that [At, ...,^4J denotes
Denote by Kt the subgroup generated by all [Cl5 ...>C[\ for Cj = A or
BJ =1,..., z, over all possible arrangements. Since A and B are normal in
G, then Kt^ G. Consider KtIKi+1. Note that
[[c1?..., cj, aZ>] = [[a, ..., cj, 6] [[c1?..., cj, a]6 e Ki+U

54 Nilpotent Groups; the Frattini Subgroup 5.2
for CjeCj, j = 1,...,/, aeA, and beB. So [Ki9AB] ^ Ki+1 and hence
KtIKi + 1 c Z(G/Zi+1). For /?, the nilpotent class of yl, and the nilpotent
class m of B, it follows that Km+n_i = 1. Let Xo = /IB. Then
is an ascending central series for AB. By Corollary 5.1.2.2, AB is nilpotent.
Corollary 5.2.2. The product of a finite collection of normal nilpotent sub-
subgroups in a group G is a normal nilpotent subgroup of G.
That the condition of normalcy cannot be relaxed on at least one of the
subgroups in Theorem 5.2.2 is exemplified by the symmetric group of degree
three.
Theorem 5.2.3. The following properties are equivalent for a finite group G:
1. G is nilpotent.
2. For each proper subgroup A of G, A is properly contained in JVG(A).
3. Each maximal subgroup of G is normal in G.
4. G is a direct product of its Sylow subgroups.
Proof. B) follows from A) by Theorem 5.1.5 and clearly B) implies C).
By Corollary 5.2.2, D) implies A). Suppose that C) holds. Denote a Sylow
/7-subgroup of G by P. If JVG{P) ^ G, then there exists a maximal subgroup
M of G containing JfG(P). By Theorem 4.2.1, G = M J^G(P) = M. A con-
contradiction is obtained. So P <a G. Let S denote the direct product of the Sylow
/^-subgroups for each primep\ \G\. Then S = G by Theorem 5.2.1. Therefore
C) implies D).
It is readily seen that this theorem is a very strong form of Theorem
4.3.5. Moreover, there are many well-defined characteristic subgroups, say
&~(G), that identify the structure of nilpotent groups and have the property
that
for example the center and the commutator subgroups. Hence the study of
finite nilpotent subgroups is essentially a study of/?-groups. Because of the
strong structural properties associated with nilpotency, it is often quite useful
to be able to identify the nilpotent subgroups in an arbitrary group. An ex-
example is that of the Sylow subgroups.
Theorem 5.2.4 (Sah). A subgroup A of a finite group G is nilpotent iff HA
is nilpotent for H the hypercenter of the group.

5.2 The Sylow Structure of a Nilpotent Group 55
Proof. Clearly HA nilpotent implies that A is nilpotent. Consider the con-
converse. By definition, there exists a least positive integer k such that
1= Zo cz Z1 cz ... cz Zk = H
such that Zi+1/Zi = Z(G/Zi). Since Zt<3 G and Zt c f-M, then
Z{G/ZV) c ZiHA/Zi) for / = 0,..., ? - 1.
Moreover, HA/H is nilpotent and so Z(HA/H) ^1. Clearly, the series can
be extended to Zk cz Zk+1 cz ••• a Zt = HA such that
for / = &,..., ? — 1.
Since Z^<i jFL4 for / = 0,..., f, it follows that {Zt}t is an ascending central
series for HA. Therefore HA is nilpotent.
Corollary 5.2.4. The hypercenter is contained in each maximal nilpotent
subgroup of a finite group G.
Theorem 5.2.5. (N. ltd). If A is a nilpotent subnormal subgroup of a finite
group G, then there exists a normal nilpotent subgroup B that contains A.
Proof. Let C be a maximal nilpotent subnormal subgroup of G that contains
A. Suppose C^fi G. Then there exist subnormal subgroups Cx and C2 such
that CcQc^, C<Q but C^a C2, and Cx <j C2. Hence there is a
subgroup C* in Cx conjugate to C but different from C1# By Theorem 5.2.2,
CC* is a normal nilpotent subgroup of C±. So it is subnormal in G and
properly contains C. The contradiction arose by assuming that C^ G.
EXERCISES 5.2
1. Verify the commutator identities
[a, be] = la, d][a, bj and [ab, c] = [_a, c~]b[b, c].
2. In Section 2.1 it was shown that if G = Gx ® ••• ® Gn, then Z(G) = Z(GX) ®
••• ® Z(Gn) and G' = G/ ® ••• ® Gn'. If G is nilpotent, can this be generalized
for determining the upper and lower central series for G?
3. Prove Corollary 5.2.2. Determine the nilpotency class of a direct product of a
finite collection of normal nilpotent subgroups in a group.
4. For subgroups A, B normal in a finite group G, if G/A and G/B are nilpotent,
prove that G/A n B is nilpotent. Show that the hypercommutator, D, is the
unique characteristic subgroup having the property that G/D is nilpotent and if
G/A is nilpotent, then D c A.
5. Show the existence of a unique characteristic subgroup Fin a finite group G
such that F is a normal nilpotent subgroup of G and if A is a subnormal nil-
potent subgroup of G, then A c: F. Moreover, the hypercenter is contained in F.
Give an example of an infinite group in which the hypercenter is not in F.

56 Nilpotent Groups; the Frattini Subgroup 5.3
6. Show that a group G is nilpotent iff the following conditions hold: Let Inn(G)I+1
= Inn(Inn(G),) where InnCG^ = Inn(G). There exists an integer k ^ 1 such
that Inn(G)fe is the identity element.
7. Let G be a finite ^-group. If G/A is cyclic for a subgroup A c Z(G), then G is
abelian. Does this hold in general for a finite nilpotent group G?
8. Let A be a maximal nilpotent subgroup in a group G. (A is not necessarily
normal in G.) Prove that JVG(A) = JTG{JVG{A)).
9. Prove that each subnormal Ti-subgroup of a group is contained in a normal
7i-subgroup. (See the proof of Theorem 5.2.5.)
5.3 THE FRATTINI SUBGROUP
Two important nilpotent subgroups of finite groups are the Frattini subgroup
and the Fitting subgroup. The structure of finite nilpotent groups is associated
with the first, whereas the second is useful in determining the structure of
finite supersolvable groups. The Fitting subgroup will be discussed in the
next chapter. As we might expect, many desirable properties of these sub-
subgroups that hold in finite groups are not valid in all infinite groups.
Definition 5.3.1
1. An element g in a group G is a nongenerator of G if G = <JC> for each
subset X in G such that G = <#, Xs).
2. The set of all nongenerators in a group G is called the Frattini subgroup
of G and it is denoted by <?(G). (<$(G) is also referred to as the ^-subgroup
of G.)
In passing, it should be mentioned that a "generator" of a group G is
defined as a non-nongenerator.
Theorem 5.3.2
1. <J>(G) is a subgroup of a group G.
2. <1>(G) is the intersection of all proper maximal subgroups of a group G
if G admits proper maximal subgroups; otherwise <?(G) = G.
3. If 0 is a homomorphism of a group G, then <D(GH c O>(G0). $(G) is
characteristic in G.
4. If G is finite, then €>(G) is nilpotent.
5. Let Y be a subset that is contained in a finitely generated subgroup of
cD(G). If there exists a nonempty subset J^ of G such that G = <7, X>,
then G =

5.3 The Frattini Subgroup 57
Proof. The identity element belongs to $(G). So suppose that a,b e <J>(G)
and that X is a subset of G for which G = (ab, Xs). Then
, X} c (a, b, X} c <Z>, *> = <X> = G.
Clearly, a eO(G) since <a, X> = <tf~\ *>. So 0>(G) is a subgroup of G,
that is, A) is valid.
For B), assume that G admits proper maximal subgroups. If for x e <?(G)
there exists a maximal subgroup M such that x?M, then G = <x, M>. But
G ^ M. Therefore <X>(G) is contained in the intersection of all proper maximal
subgroups of G. Consider an element x e G for which there exists a subset
X such that G = <x, Z> but <Z> # G. By Zorn's lemma there exists a
subgroup M maximal with the property that M contains {X) and x$M.
If M is not a proper maximal subgroup, it follows that M is properly con-
contained in a subgroup H / G. But then ieH implies that H = G. Therefore
M is a proper maximal subgroup of G. Consequently, the intersection of the
proper maximal subgroups must be contained in ^(G). Equality follows. If
G contains no proper maximal subgroups, then the last argument leads to a
contradiction. This implies that each element of G is a nongenerator.
As for C), it is enough to apply B) to the homomorphic image G6. Note
that <?(G0) is the image of the intersection of the maximal subgroups of G,
if they exist, that contain Ker@).
D) is a restatement of Lemma 4.2.2.
For E), let Y be contained in a finitely generated subgroup
<xl5 ..., xn}^ <D(G). Suppose that G = < Y, X} for a nonempty set X. Then
g = <y,x> c «x1? ...,xny,x> = <x1? ...,*„,*> = <x>.
Corollary 5.3.2. If N is a subgroup normal in a group G and N c <I>(G),
then <D(G/JV) = <D(G)/iV.
EXAMPLE 1. That a Frattini subgroup can coincide with the group itself
is exemplified by the additive group Q of the rational numbers. Suppose
there exists an element x e Q and a nonempty set X ^ Q for which x $ X,
(Xy c Q, and Q = <x, Xy. For each element yeX, there exist nonzero
integers n and m such that nx = my. This implies that nx e <(^>. For the
element (l/«)x there exist an integer k and an x*e<X> such that(l//?)x =
kx + x*. Therefore x = k{nx) + «x* e <X>. Hence G = <X>. A contra-
contradiction arises. Therefore ^>(Q) = Q. By our previous results, this also implies
that Q has no maximal subgroups. The significance of Theorem 5.3.2E)
is indicated by the fact that Q = <l/w!|«=l,2, ...> and so Q can be gen-
generated by a proper subset from which any finite subcollection can be deleted.

58 Nilpotent Groups; the Frattini Subgroup 5.3
EXAMPLE 2. Not every infinite group coincides with its Frattini subgroup.
Consider the additive group of integers, Z. Each prime integer generates a
maximal subgroup and their intersection is the zero of Z.
EXAMPLE 3. Consider the group G = [<a>] <6> such that a5 = b4 = 1 and
b'xab = a2. Then $(G) = 1. For the epimorphism 6: G -> Gj{a) ^ <Z?>,
O>(G0) has order two and d>(G)e is the identity. Hence C) in Theorem 5.3.2
seems to be the best possible result unless other conditions are imposed.
EXAMPLE 4 (P. Hall). In general, <X>(G) is not nilpotent. Let G be the group
C5oo and Gn be the subgroup of order 5". Clearly, Gx has an automorphism
Qx that is cyclic of order four. Inductively, if Gn has an automorphism 6n
of order four, then there exists 6n+l e Aut(Gn+1) such that |0W+1| = 4 and
0n+1\Gn = 6n. It can be concluded that G has an automorphism 6 of order
four. Set G* = [G] <#> for <#> isomorphic to itself in Aut(G). Without
loss of generality, assume that G and <#> are contained in G*. If A is a sub-
subgroup of G* such that A n G = Gn, then A c ^4Gn+1 c G*. Therefore each
maximal subgroup of G* contains G. This implies that <I>(G*) contains G.
Since [G*, G*] c G, then each maximal subgroup M of G* is normal in G*.
Moreover, M n <#> is a maximal subgroup in <0>. Hence <I>(G*) = [G] <#2>
and it is solvable but not nilpotent.
EXAMPLE 5. There are groups in which the Frattini subgroup is proper,
but this does not imply that every subgroup is contained in a maximal sub-
subgroup. For example, consider G = (x \ xA = 1> nQ, Q the additive group of
rationals. The subgroup {A<X>,#) \qeQ] that is isomorphic to Q must be
contained in each maximal subgroup M, for otherwise its intersection with
M would indicate that Q had a proper maximal subgroup. So <D(G) ^
<x2> ITQ. We should be led to a similar contradiction if we assumed that a
subgroup isomorphic to <(x, 0Q)> is contained in a proper maximal sub-
subgroup of G.
Theorem 5.3.3. If N is a normal subgroup of a group G such that N is con-
contained in a finitely generated subgroup of ®(A) for some subgroup A of G,
then JV <= O(G).
Proof. If G = <D(G), then the result follows. So assume that <D(G) c G
properly. This implies that G has proper maximal subgroups. If N ^ M,
M a maximal subgroup of G, then G = iVM. By the modular identity,
A = AnG = An NM = N(A n M).
Since A n M is proper in A and N = A n M by Theorem 5.3.2E), then a
contradiction is reached. Therefore N must be contained in each maximal
subgroup of G, that is, JV c <3>(G).

5.3 The Frattini Subgroup 59
Corollary 5.3.3. If N is a normal subgroup of G and ®(JV) is finitely genera-
generated, then <D(iV) c 0>(G).
Just note that <D(iV) is characteristic in N and hence it is normal in G.
Corollary 5.3.3 does not always hold if N is not normal in G. Consider
G = [<tf>] <6> of Example 3. 0>(G) = 1 and <?«&» has order two.
Theorem 5.3.4. For a group G, G' n Z(G) c <D(G).
Pro<?/. This is clearly valid whenever G = ®(G). Assume otherwise. If
there exists a proper maximal subgroup M such that G' n Z(G) ? M, then
G = (G' n Z(G)) M and therefore M^G. This implies that (Gf n Z(G)) c
G' ^ M. So G = M and a contradiction arises. Hence G' n Z(G) c O(G).
Theorem 5.3.5. If iV is a normal subgroup of a group G and N is contained
in a finitely generated subgroup of 3>(G), then Inn(iV) c <D(Aut(iV)).
Proof. Conjugation by elements of G induces automorphisms in N. So there
exists a mapping 6: G -> Aut(iV) such that JV0 = Inn(JV). Since JV c O(G),
then Inn(iV) c (J>(GH and Inn(iV) is contained in the image of a finitely
generated subgroup. Therefore the image is finitely generated. Inn(JV)<a
Aut(N) implies that Inn(N) c O(Aut(N)) by Theorem 5.3.3.
Theorem 5.3.5 can be used to show that not every /?-group can be the
Frattini subgroup of some other />-group. Consider the quaternion group
Q8. The automorphism group of Q8 is isomorphic to S4, S4 the symmetric
group of degree four, and <D(S4) = 1. But, if Q8 is the Frattini subgroup of
some group Q*9 then Inn(Q8) must be the identity. This is a contradiction.
More generally, no nonabelian p-group of order p3 can be the Frattini sub-
subgroup of any group. That proof will not be given.
Theorem 5.3.6. Let G = JJI Gt. Suppose that for each iel, Gt satisfies one
of the following conditions:
1. Gt is finitely generated or
2. ®(Gt) is finitely generated.
Then
i
Proof. If G; has a proper maximal subgroup M^ for some /, then
},) for J = I\i
is a proper maximal subgroup of G. So

60 Nilpotent Groups; the Frattini Subgroup 5.3
Hence
For each / such that O(G^) is finitely generated it follows that ®(Gt) U (Ylj 1 j),
for lj the identity element of GJ? is contained in 3>(G) by Corollary 5.3.3.
Hence, suppose that Gt is finitely generated for some iel. Since Gt has
proper maximal subgroups, then, as noted in the first part of the proof, G
has proper maximal subgroups. Therefore <1>(G) is contained in G properly.
Suppose that
O(H) ? O(G) n H for H = GtU (l\lj)'
Then there exists a proper maximal subgroup M of G such that G = <$(H)M.
So, by the modular identity, H = O(H) (H n M). Clearly // n M # 1 or H.
Hence H n M is a proper subgroup of //. Since H is finitely generated, then
H n M is contained in a proper maximal subgroup /f * in H. (Apply Zorn's
lemma.) Therefore H = <I)(H) H* = H*. This contradiction came about by
assuming that ®(H) ? <D(G) n H. Hence fl/ $(Gf) ? O(G). Consequently,
One might expect that Theorem 5.3.6 could be proven without any con-
conditions on either Gt or O(G,-). Unfortunately, the proof of such a theorem is
dependent upon whether or not there exists a simple infinite group Gt such
that <D(G;) = Gt. This is not known.
The theorem is helpful in the study of finite nilpotent groups G, since
d>(G) = O(Pi) ® ••• ® O(Pn), Pj the Sylow ^-subgroup of G, for which
G = Px ® ••• ® Pn. This, together with the next theorem, explains in part
why attention is directed toward the study of />-groups.
Theorem 5.3.7. If a group G is nilpotent, then G' c <I>(G).
Proof. Suppose G = G'^4 for a proper subgroup ^4 of G and consider the
upper central series
1= Zo c Zx c ••• cZn = G.
Form {^4jw such that ^4^ = Z^. For some least integer k, G = Ak and ^4fc_!
is a proper normal subgroup of G such that G\Ak_x is abelian. Therefore
G' c Zfc_!74 and G'A c y4fc_x is a proper subgroup of G. The contradiction
arose by assuming that G
Corollary 5.3.7. A finite group G is nilpotent iff G' c 0>(G).
Proof. Exercise.
The corollary does not hold in general for infinite nilpotent groups.
For example, let G* = [C5oo] <#> be defined as in Example 5.3.5. Then
[G*, G*] ? ^>(G*M but G* is not nilpotent.

5.3 The Frattini Subgroup 61
Theorem 5.3.8. Let G be a finite group. A subgroup JV <a G is nilpotent iff
the hypercommutator D of JV is contained in
Proof. Suppose that D c $(G). Since D is characteristic in JV, then D <a G.
Denote a Sylow /^-subgroup of JV by P and consider DP/D. Since DP/D is a
characteristic subgroup of JV/D, it follows that DP<a G. By Theorem 4.2.1,
G = (DP)JTG(P) = PJTG(P). Hence P<G. Since each Sylow /?-subgroup
of JV is normal in JV, then JV is nilpotent and D = 1. On the other hand, if JV
is nilpotent, then JV 0>(G)/<D(G) ^ N/N n <X>(G) is nilpotent. This implies
that D <= JV n O(G) <= O(G).
Corollary 5.3.8. If yl is a normal subgroup in a finite group G and AjA n €>(G)
is nilpotent, then A is nilpotent.
The next two theorems depend on Theorem 5.2.5. Their proofs are left
as exercises.
Theorem 5.3.9. A subnormal subgroup A in a finite group G is nilpotent iff
A/A n <D(G) is nilpotent.
Theorem 5.3.10. A subnormal subgroup A of a finite group G is nilpotent
iff A' c
Theorem 5.3.11. Let P be a Sylow /^-subgroup of a finite group G for a prime
/> dividing |O(G)|. The Sylow ^-subgroup of <?(G) is contained in the normal
subgroup JV of least order that contains <P(P).
Proof. If JV = 1, then P is an elementary abelian /7-group. Hence P splits
over the Sylow/^-subgroup P* of O(G). By Corollary 4.1.4, G splits over P*.
Hence P* =1. If JV ^ 1, then G/JV has elementary abelian Sylow /^-subgroups.
Apply the first part of the theorem to obtain NP^ c JV. Therefore P* ^ N.
Before leaving this section, several comments should be made concerning
cyclic groups. It is evident that a finite/>-group G is cylic iff GI®(G) is cyclic.
An infinite cyclic group is isomorphic to the additive group of integers, Z.
Since each prime integer generates a maximal subgroup of Z, then O(Z) is
the zero element. But this is not necessarily the case for a finite cyclic group.
The finite cyclic group G can be divided into two classes: A), G is a power
of a prime p and B), G is divisible by two or more primes. The first class is
distinguished by the fact that G has precisely one maximal subgroup. In
B), G is a direct product of its cyclic Sylow subgroups. The next theorem
relies on these properties and it is needed in Section 6.2.
Theorem 5.3.12. If A is a finite cyclic group, then A contains a characteristic
series 1= Ao a A1 c ••• a An = A such that Aj+l/Aj is prime cyclic for
.7 = 0, ...,h-1.

62 Nilpotent Groups; the Frattini Subgroup 5.4
Proof. Since A is abelian, then A = Sx ® -•• ® St, for {Sj}t the distinct
Sylow subgroups of A. Each Sj is a characteristic subgroup of G, Sj is the
power of a prime pj9 and Sy is a cyclic subgroup. Moreover, Sj contains a
series
such that Sjj is a maximal subgroup of Sjt i+1, \SJti+1ISjti\ = pj9 and SJti is
a characteristic subgroup of Sj. Hence Sjfi is a characteristic subgroup of A.
The series
c ... cz A
satisfies the conclusion of the theorem.
Note that the series formed in the proof of Theorem 5.3.12 is not unique
unless A is a power of a prime integer. However by the Jordan-Holder
theorem, all such series have the same length and there is a bijection between
the factors in isomorphic pairs.
EXERCISES 5.3
1. Prove Corollary 5.3.2.
2. Prove Corollary 5.3.7.
3. Prove Corollary 5.3.8.
4. Prove Theorem 5.3.9.
5. Prove Theorem 5.3.10.
6. Show that if TV is a normal subgroup of order;?2 contained in <D(G), in a finite
p-group G, then N is contained in the center of <I>(G).
7. Show that ifp divides |O(G)| for a finite group G, then/? divides |G/$(G)|.
8. Prove that <J)(H) cz cp(G) for each subgroup H of a finite nilpotent group G.
Find an example in which <!>(//) = ^(G) whenever H is a proper subgroup of G.
9. For a finite nilpotent group G, show that G/<I>(G) is a direct product of elemen-
elementary abelian ^-groups, There is one direct factor for each prime p dividing G.
10. Given that G is finite and G/Q)(G) is cyclic, then show that G is cyclic. Show
that this conclusion is not necessarily valid whenever G is infinite by consider-
considering g = c5 n c5oo.
11. Given that G is finite, nilpotent, and generated by the conjugacy class of an
element, then show that G is cyclic.
5.4 ADDITIONAL REMARKS ON THE FRATTINI SUBGROUP
The precise nature of the Frattini subgroup seems to be elusive and at times
creates problems rather than helps solve them. Some of its characteristics
will be brought out in this section. The theorems will be concerned primarily

5.4 Additional Remarks on the Frattini Subgroup 63
with finite groups but a few of them can be extended to certain classes of
infinite groups.
Basis Theorem (Finite Groups). If a set X together with ®(G) generates G,
then X generates G.
This is just Theorem 5.3.2E) applied to finite groups.
Definition 5.4.1. A subset lofa group G is a minimal generating set (or an
irreducible set of generators of G) if G = <X> and for each proper subset
X* of X, <X*> c G properly.
Clearly, one can obtain a minimal generating set for any finite group.
However, for infinite groups this is not always the case, as Example 1 in
Section 5.3 points out. The proof of this is almost identical to that given
in the example and it is left to the reader as an exercise.
The word "conjugate" has been used in the past with respect to the group
of inner automorphisms. An extended use of the word is that x is conjugate
to y with respect to an automorphism group A if A contains an element a
such that xa = y.
Theorem 5.4.2. If a finite group G has a minimal generating set with n ele-
elements, then |Aut(G)| divides |<D(G)|" |Aut(G/O(G))|.
Proof. Consider a e Aut(G), x e G, and the coset x $(G). Then xa can be
associated with xa <1>(G). This association defines a homomorphism
6: Aut(G)-* Aut(G/O(G)). Ker@) fixes the cosets of G with respect to
<D(G) and furthermore Aut(G)/Ker@) is isomorphic to a subgroup
A <= Aut(G/O(G)). Hence |Aut(G)| divides |Aut(G/O(G))| |Ker@)|. A set
gu ...gn of distinct coset representatives, for n = (G: <t>(G)), generates
G. There are as many distinct systems xu ..., xn conjugate to gu ..., gn under
Ker@) as there are elements in Ker(#). The |O(G)|w representative systems
decompose into classes conjugate to one another under A such that each
class contains |Ker@)| elements. Therefore |Ker@)| divides |O(G)|w. Con-
Consequently, |Aut(G)| divides \®(G)\n |Aut(G/O(G))|.
Theorem 5.4.3 (Burnside Basis Theorem). For a finite p-group G let
|G/<D(G)| = p\ G/(D(G) = <*! O(G), ...,xB®(G)>
and G = (yu ...,yr}. Then
1. G = <*!, ...,*„> and
2. G/O(G) = <j;I.1®(G),...,^BO(G)>.
Proof. Statement A) follows from the Basis Theorem. Since each subgroup
H of G is subnormal in G, then <?(#) c €>(G) for all subgroups of G. There-

64 Nilpotent Groups; the Frattini Subgroup 5.4
fore G/<i>(G) is elementary abelian and this implies that there are precisely
n elements in a minimal generating set. So if G = <j/l9 ...,yr}, then elimina-
elimination of superfluous elements in y1 ®(G), ...,jr<l>(G) has to yield n distinct
cosets.
Theorem 5.4.4. If \G\ = pn and |O(G)| = /?"~r, then |Aut(G)| divides
Pr{n~r)l\{pr -Pl) for / = 0,...,r-l.
i
Proof. For an elementary abelian p-group of r generators there are
possible generating systems of r elements. This implies the same number of
automorphisms and only that number. So, by Theorem 5.4.2, |Aut(G)|
divides
|O(G)|'|Aut(G/O(G))| = (y'-T(/-i)...(/_^-i).
Corollary 5.4.4. An automorphism of a p-group G that induces the identity
automorphism on G/cD(G) has order a power of p.
Another result on the automorphisms of finite /?-groups, due to W.
Gaschiitz, is stated without proof: Each finite nonabelian p-group has an
automorphism of/?-power order that is not an inner automorphism.
A useful result for finite />-groups is the following:
Theorem 5.4.5. For a finite p-group G, <I>(G) = G' Gp, Gp the subgroup of
G generated by the pth power of all the elements of G.
Proof. By Theorem 5.3.7, G' c <D(G). Then
) c O(G)
since G/G' Gp is elementary abelian. Therefore, Gp c <D(G). Since G/G' Gp
is elementary abelian, then <1>(G) c G; Gp. Hence O(G) = G' Gp.
The reduced product was introduced in Section 4.2.
Theorem 5.4.6. For each normal subgroup N of a finite group G, a reduced
product G = N.4 exists. Moreover, N n A ^ Q>(A).
Proof The result is clearly valid if N ^ O(G). Assume A^ ^ ^(G). Then
there is a maximal subgroup M such that N ? M and G = NM. The finite-
ness of G implies that M contains a subgroup ^4 such that G = Ny4 is a reduced
product. If N n A ? $>(A) then A contains a proper maximal subgroup A*
such that G = NA*. Hence a contradiction is reached.
Reduced products are not unique and even the factors A do not have to
be conjugate, as the next example points out.

5.4 Additional Remarks on the Frattini Subgroup 65
EXAMPLE. Consider the p-gvoup G = (a,b \ a9 = b3 = 1, b~lab = a4}.
G = [<#>] {b} is a reduced product. Since {a3, b} is a characteristic subgroup
of G containing all elements of order 3, then every other maximal subgroup
M is cyclic of order 9. If M ^ <#>, then G = <a> M is a reduced product,
M<3 G, and there are two possibilities for M.
The existence of the reduced product for each normal subgroup of a
finite group gives rise to an "interpolation theorem" for Theorem 5.3.2C).
If the structure of each subgroup of a given finite group G is known, then the
image of <D(G) under a homomorphism 6 is completely determined by the
structure of Q)(A)e for some subgroup A of G.
Theorem 5.4.7. For each homomorphism 6 of a finite group G there exists a
subgroup A of G such that €>(G0) = ®(A)d.
Proof. If Ker@) c= $(G), then A = G since O(G/Ker@)) = O(G)/Ker@).
If Ker@) ? O(G), then the reduced product G = A Ker(#) exists for some
subgroup A of G. Ker(fl) n ^ c #(,4). So
®(A Ker@)/Ker@)) ^ <t>(,4/Ker(fl) n A) = O(X)/Ker@) n A
Therefore 0 e
Definition 5.4.8. A property 0* defining a class of groups will be called
hereditary if
1. each subgroup of a group Ge^ has property ^, and
2. ^ is preserved under homomorphisms.
For example, the concepts of nilpotence and solvability are hereditary
properties.
Theorem 5.4.9. Consider the hereditary property 0> that satisfies the con-
condition: The group H e 0> if H/<5)(H) e0>. If Ge e & for a homomorphism 0
of a finite group G, then G contains a subgroup A such that ^4 e 0*.
Proof. Suppose that Ker@) c <?(G). Then
G/O(G) s (G/Ker@))/(O(G)/Ker@))
implies that G/<D(G) e ^. So G e ^. On the other hand, if Ker(#) ? ?>(G),
then there exists a reduced product G = A Ker@) for some subgroup A of
G and ^ n Ker@) c d>(^). Since
G/Ker@) ^ ^/^ n
then
Hence, if a group G has a subgroup N <3 G such that G/iV is solvable
(nilpotent, cyclic, supersolvable) and A is a subgroup of G such that G =

66 Nilpotent Groups; the Frattini Subgroup 5.4
is a reduced product, then A is solvable (nilpotent, cyclic, super solvable).
Strong as the hypothesis may seem, the reader will find that many group
properties possess it.
The remaining comments supplement Section 4.2.
Theorem 5.4.10. If A is an abelian normal subgroup of a finite group G and
A n <X>(G) = 1, then G = \_A~]B for some subgroup B of G. Moreover, A is
a direct product of minimal normal subgroups of G.
Proof. Use induction on |G|. If A* is a minimal normal subgroup of G con-
contained in A, then A* n <?>(G) =1. This implies that G = [y4*]fl* for some
subgroup J3* of G. Moreover, v4 n 5* <i G, since inB*<B*. Further-
Furthermore, ,4 n O(JB*)<i G implies that A n $(?*) c O(G). So ,4 n?* n $(?*) = 1.
By induction, ?* = [A n ?*]? and i n B* is a direct product of minimal
normal subgroups of 5*. These are also minimal normal subgroups of G,
since A is abelian. Therefore G = [4*]B* = A*(A n B*) B = \_A~\B.
Consequently, in a finite group having $(G) =1, G splits over each
normal abelian subgroup. However, this does not imply that G splits over
each normal nonabelian subgroup. This and the study of semidirect products
in Chapter 4 raise the question of the existence of criteria for determining
whether or not a group will split over each normal subgroup. One such
criterion is developed in Exercise 5.4.6.
EXERCISES 5.4
1. Prove the statement made after Definition 5.4.1 that the additive group of
rationals does not have a minimal generating set.
2. Prove Corollary 5.4.4.
3. Let A be the intersection of the self-normalizing maximal subgroups of a finite
group G. Prove that ,4/<D(G) = Z(G/<I>(G)). Hence A is nilpotent.
4. Show that the automorphism group of the quaternion group is isomorphic
with the symmetric group of degree four.
5. Using the result of Exercise 4, show that equality may occur in Theorem 5.4.4.
Does equality also hold for the dihedral group?
6. For the finite group G, denote the normal subgroup generated by $>(H) for all
subgroups H c: G by E(G). Then E(G) is a characteristic subgroup of G called
the elementary commutator of G. Prove the following statements:
a) If G splits over M = N n E(G) for a subgroup N normal in a finite group G,
then G splits over N. (Use Theorem 5.4.6.)
b) A finite group G splits over each normal subgroup of G iff G splits over each
normal subgroup of G that is contained in E(G).
c) In a finite nilpotent group, E(G) = <D(G).
d) The converse of (a) is not valid. Consider the dihedral group.
e) E(G) is not always O(G) in a finite group G. (Find an example.)

6
THE FITTING SUBGROUP;
SUPERSOLVABLE GROUPS
6.1 THE FITTING SUBGROUP
The notion of a maximal normal nilpotent subgroup of a group is suggested
by the fact that the product of two normal nilpotent subgroups is a normal
nilpotent subgroup. If it exists, then clearly it is unique.
Definition 6.1.1. The maximal normal nilpotent subgroup in a group G, if
it exists, is called the Fitting subgroup of G. It is denoted by F(G).
Not every group has a Fitting subgroup. For each positive integer n,
the collection of nilpotent groups of class n is not empty. (See Exercise
6.1.2.) Let Nn denote a member from each collection for each integer n. Form
G = \\lNn for / the set of positive integers. This group is not nilpotent.
For if G was nilpotent of class k, then each subgroup has class equal to or
less than k by Theorem 5.1.3B). A contradiction is obtained.
Theorem 6.1.2. For each prime p dividing the order of a finite group G, the
intersection of all Sylow ^-subgroups of G is the Sylow ^-subgroup of F(G).
Proof. For a fixed prime p, the intersection Fp of all the Sylow /?-subgroups
of G is a characteristic subgroup in G and hence it is normal in G. Therefore
Fp ^ F(G). The Sylow ^-subgroup P of F(G) is contained in a Sylow /'-sub-
/'-subgroup of G. The Sylow ^-subgroups of G are in the same conjugate class.
Hence P is in the intersection of all the Sylow /^-subgroups, that is, P c Fp.
Therefore P = Fp.
Corollary 6.1.2. The Fitting subgroup of a finite group G is the direct product
of its Sylow/^-subgroups, each direct factor being the intersection of all Sylow
/?-subgroups in G for the respective prime p.
Theorem 6.1.3. In a finite group G,
1. <D(G) <=F(G),
2. if G is a solvable group, then VG(F(G)) <= F(G),
3. if M is a minimal normal subgroup of G, then F(G) c= ^G(M), and
4. F(GI$>(G)) = F(G)/$>(G).

68 The Fitting Subgroup; Supersolvable Groups 6.1
Proof. A) is the result of ®(G) being nilpotent. For B), first note that
«g(F(G))<i g- Suppose that %G(F(G)) ? F(G). Denote by H a subgroup
normal in G such that H c: <^G(F(G)) and minimal with respect to the prop-
property of properly containing A = <#G(F(G)) n F(G). Then HjA is a minimal
normal subgroup of GjA and hence it is abelian. Therefore [//, H~] ^ A
implies that [H,H, H] =1. So H is nilpotent of class at most two and
H c F(G). The contradiction implies that #G(F(G)) c F(G). Next consider
C). If M n F(G) =1, then [M, F(G)~] cz M n F(G) =1. Hence F(G) c
#G(M). Otherwise M c F(G) implies that MnZ(F(G)) ^1. Therefore
the minimality of M being normal in G leads toMc Z(F(G)) and the desired
result. Finally, consider D). If M is the subgroup of G such that F(G/<I>(G))
= M/<D(G), then M is nilpotent by Corollary 5.3.8. Hence M c F(G).
Therefore F(G/O(G)) = F(G)/<t>(G).
Since the hypercenter of a finite group G is nilpotent, then it is always
contained in F(G). However, it doesn't have to coincide with F(G). Just
consider the symmetric group on three symbols, S3; then Z(S3) =1, whereas
F(S3) = A3, A3 the alternating group on three symbols.
In finite solvable groups, F(G) always exists nontrivially and permits a
grasp on the group with respect to some well-defined properties.
Theorem 6.1.4. Consider a finite group G having $(G) =1 and F(G) #1.
Then G = [F(G)~]C for some subgroup C of G such that
1. C^C* c Aut(F(G)) and
2. if a Sylow /^-subgroup P of F(G) has order pn, then C is homomorphic
to a subgroup of the general linear group GL(k,p) for some k ^ n.
Moreover, if P = F(G) is a unique minimal normal subgroup of G, then in
B), k = n and the mapping is a monomorphism.
Proof. Since ®(G) = 1, then <b{F{G)) = 1. From Corollary 5.3.3, this implies
that F(G) is an elementary abelian group. By Theorem 5.4.10, there exists
a subgroup C in G such that G = IF{G)]C. Furthermore, %G(F(G)) c F(G)
by Theorem 6.1.3. So %G(F(G)) = F(G). Hence, under conjugation by
elements of C, there is induced a monomorphism of C into Aut(F(G)) since
the kernel of the mapping is in #G(F(G)). This proves A). B) also follows
out of Theorem 5.4.10 in that F(G) is the direct product of minimal normal
subgroups H, each being- an elementary abelian /?-group for some prime p
dividing the order of F(G). Suppose \H\ = pk. Then Aut(H) is isomorphic
to GL(k,p), since H can be considered as a vector space over a field of
characteristic^. Therefore H <3 G implies the existence of a homomorphism
of C into Aut(H). And of course if P = F(G) is a unique minimal normal
subgroup of G, the remaining portion easily follows.

6.2 Supersolvable Groups 69
The dual of the notion of the Fitting subgroup would be the subgroup
D of G having the properties that GjD is nilpotent and if GjN is nilpotent for
s subgroup N c G, then D c N. By Exercise 5.2.4, if N and M are normal
in a group G and both G/N and G/M are nilpotent, then G/N n M is nil-
potent. So, at least for finite groups, the dual of F(G) would be the hyper-
commutator.
EXERCISES 6.1
1. Give an example to show that GjF(G) nilpotent does not imlpy that G is nilpo-
nilpotent.
2. Prove that the collection of nilpotent groups of class n > 1 is not empty by con-
considering the group G = <x, y\x2n = y2 =1, yxy — x). For n =1, any
abelian group will do.
3. The sockel, S, of a group G is the subgroup generated by the minimal normal
subgroups in G that are different from the identity. Prove the following state-
statements for finite groups G.
a) S is a characteristic subgroup of G.
b) S is a direct product of a subcollection of the minimal normal subgroup in G.
c) Let SA denote the subgroup in S generated by the abelian minimal normal
subgroups in G that are different from the identity element, if such subgroups
exist. Then SA is a characteristic subgroup of G that is a direct product of a
subcollection of the abelian minimal normal subgroups in G.
d) Let SN denote the subgroup in S generated by the nonabelian minimal
normal subgroups in G that are different from the identity element, if such
subgroups exist. Then SN is a characteristic subgroup of G that is the direct
product of all nonabelian minimal normal subgroups in G.
e) In a group G, S = SA(g) SN. (SA or SN may be the identity element.)
f) In a solvable group G,Sg F(G). Furthermore, if 0>(G) = 1, then S = F(G) =
sA.
g) F(G) c
6.2 SUPERSOLVABLE GROUPS
It has been noted that the class of abelian groups is a proper subclass of the
class of nilpotent groups, which in turn is a proper subclass of the class of
solvable groups. For finite groups, there is a class intermediate to the nil-
potent and solvable classes. However, it will be defined more generally.
Definition 6.2.1. A group G is said to be supersolvable if G contains an in-
invariant series of finite length having the property that each factor group of
the series is either prime cyclic or infinite cyclic.

70 The Fitting Subgroup; Supersolvable Groups 6.2
The symmetric group of degree three is an example of a supersolvable
group that is not nilpotent, whereas the symmetric group of degree four
indicates that there are solvable groups that are not supersolvable. The
second example also points out that the extension of a supersolvable group
by a supersolvable group need not be supersolvable.
Theorem 6.2.2
1. Each subgroup of a supersolvable group is supersolvable.
2. Each epimorphic image of a supersolvable group is supersolvable.
3. Each normal subgroup H of a supersolvable group G is a term in an
invariant series of G having the property that each factor group of the
series is either prime cyclic or infinite cyclic.
4. A direct product of a finite collection of supersolvable groups is super-
solvable.
5. Each maximal subgroup of a supersolvable group G has prime index in
G.
Proof. Suppose that l=NoaN1cz~-c:Nn = G is an invariant series
such that Nj/Nj _ l is either prime cylic or infinite cyclic for j = 1,...,«. Con-
Consider a subgroup H of G and the series 1 = Ho c Ht c ... c Hn = H de-
defined by Hj = Nj n H. Then HJ+1 c Nj+1 and Hj = Nj nHJ+1. Hence
NjHj+1INj s HJ+1/Nj n Hj+1 = HJ+1/Hj.
Since NjHj+1INj c Nj+1/Nj, then NjHj+1INj is a subgroup of a cyclic
group. Therefore HJ + l/Hj is the identity, prime cyclic, or infinite cyclic.
Since Hj o H for all j, then by deletion of repetitious elements in the
//-series, A) is obtained.
For B), consider an epimorphism 0 of G and suppose that G has the
invariant series {N^ in A). Then {iVf}n is an invariant series for Ge. If
Ni+1INi is prime cyclic, then Nt6+ JN? is either a one-element group or prime
cyclic. If Ni+1INi is infinite cyclic, then A = Nie+1/Nie is a one element
group, or infinite cyclic, or cyclic of finite order. Suppose that A satisfies
the last case. By Theorem 5.3.12, A contains a characteristic series
1 = Ao c At c ••• cz At = A
such that Aj+l/Aj is prime cyclic for j = 0, ...,n — 1. Hence there exists a
refinement N? = Ho a Ht a -- c Ht = Ndi+1 such that HJ+1/Hj is prime
cyclic and //j <i G0 for 7 = 0,..., t — 1. Now, by combining all cases, one
concludes that the series {N?}n may be refined to an invariant series {Mt}s
such that Mi+l/Mi is either prime cyclic or infinite cyclic. This proves B).
Next, suppose that H is a normal subgroup of G and consider the series
{Hj}n formed in A). Clearly Hj = H n Nj o G. Let 6 be the epimorphism

6.2 Supersolvable Groups 71
of G defined by G -» G/H. Combining the series {Hj}n with that formed in
B), an invariant series is obtained for G that defines G as a supersolvable
group. So C) is valid.
Induction is used on D) and it is enough to consider H ® K for two
supersolvable groups H and K. Suppose that {ifjn and {Kj}m are invariant
series of H and K, respectively, that determine their supersolvability. Then
the series
1 = H0^H1cz ... czHn = H®Ktcz H®K2a ... a H ® Km = H ®K
is an invariant series satisfying the definition of supersolvability.
Finally, for E) note that the result is valid for each normal maximal sub-
subgroup of G. Consider a self-normalizing maximal subgroup M, if it exists,
and denote the intersection of its conjugate class by N. (N is called the core
of M.) By B), G/N is supersolvable and MjN is a self-normalizing maximal
subgroup of GjN. There exists H/N <a G/N such that H/N is prime or infinite
cyclic. Each subgroup of an infinite cyclic group is a characteristic subgroup,
since the only nontrivial automorphism of an infinite cyclic group sends each
element into its inverse. Hence if H/N n MjN is not the identity element,
then if n M is a normal subgroup of G contained in the conjugates of M and
properly containing N. This contradiction implies that H/N n M/N is the
identity element. If H/N is not prime cyclic, then H/N contains a proper
subgroup H*/N<l G/N such that MjN c H*MjN cz HMjN = G/N. This
contradicts the maximality of M. So H/N is prime cyclic and G/N =
(H/N) (M/N) = HM/N. Then (G:M) is a prime integer since (HM:M) =
(H:HnM) = (H:N).
The example mentioned immediately after Definition 6.1.1 indicates
that D) is not always valid if the index set is infinite.
The maximal condition on subgroups in a group is the condition that
for each ascending chain of subgroups H1 c: H2 ? ••• in a group G, there is
a least integer k such that Hk = Hj for all j ^ k. This is also referred to as
the ascending chain condition on G, or it is simply said that G has the a.c.c.
condition. As the reader can easily verify, each subgroup of G would also
satisfy the a.c.c. condition on its subgroups. Also, G is finitely generated.
Moreover, a necessary and sufficient condition that G has the maximal con-
condition is that G and each subgroup of G are finitely generated.
Theorem 6.2.3. If G is supersolvable, then
1. G satisfies the maximal condition on subgroups,
2. G is finitely generated, and
3. F(G) exists nontrivially in G.

72 The Fitting Subgroup; Supersolvable Groups 6.2
Proof. Exercise.
Theorem 6.2.4. A group G is supersolvable iff G possesses an invariant series
{Nt}n of finite length such that Ni+1INt is cylic for / = 0, ...,n — 1.
Proof. A supersolvable group satisfies the conditions. Conversely, let
1 = No cz • • • c Nn = G be an invariant series of finite length satisfying the
conditions. Suppose that A = Ni+lINt is finite cyclic. Using Theorem 5.3.12,
A has a finite characteristic series
1 = Mo cz • • • a Mk = A
such that Mi+1/Mi is prime cyclic for / = 0,..., k — 1. This defines a series
{Mf}k between Nt and Ni+ x such that Mf o G and Mf + i/Mf is prime cyclic.
Apply this argument to each finite factor group in {NJM. The insertions yield
a refinement {iV*}f such that Nf + 1jNf is either infinite cyclic or prime
cyclic. Hence G is supersolvable.
Theorem 6.2.5. Consider a finite group G in which the index of each maximal
subgroup is a prime. If p is the largest prime dividing the order of G, then
the corresponding Sylow /?-subgroup is normal in G.
Proof. By hypothesis, each maximal subgroup has prime index in G. Let p
denote the largest prime dividing \G\ and consider a Sylow /^-subgroup P of
G. If P ^ G, then JrG{P) is contained in a maximal subgroup M of G.
Since
and /M(P) = /G(P),
then, by a Sylow theorem, 1= (G:M) modulo p. But (G:M) =1+ A77,
A: 7^ 0, is a prime q < /?. However kp = q —I implies that /? divides # — 1.
So p < q. The contradiction implies that P must be normal in G.
Corollary 6.2.5.1. For the largest prime p dividing the order of a finite
supersolvable group G, G contains a normal cyclic subgroup of order p.
Corollary 6.2.5.2. For a finite supersolvable group G, there exists an invariant
series \=N0<^N1c:---c^Nn = G such that
l^+i/NJ< WJVi-il for /=1,...,/!-1.
It is evident that for the largest prime p dividing the order of a finite
supersolvable group G, the Sylow /?-subgroup of G is the Sylow /^-subgroup
of F(G). This is not valid in general for an arbitrary solvable group and it
indicates one of the reasons why the role of the Fitting group becomes im-
important in the study of supersolvable groups. Theorem 6.2.5 also points
out another distinguishing element between solvable and supersolvable

6.2 Supersolvable Groups 73
finite groups. By Theorem 4.3.5, there exists a Sylow basis for a solvable
group G such that
G = S1S2...Sn and StSj = SjSt for ij = 1, ...,n.
The symmetric group of degree four satisfies this condition but of course is
not supersolvable. However, for finite supersolvable groups, the Sylow
subgroups can be so ordered such that
Sx <a G, S2 <i S2 ... Sn, ...;
this is a much stronger condition.
Corollary 6.2.5.2 can be strengthened. It can be shown that if G is an
infinite supersolvable group, then G has an invariant series
1= No a Nt c •.. a Nn= G
such that for a least positive integer k, Ni+1jNt is infinite cyclic for / < k
and \Ni+1jNt\ < |iVf/iVi_il for / > k. Of course, for the finite case, one can
always assume the existence of a prime cyclic normal subgroup for the
largest prime occurring in the order of the group. This is helpful in initiating
inductive proofs.
Theorem 6.2.6. A finite group is supersolvable iff each maximal subgroup
has prime index.
Proof. The necessity follows from Theorem 6.2.2E). Assume that each
maximal subgroup of G has prime index and then use induction on \G\. By
Theorem 6.2.5, the Sylow /^-subgroup P for the largest prime/> dividing \G\
is normal in G.
Denote a minimal normal subgroup of G contained in P by N.
Inductively, G/N is supersolvable and of course if N is cyclic, then N is
prime cyclic and hence G is supersolvable. So assume that each minimal
normal subgroup N of G contained in P is elementary abelian and not prime
cyclic. Then |JV| > p and N <= Z(P). If N = P, then N ? O(G) by Corollary
4.1.4B). Thus there exists a maximal subgroup M of G such that G = [7V]M
by Theorem 5.4.10. But (G:M) a prime implies that N is prime cyclic and a
contradiction is reached. Therefore TV # P.
Consequently, there exists a subgroup M<i G such that N c M c p
and (M:N) = p by the supersolvability of G/N. Then M = <N, x} is ele-
elementary abelian for some generator xeM. Otherwise <D(M) = N implies
that M is cyclic by the Burnside Basis Theorem (Theorem 5.4.3) and
17^ <D(M) c= iV properly contradicts the minimality of N. If P $?
then there exists a subgroup P* ^ P such that
P* <i G, (P*:^G(M) nP)=p, and P* = (#G(M) n P)

74 The Fitting Subgroup; Supersolvable Groups 6.2
Note that M/N <= Z(P*/N). Hence
[M, P*] <= JV and [M, P*] = [N <x>, («G(M) n P) < j>] = <{[*', /]}>.
(Use the identities lab,'] = [a, c]\b, c] and [a, be] = [a, c] [a, fr]c.) Since
iV c Z(P), another application of these identities together with an inductive
argument yields tx\yj] = [x,yJJ. So IM,P*2 = <[x, j]> has order p.
Since M, P* <a G, then [M, P*] <3 G. This contradicts the minimality of N.
SoPg %g(m)' BY applying Theorem 4.4.1, M = N ® L, L<G, and
|L| = ^ for some subgroup L of M. This contradiction implies that at least
one minimal normal subgroup of G is cyclic of order p. Therefore G is
supersolvable.
Corollary 6.2.6. A finite group G is supersolvable iff G/<I)(G) is super-
solvable.
By Theorem 5.3.7, a nilpotent group G has G' c <1>(G). An analogous
theorem for supersolvable groups is the following:
Theorem 6.2.7. The commutator subgroup of a supersolvable group G is
nilpotent, that is, G' <= F(G).
Proof. If 1 = No a Ni a -•> a Nn = G is an invariant series defining a
supersolvable group G, form Hj = Nj n G' for j = 0,...,«. Then conjuga-
conjugation by the elements of G/Hj on Hj+1IHj induces a homomorphism
of G/Hj -> Aut(Hj+1IHj) having kernel C/Hj = Vg/Hj(Hj+1IHj). G\C is
isomorphic to a subgroup of A\it(Hj+1IHj). Hence GjC is abelian, since
Hj+JHj is cyclic. Therefore G' c C. This implies that Hj+JH,- ? Z{GfIH3).
In turn, this implies that {#,-}„ is an ascending central series for G'. So Gr
is nilpotent.
The converse of Theorem 6.2.7 is not necessarily valid. For example,
the commutator subgroup [y44, ^44] of the alternating group of degree four,
^44, is F(A4). However, AA is not supersolvable.
In general, a nilpotent group is not supersolvable, even when it is
abelian. The additive group of the rationals, Q, provides an example of this.
Since a group satisfying the maximal condition is finitely generated and Q
is not, then, by Theorem 6.2.3, Q cannot be supersolvable.
Theorem 6.2.8. A nilpotent group G is supersolvable iff G satisfies the maxi-
maximal condition on subgroups.
Proof. If G is supersolvable, then G satisfies the maximal condition
on subgroups (see Theorem 6.2.3). Consider the converse. Each factor
group in the upper central series would be a finitely generated abelian group.
This in turn implies that for two consecutive terms Zi and Z?+1 in this series,

2
0
0
-2
and
b =
0
-1
1
0
6.2 Supersolvable Groups 75
there exists a cyclic series
Zi = HoczHicZ'"c:Hn = Zi + 1
such that Hj/Z^ GjZi for j = 1,...,/?. Therefore //,<! G. This is valid
for / = 0,...,«— 1. So G has an invariant series that satisfies the hypothesis
of Theorem 6.2.4.
The product of two normal nilpotent subgroups of a group is nilpotent
(see Theorem 5.2.2). As an exercise, one can prove that the product of a
nilpotent normal subgroup and a supersolvable normal subgroup is super-
solvable. The next example will show that it is not generally true that the
product of two normal supersolvable groups is supersolvable.
EXAMPLE. (Huppert). Consider the elements
a =
of GLB, 5). Then Q = <a, b} is nonabelian of order eight and b xab = a x
implies that Q is the quaternion group. Let A be an elementary abelian group
of order 52 and Q be isomorphic to a subgroup of Aut(,4) ^ GLB, 5) with
respect to the isomorphism 9. Form G = [^4]02. Without loss of generality,
we can assume that A, b, and a are in G. Consider two subgroups C = [A] <a>
and D = [A\ <Z?>. Both C and D are supersolvable and normal in G. More-
Moreover, G = CD. However, even though G has a normal subgroup A of order
52, there are none of order 5. Therefore G is not supersolvable.
EXERCISES 6.2
1. Prove that the product of a normal nilpotent subgroup and a normal super-
solvable subgroup is supersolvable.
2. Prove that if G/N is supersolvable and N is cyclic, then G is supersolvable.
3. Prove that a group with maximal condition is finitely generated.
4. Prove that all invariant series defining a finite supersolvable group have the
same length.
5. Verify the example.
6. Prove Theorem 6.2.3.
7. Prove Corollaries 6.2.5.1 and 6.2.5.2.
8. Prove Corollary 6.2.6.
9. Show that if G is an infinite supersolvable group, then G has an invariant series
1 = jv0 c: Nl cz ••• a Nn = G such that for a least integer k, Ni+1/Nt is in-
infinite cyclic for / < k and \Ni+ 1/Ni\ ^ \NJNi_ -J for i> k. Then show that any
other cyclic invariant series has precisely k factors that are infinite cyclic.

7
GENERAL EXTENSION THEORY
7.1 EXTENSIONS AND FACTOR SETS
Let us return to the notion of a group G being an extension of a group A
by a group B as presented in Chapter 3, namely, 1 > A —^ G -$-+ B > 1
as a short exact sequence. We will examine the general case in which this
may not be a splitting extension. As before, we are motivated by the fact that
if A is a normal subgroup of a group G, then GjA = B. Hence there is a
mapping of B into Aut(^4) and a mapping of B into the coset representatives
of G with respect to A. For the splitting extension, these mappings could be
taken as homomorphisms. In general, this is not possible and how to com-
compensate for it is the crux of this chapter. However, the only formidable
problem seems to be that of notation and patience overcomes this.
Consider a group G and a subgroup i<G such that GjA ^ B. Denote
a left coset representative of A in G that corresponds to the element b e B
by ba, with the understanding that the identity element of G will correspond
to the identity element of B. Thus G = [JBbaA and a defines an injective
mapping of B > G. If/? is the epimorphism of G > B having Ker(/?) = A,
then a/} = iB. Moreover, the mapping an {ba)~1aba = ab, for all aeA, is
an automorphism of A. So there exists a mapping (not necessarily a homo-
morphism) of B -» Aut(A). Then note that
b°Ab\A = bab\A = (bb^'A
implies that for each ordered pair b,b1eB, A contains an element (b, bx)
such that
6*6? = F6!)* (Mi).
The set ? = {(&, ^i)} determined by coset multiplication for all ordered
pairs b,b1sB is called the factor set with respect to the mapping a for the
short exact sequence 1 > A —2-> G —?-> J5 > 1, where in this case a = iG\A.
Clearly, the factor set defines a mapping ofBxB into A. We can expect that
a different choice of coset representatives may lead to a different mapping.
For an element aeA,

7.1 Extensions and Factor Sets 77
By the associative law, (b° b\) b\ = ba {b\ ba2). So
(b*bl)b°2 =((bbiy(b,b1))b°2
and
b°(bib2:) = b°{(b1b2y(bub2))
= (bbxb2y (b, bxb2) (bl9 b2)9 for b, bj e B.
Therefore (bbu b2) (bl9 bf2 = (b, b±b2) (bl9 b2).
For a, ax e A and b, bx e B, one has
ba ab\ax = bab\ahxa± = (bb^ {b.b^a^a^.
Since (lB)ff = 1G, then (lB)aba = ba AB, b). Hence AB, 6) = 1G. Similarly,
F,lB)=lG,V6eB.
Summarizing, the elements of the factor system satisfy the following
conditions:
1. (ab)&* = (Mi)~V&1(Mi),
2. (bbl9 b2) (b, b.f2 = (b9 bxb2) (bl9 b2)9
3. F,lB) =
4. (b,b1)=
With several further observations, there is now enough information for
a converse to hold in the sense that given groups A and B there exists a group
G such that 1 > A-^-+ G—^-> ? > 1 is a short exact sequence. In the
preceding discussion a was the inclusion mapping. But for the extension, a
will be a monomorphism. The mapping a of B -> G had the elements 6*
as coset representatives of Im(a) = Ker(/f) in G.
In general, there must be a mapping of B -> Aut(yl) and a mapping of
5x B-^i such that the images of B x B, namely {(b, bj}, satisfy the con-
conditions (l)-C) with ab denoting the image of a e A under the automorphism
associated with b. We will show that if these requirements are met, then the
extension G exists, and that there is a mapping a of B -> G such that
a'. (b,b1y=((bb1yylb°bi.
Let A and B be two groups for which the mappings in the preceding
paragraph exist and conditions (l)-C) are satisfied. Consider the set
G = B x A= {[_b,a]\beB, aeA)
and define a binary operation on G by

78 General Extension Theory 7.1
In A), set b = bl=lB=l. Then
(a1I = (U)*1 A,1) = 121a1lA = a1, VaeA.
Hence y: a !—> a1 is an automorphism such that y2 = y. Therefore, y is the
identity automorphism on A.
The operation is associative since
= \bb,b2, (bbu b2) (b, btf* (a*1)* V*fl2]
= \bb,b2, (bbu b2) (b, b^ (bu b2ylab^ (bu b2) a^a2-\, by A),
= \bbxb2, (b, bxb2) (bu b2) (bu b2y'ab^ (blt b2) aib*a2], by B),
= \bb,b2, (b, bxb2) a"^ (bu b2) a^a2-\
= Lb,a-][b1b2,(b1,b2)a1b>a2]
Note that [1,1] [b,d] = [b,(l,b) 1"a] = [6, a] and so [1,1] is a left
identity for G. Similarly, [1,1] is a right identity for G.
Since b corresponds to an automorphism of A, then for each element
a e A and each element (b'1,^ there exists an element a1eA such that
a\a = {b-\by\
So for [b, a], one has
[b-\a{\[b,ci] = i\,{b~\b)a\a-] = [1,1].
Therefore each element of G possesses a left inverse. This element is also a
right inverse.
Consequently, the set G, together with the operation defined above, forms
a group.
Now to prove that G satisfies the required extension. One can readily
verify that the mapping [Z>, d]\-^b defines an epimorphism fi'.G-+B. Then
Moreover,
[I,a][l,a1] = [l,(l,l)a1a1] = [l,aa1].
Consequently, the mapping a: ai—> [1, a] defines an isomorphism of A with
Ker(/?). These two mappings give rise to the short exact sequence
1 > A-^-* G-*-> B >1,
that is, G is an extension of A by B. Don't overlook the fact that a:b\-^[b, 1]
is a mapping of B -» G such that cr/? = iB and that the set {[&, 1] | b e B] can

7.1 Extensions and Factor Sets 79
be taken as a set of coset representatives with respect to Ker(j8) in G. More-
Moreover, from the definition of the operation, it follows that
[6,1] U>l91] = [bbl9 (b9 bj] = [bbl91] [1, (b, b,)l
Therefore,
that is, D') is satisfied.
The following theorem due to O. Schrier is a summary of the preceding
material in this section.
Theorem 7.1.1. For each extension 1 > A—^-» G —?-> B >l and in-
jective mapping a: B -> G such that crjS = iB there is a mapping i:B x B-+A,
with images denoted by {(b, bj}, uniquely determined by a, and a mapping
<da:B -+ Aut(A), with a& denoting the image of aeA under the auto-
automorphism associated with beB, such that for all b, bl9 b2eB and for all
aeA,
1. (*>)>' = (Mi) a**1 (Mi),
2. (bb19b2)(b9b1y* = frb^ib^^i
3. (ft,lB) =
4. (b9b1y
Conversely, if for given groups A and B there exist a mapping
0ff: B -> Aut(^4) and a mapping t : B x B -»^4 having the images under 0^
and t satisfy conditions (l)-C), then there is an extension
1 > A -2-* G -^-> 5 > 1
and an injective mapping a:B -> G such that cr/? = tB and satisfying condition
D). Moreover, the factor sets and the mapping of B -> Aut(^4) determined
by o coincide with the images under t and the mapping ©ff, that is, G is
unique up to isomorphism with respect to t and 0ff.
The proof of the last statement is left as an exercise.
Given the subgroup i< G, one can take any element of A as a representa-
representative for A in the extension and obtain the same result in a similar manner
but making appropriate adjustment for an identity element and the inverses.
For example, the identity element would then bel<r(l,l)~1. In order to
firm up the ideas involved in Theorem 7.1.1, this will be left to the reader
as an exercise.
Note that nothing in the proof demands that the groups involved need
to be finite. Even in the finite case, it appears evident that the selection of
the elements would be a tedious task in order that the converse go through.
As a result, this theorem is seldom applied to specific extensions, but rather
it is used to prove further results that are more easily applied.

80 General Extension Theory 7.2
Why should extensions be looked at in two distinct manners? Note that
the method using extension theory is concerned with the property of embed-
embedding in the "large." The method of factor sets looks at embedding in the
"small."
EXERCISES 7.1
1. Prove the analogy to Theorem 7.1.1 whenever an arbitrary element a* e A is
used as the coset representative corresponding to the identity element of B.
2. In the converse of Theorem 7.1.1, prove that in the extension G, the factor set
and the mapping of B -> Aut(A) determined by a coincide with the images of
t and the mapping 0a. (Hence G is unique up to isomorphism with respect to %
and 9a.)
3. Examine the symmetric group of degree three as an extension of a cyclic group
of order three by a cyclic group of order two with respect to the factor sets.
4. Examine the quaternion group as an extension of a cyclic group of order four
by a cyclic group of order two with respect to the factor sets.
5. Find an example in which the mapping B -» Aut04) is not a homomorphism.
7.2 EQUIVALENT EXTENSIONS
By Theorem 7.1.1, for each extension 1 > A—^-* G—2—> B > 1 and each
injective mapping a\B -+ G such that ofi = iB, there is uniquely associated
both a factor set, I = {(b,b{)}9 and a mapping <dG:B -> Aut{A), where ab
denoted the effect on an element aeAby the element b@a, that satisfy condi-
conditions (l)-C). Denote this pair of mappings related by conditions (l)-C)
by E(Ga, I). Call it an extension system. Conversely, for each extension
system E(Qa, X) there is an extension G of A by B having E(Ga, li) as an
extension for G. However, for a given extension, the extension systems are
not necessarily unique, since they are dependent upon the injective mappings
of B -> G. This raises the question: What possible relationship can there be
among the extension systems? For convenience, retain the identity element
of G as the coset representative of A.
Again, consider a group G exists having a subgroup A <i G and G/A = B.
Assume that the extension system E(Oa, I) exists. Select a different set of
coset representatives {ba*}, having a factor set ?* = {(b, b^)*}, and denote
the image of an element a e A under the automorphism associated with b by
ab*. Denote this extension system by E(Oa*, S*). Then for each b e B there
is an element abeA such that ba* = baab. Moreover, la* = lff = 1 and aiB =1.
Consequently,
cF = {bayxab^ = a;1 (b'y'ab'a, = a^abab.

7.2 Equivalent Extensions 81
The factor set Z* is related to Z as follows:
b°*b? = (bb.r^b.r = {bbxyahhl (b9bt)*
= b°abblabl = Vb\d?ah,
= (bbiy(b,b1)abb%1.
So
{b,b1)* = ab-b\(b,b1)abb>abl.
This motivates the next definition.
Definition 7.2.1. Two extension systems E(Sa, Z) and E(®a*, Z*) are equi-
equivalent, E(Gff, Z) = E(Sa*, Z*), if there exists an element ab e A, for each
beB, such that the following relations are satisfied:
1. ab* = a6~Vtf6 and
2. (Mi)* = ^MMiK1^,
for which <z1b = 1^.
This definition defines an equivalence relation on the set E{ —, —),
(Exercise 7.2.1).
Since for each extension system there is an extension G of A by B, then
it seems reasonable to relate the notion of equivalent extension systems to
that of equivalent extensions. By Definition 3.2.2, two extensions
1 ,i_L,G-i^5 >1 and 1 >iJ^G*-^>jB >1 are equi-
equivalent iff there exists aye Hom(G, G*) such that the diagram here is
commutative:
A—2-> G—0—> B
In fact, y is an isomorphism. (See Theorem 3.2.3.)
Consider the extension system E(Sff, S). This gives rise to the extension
1 > A—Z-+ G-2—> B >1 for which there exists an injective mapping a
such that ofi = iB and baaa^b. Moreover, {abf = (a*)b. Similarly, for
E(®a*9 Z*), there exists an extension G*, an epimorphism /?*, an injective
mapping a*: B -> G* such that a*fi* = iB, and a monomorphism a* forming
a short exact sequence 1 > A -Z—> G* -^—> B > 1.
Theorem 7.2.2. E(Gff, Z) = ?@ff*, 2*) iff the extensions 1 > A^*-+ G-^
B > 1 and 1 > A -^ G* -^-> B > 1 are equivalent.
Proof. Suppose that the extensions are equivalent. Then there exists an

82 General Extension Theory 7.2
isomorphism y: G -* G* such that the above diagram is commutative. There-
Therefore a* = ay and ft — yfi*.
For &WbeB,b = blB= b"^* = ba"f'lp. Hence for each b e B there exist
elements ab e A and b"eG such that b°*y~' = b"ab. Clearly, alB= lA. Then
= ((air1 (Jydff-1 = ((air1 (b°rla"b°al)«-1
= (/)•••'¦' = /, VZ>eJ5 and
Next note that
(F,fe1)*)a*= {{bb^fY^bX, by Theorem 7.1.1,
= (W7«S1)'1(
Therefore
Since a* = ay, then @, &1)*f-la = (b, bj* = abb\ (b, b,) a;1 abl.
Hence the conditions of Definition 7.2.1 are satisfied and
E(@e91) s ?@^, Z*).
Next, assume that E@ff, X) s ?@ff*, S*). Consider the extensions
associated with each extension system in the paragraph preceding the theorem
and in the diagram.
G—?—> B
We have to show not only that y exists as an isomorphism but also that the
diagram is commutative.
For each g e G, g = baa* for some b e B and aeA. Define the mapping

7.2 Equivalent Extensions 83
Note that (b'tf) (b1a\) = (bbtf ((b, bt)abiax)\ by Theorem 7.1.1. So
(F<fbldS> = (bb.r (a^r ((b, bj a»>aiy
= (bb.r ((b, b,r of/ {alT1 a^a.Y, by B),
= {bb,r ((Mi)*** Kr'a"^ {a^a^y
= (bbtr ((Mi)***1 (abbrlabla^ab>abl KV))"
= (bb.r {(b, bt)* {dft-Wa^aJY, by A),
Therefore, y is a homomorphism. Since lff* = 1G*, then the set of all g e G
mapped onto la* is V =1G. Consequently y is an isomorphism.
Moreover, (fy = (lacf)y =V* (a['1a)<x* since it is assumed in Definition
7.2.1 that alB = \A. Therefore, the left-hand portion of the diagram is com-
commutative. For g = bacf e G, one has
QfdyP = {b'Xa^afy* = b"P {a^dfP = bl*lB = b.
Hence fi = y/?*, that is, the right-hand portion is commutative. So the two
short exact sequences are equivalent.
Corollary 7.2.2. If 1 > A -?-* G -?-* B > 1 is an extension having two
extension systems E(ea, S) and ?@ff*, S*), then ?@ff, X) = ?@ff*, X*).
Proof. First note that iB = afl = g*(I. Then apply Theorem 7.2.2 since the
given extension is equivalent to itself.
Even though it can be proved directly that the equivalence of extension
systems is an equivalence relation on E( —, —), it is also possible to prove this
by using Theorems 7.2.2 and 3.2.4.
The next theorem brings together the present theory and the notion of a
splitting extension (see Definition 3.3.1).
Theorem 7.2.3. An extension 1 > A-*-+ G-*-> B >l with an ex-
extension system ?@^, X) is a splitting extension iff there exists an equivalent
extension system E(®a*, X*) having (b, b^)* = lA, VZ>, bt eB.
Proof. Suppose that the given extension is a splitting extension. Then there
exists a monomorphism o*:B -» G such that iB = a*/?. This implies the
existence of an extension system EFa*, X*) having the property that
y^bf=\A.
By Corollary 7.2.2, the two extension systems are equivalent. The converse
readily follows.

84 General Extension Theory 7.3
As noted in Chapter 3, the monomorphism a is not unique and a split-
splitting extension may lead to more than one representation as a semidirect
product. Also in Chapter 3, it was indicated that any extension equivalent
to a splitting extension is also a splitting extension.
The next example should help to clarify the preceding material.
EXAMPLE. Consider the 3-group
G = (x,y,z\x9=y3=z3=h
yxy = x, z~~1xz = xy, z~~xyz = x~3yy.
G contains precisely one abelian maximal subgroup <jc) ® <j> and three
nonabelian maximal subgroups having exponent three. Denote one of the
nonabelian maximal subgroups by M. Then 1 > M > G > GjM > 1
is a short exact sequence with GjM cyclic of order three. Note that x$M.
So the coset decomposition G = M + xM + x2M exists and yields a non-
trivial factor set. On the other hand, there exists an element w of order three
such that G = [M] <w> and G = m + wM + w2M has a trivial factor set.
The two extensions are equivalent by Corollary 7.2.2. It is suggested that the
reader determine the elements ab of Definition 7.2.1 in order to show this.
EXERCISES 7.2
1. Prove that the equivalence of extension systems is an equivalence relation via
the definition of equivalent extension systems. Then prove it using Theorems
7.2.2 and 3.2.4.
2. Give an example of extensions having two inequivalent extension systems for
fixed groups A and B.
3. Carry through the details in the Example.
4. Reword and prove Theorem 3.2.5 with respect to extension systems.
5. Examine the nonabelian p-groups of order p3 for p = 2, 3 in regard to exten-
tions of a maximal subgroup and determine the extension systems, equivalent
extension systems, etc.
6. In order to demonstrate further the vagaries of equivalent extensions, consider
the symmetric group of degree three, S, and the extension l-> S -*-+ SU S-^-»
S ->1 where a:^h-> (s, 1) and /?:(>, .s^i-* sv If the diagonal D is defined to be
D = {0, s)\se S}9 then D ? S, SUS = [5II <1>] D and D -> AutEII <1»
is a monomorphism. The two extension systems are equivalent and the factor
sets are the identity element.
7.3 EXTENSIONS OF ABELIAN GROUPS
The theory of extensions through extension systems permits us in theory to
examine a general method for the construction of all solvable groups. Note
that this reduces to the case of an extension 1—> A —^-> G —?-> B > 1

7.3 Extensions of Abelian Groups 85
having both A and B abelian. One could attach another property in the case
for nilpotency such as Aa c Z(G).
If in an extension A is abelian, but not necessarily B, several immedi-
immediate consequences can be developed. Consider the extension system E(Qa, S).
Then
(abf> = (Mi)~V61 (Mi) = abb\
Therefore the automorphisms associated with the set {ba} form a group.
Whereas in the general theory there is a mapping 6a: B -* Aut(^4), in this
case Qa is a homomorphism. Also suppose that E(Sa, S) = E(®a*, 2*).
In the notation of Definition 7.2.1,
ab* = a^abab = ab, Mae A and VbeB.
Consequently, 0CT = 0ff* for two equivalent extension systems.
Summarizing:
Let E(Ga, li) be the extension system for the extension G of an abelian
group A by a group B.
1. Sa is a homomorphism.
2. If ?@,, I) = ?(©„, I*), then 0, = 0.,
Now let F@) consist of all factor sets {(b, bx)} from ?@, -). The ele-
elements in {(b, b^)} satisfy the relationship
for all b,bub2eB. Consider two factor sets {{b,b^} and {(b,bj*}. Let
r = {(b^F2)} be defined by
Then
(P^bi) = (b, b^y1 (bbu b2) (b, b.f2 (b, b^)*-1 (bbl9 b2r (b,
b2) (bbu b2)*) (F,6J F,
So, with respect to 0, the set {(bu b2)} satisfies the condition of being a factor
set. By Theorem 7.1.1, there exists an extension 1 > A —2-> G —?-> 5 > 1
with respect to the extension system E(Q, T) for y the mapping of B -> G
for which y? = iB. Hence ?@,r)e?@, -) and {(Ml)}eF@). In G
the mapping 5x5-^i having images {(?, fej} was fully defined by (b1, Z?2) =
(Z?1? Z?2) (bl9 b2)*, i.e. the addition of functions. This sum of functions can be
taken as an operation on F@). The inverse of a factor set {(b, b^} is the
factor set {{b^bxY1} and the identity element is the factor set in which all
elements are the identity element of A. Clearly, the operation is commuta-
commutative. Hence F@) is an abelian group.

86 General Extension Theory 7.3
Next, consider the subset of F@) consisting of all factor sets that arise
from splitting extensions and denote it by S@). If {F,61)}eS@), then
there exist elements ab, abl9 and abbleA such that
or
Let {(b, b})*} denote another element of <S@) for which there exist elements
at, a*,, a%bl such that
It follows that (b, bx) (b, b^*'1 e S@). Consequently, S@) is a subgroup
of F@).
Suppose that ?@, ?) and ?@, X*) are equivalent and have factor
sets {(Mi)} and {(Mi)*}- By B) in Definition 7.2.1,
The commutativity of A implies that
(b, b±r (b, b.y1 = abh\ ft abl = (b, bj
and this defines a factor set T={(b, b^'} e F@). But then 1 = abb\ (b, bj a\l abl
indicates that ?@, F) is a splitting extension. Hence {(b, bx)'} e 5@).
Therefore {(&, &i)} and {(&, &i)*} belong to the same right coset of S@) in
F@). Consequently, F@)/Sr@) is an abelian group bijective with the set
of inequivalent extension systems for an abelian group A by a group B with
respect to the fixed homomorphism 0.
This last conclusion can be extended. E@, ?) = ?@, ?*) iff the
respective extensions 1 > 4-2-> G-^ B »1 and 1 >A-?-+ G*-^U
B > 1 are equivalent. Hence, for a fixed 0, F@)/5@) is bijective with
the inequivalent extensions of an abelian group A by a group B.
The next theorem states the results that have just been proved.
Theorem 7.3.1. Consider the collection of extension systems ?@, —) for
an abelian group A by a group B.
1. The collection F@) of all factor sets from ?@, -) forms an abelian
group.
2. The collection 5@) of all factor sets from extension systems in ?@, —)
associated with splitting extensions of A by B is a subgroup of F@).
3. F@)/S@) is bijective with the equivalence classes of extension systems
of A by B with respect to 9.
4. F@)/Sr@) is bijective with the equivalence classes of extensions of an
abelian group A by a group B with respect to 0.

7.4 Cyclic Extensions 87
Next consider the case that A* c Z(G) for the extension
1 > A^-> G-Z-+ B >1.
Clearly, this means that A is abelian and so Theorem 7.3.1 is applicable.
This extension will be called a central extension.
Theorem 7.3.2. Let G be the extension of an abelian group A by a group B
associated with an extension system ?@,,, ?). G is a central extension iff
ab = a, VaeA and MbeB, that is, Qa:B -» iA.
Proof. If G is a central extension, then
(aby = (aaf = {ba)~1a(lba = a*
implies that ab = a, MbeB. So <da\B-* iA. Conversely, suppose that
Ker@ff) = B. Then (a*)b = (abf = aa. Since A is commutative and a*
commutes with all b% it follows from \)BbaA* = G that A* c Z(G).
Corollary 7.3.2. The collection of all inequivalent central extensions is bi-
jective with .F@)/S@), where G:B -+ iA.
An extension 1 > A > G > B—-> 1 having G abelian is called
an abelian extension and of course Theorem 7.3.1 and Theorem 7.3.2 are
immediately applicable. However, this type of extension will not be dis-
discussed any further. If B is cyclic, then the extension is called a cyclic exten-
extension and such extensions will be discussed in the next section.
EXERCISES 7.3
1. Let a be a bijection of a group G with a set H. Prove that a unique binary op-
operation can be defined on H such that H is a group and a can be extended to
an isomorphism.
2. Prove that the equivalence classes of the extension systems, ?@,—), of an
abelian group A by a group B from an abelian group isomorphic to F@)/S@).
Then prove that the equivalence classes of the extensions of an abelian group A
by a group B with respect to a fixed 0 also from an abelian group isomorphic to
F@)/5@). (Use C) and D) of Theorem 7.3.1 and Exercise 1.)
3. Prove that the equivalence classes of central extensions of an abelian group A
by a group B from an abelian group.
4. Consider the collection of extension systems ?@, —) for an abelian group A
by a group B. Prove that if A has finite exponent n, then the exponent of
F@)/5@) divides n.
IA CYCLIC EXTENSIONS
Consider the extension 1 > A—^- G—^-» B >1 and suppose that
B = (by is cyclic of order n. Then there exists a mapping o\B -> G such that
G = \JB (bayAa and (bG)n = a* for some as A. For the corresponding

88 General Extension Theory 7.4
Ga:B -> A\xt(A), defined by a^^T V6* = (aa)\
(a*fn = {b°yna{ba)n = ET V5* = (a^aaf.
So abn = aT1 da. Moreover, since (b0)'1 (ba)"ba = (by, then ab = a, that
is, a is fixed by the automorphism associated with b. These two conditions
are enough for a converse to be valid.
Theorem 7.4.1. If B = <6> is cyclic of finite order ti, then the extension
1 > 4 -JL-+ q _L_* ^ y i exists iff
1. there is an automorphism of A defined by y: a i—> afe,
2. an element aeyl such that ay" = a~laa, and
3. ay = a.
Proof. The necessity of the conditions has been shown. So assume that A)
and C) are satisfied. Then set abl = ay\ for / =1,...,/? — 1, and ab" = a~1aa,
A factor set can be defined by (b\ bj) = 1, if / + / < n, and (b\ bj) = a, if
i + j > n, for 0 ^ / < n and 0 ^ y < /?. It remains to show that conditions
(l)-C) of Theorem 7.1.1 are satisfied. Condition C) is a consequence of the
definition of the factor set. For condition A) of that theorem, note that
(abJf = (b\ bj)~ * abi + J (b\ bJ), for / + j < n,
since (b\bj)=l. Then
(abJ)bi = abi+J = a~1abka, for i+j = n + k9
leads to (abJf = (b\ tiyxabh (b\ bj) whenever i +j> n. So A) is satisfied.
Next consider
(bi+J\bk)(b\bJf and (b\bj+k)(b\bk).
If / + j + k < /?, then all factors reduce to the identity element and the ex-
expressions are equal. Of course, restrict values on /, j, and k to 0,...,/? — 1.
Then, whenever / + j + k ^ 2/7, both expressions reduce to a1 and equality
again results. Finally, consider n < / + j + k < In. Noting that whenever
/ + j > /7, bi+j = b\ for 0 ^ r < /?, and similarly for j + k > /?, an exami-
examination of the possibilities for /, /, and k will leave at least one factor in each
expression the identity element and the other as a. Consequently, B) of
Theorem 7.1.1 is satisfied. Therefore the extension exists.
Because of Theorem 2.2.5, it can always be assumed that there is an
equivalent extension H such that H contains A. This is used extensively in
the construction of examples.
Theorem 7.4.2. If B is an infinite cyclic group, then the extension
1 > A—?-* G—?-» B > 1 exists iff it is a splitting extension.
Proof. Exercise.

7.4 Cyclic Extensions 89
Theorem 7.4.3. A finite group G having a cyclic normal subgroup A =
of order n and GjA = B cyclic of order m, has generators a and b satisfying
the following defining relations:
1. an = 1, bm = a\ for n,rn>l, and
2. b^ab = aJ\ for jm =1 and /(/-I) = 0 modulo n.
0
Proof. Since jB is cyclic, then G contains a coset representative b with respect
to A such that bm = a1. The automorphism of A associated with b yields
b~1ab = «J since A is abelian and normal in G. However, by Theorem 7.4.1,
baib = a1. So aij = a1, that is, i(J -1) = 0 modulo w. Finally, 6~wa6m =
ajm = a implies that jm = 1 modulo /?. Hence the conditions are satisfied.
The converse of Theorem 7.4.3 is also valid and its verification is left to
the reader.
At least in theory, one should now be able to construct all solvable
groups.
A special case of the extension of a cyclic group by a cyclic group is the
metacyclic group having G' and GjG' both cyclic.
Theorem 7.4.4. A finite group G is metacyclic iff G = <a, b} such that
1. a" = #»=!,
2. b~1ab = aj,
3. (nj -1) =1, and
4. jm = 1 modulo n.
Proof. Exercise.
As mentioned above, a finite solvable group can be formed through
cyclic extensions. However, does there exist one in which the factors of a
derived series are cyclic? The next theorem points out that this is possible
only if G is metacyclic.
Theorem 7.4.5. If in a group G, there is an integer i > 0, such that Gi~1jGi
and G7G/+1 are cyclic, then Gl = Gi+1.
Proof. Set G* = Gl'/G/+1 for which G° = G. Then G*' = G1'1/^1 and
G*" = G7G?+1. If x is a generator of G*", then G*l%G*(x) is monomorphic
to a subgroup of Aut(G*"). Hence G*/^G*(x) is abelian and G*' c ^G*(x).
Therefore G*" =1.
From the earlier work on semidirect products, it is evident that each
extension of a finite cyclic group A by a finite cyclic group B of relatively

90 General Extension Theory 7.5
prime orders is a splitting extension and these are completely determined by
Hom(?,AutG4)).
Several further results will be given in Section 8.2. Among them will be
that each finite group G having all Sylow subgroups of G cyclic is metacyclic.
EXERCISES 7.4
1. Prove Theorem 7.4.2.
2. Prove the converse of Theorem 7.4.3.
3. Prove Theorem 7.4.4.
4. Determine a metacyclic group of order 54 that does not have all the Sylow sub-
subgroups abelian. Hence not all metacyclic groups have every Sylow subgroup
cyclic.
5. Determine all nonabelian groups of order 12.
6. A p-group G is called bicyclic if G has a cyclic normal subgroup A such that
G/A is cyclic. (Note how this differs from the definition of metacyclic given
above. By the Basis Theorem, the only ^-groups satisfying that definition would
be cyclic.) Show that G has two generators x, y that satisfy the following condi-
conditions :
a) xPn = 1, yPm = xPk, y~x xy = xj, for k ^ 0,
b) jpm = 1 modulo pn,
c) pk(j — 1) = 0 modulo pn, and
d) \G\ = pn+m.
7.5 FINITE EXTENSIONS OVER A CYCLIC
MAXIMAL SUBGROUP OF PRIME POWER ORDER
If 1—-+ A -^-» G —?-> B > 1 has A and B finite cyclic and A is a maximal
subgroup in G, then G is called a finite extension over a cyclic maximal sub-
subgroup. The main purpose here is to characterize G whenever G is a p-group.
If A and B have relatively prime orders, then the extension is a splitting
extension and so it can be represented as a semidirect product. Clearly, if
A is a maximal subgroup in G, then B must be cyclic. For the remaining
portion of this section it will be assumed that A is cyclic of order p"'1,
A^ G, and (G:A) = p.
Suppose that G is abelian and that the exponent of G is pk for k ^ 1.
If k = n, then G is cyclic and it has precisely one element of order p. If G
is not cyclic, then the exponent of G is pn~x and G is expressible as a direct
product of cyclic /^-groups and having at least two nontrivial direct factors.
Therefore, whenever G is not cyclic and has one element of order p, there
exist at least two such elements. Consequently, either G is cyclic or
G = (ala**1-1 =1> ® (c\cp =1> where A = <a>.

7.5 Finite Extensions over a Cyclic Maximal 91
Next, consider G to be a nonabelian p-gioup. Since A is to be cyclic,
then G has at least one subgroup of order p. If G has a second subgroup of
order p, then this subgroup is not contained in A and a splitting extension
exists. Therefore G can be expressed as a semidirect product.
The class of nonabelian/^-groups having precisely one subgroup of order
p is not empty. For example, the generalized quaternion group
G = (a,b\a2n-1 =1 b2 = a2-2 b~xab = a'1)
of order 2n has this property. Since each element not in <a> has the form
alb, then
(a'bJ = a'ba'b = aV (b'Wb) = aVa'1 = b2.
The generalized quaternion group is just the quaternion group whenever
\G\ = 23. It has a unique property given in the next theorem.
Theorem 7.5.1. A finite p-group has two cyclic maximal subgroups and only
one subgroup of order p iff it is the quaternion group.
Proof. Clearly, the group G is not abelian. Suppose that \G\ = pn and that C
is the intersection of the two cyclic maximal subgroups. Then Z(G) = C.
Moreover, G' c C since (G:C) = p2. Therefore, G is nilpotent of class two
and GjC is elementary abelian. Hence, for each pair of elements a,beG,
\bp, a] = [b, aY =1. Furthermore, (ab)p = apbp[b, ayip~1)/2. Then (ab)p =
apbp, whenever p is odd. Therefore, the mapping g h-» gp, Vg e G, defines an
endomorphism e.G^C having |Ker@)| ^p2 since |G/Ker@)| ^pn~2.
This contradicts the hypothesis, since Ker(Q) contains only elements of ex-
exponent^. In the case;? = 2, (abJ = a2b\b, a]. Then {abf = a*b\b9 a]2 =
a4b4. Hence, the mapping g h-» g4, Mg e G, defines an endomorphism
0:G -» C2 having |Ker@)|^ 23. So there are at least two cyclic subgroups
<a> and <Z?> of order four. One can assume that <a> is contained in one of
the two given cyclic maximal subgroups. If<a> c C, then <a><6> is an abelian
group having at least two elements of order two. Therefore, <a> is maximal
and G = <«> <&>. Since there is only one subgroup of order two, then
a2 = b2 = (abJ gives the defining relations a4 = 1, b2 = a2, and b~xab = a3.
This is the quaternion group. The converse follows out of the remark pre-
preceding the theorem.
Theorem 7.5.2. A finite p-group G has precisely one subgroup of order p iff
either
1. G is cyclic or
2. G is the generalized quaternion group.
Proof. Suppose that G contains precisely one subgroup of order p. Then
each proper subgroup of G has precisely one subgroup of order p and so it

92 General Extension Theory 7.5
is either cyclic or a generalized quaternion group, by induction on |G|.
If p is odd, then each maximal subgroup is cyclic and G must be cyclic
by Theorem 7.5.1.
Assume that p = 2. Then G contains a normal subgroup A ^ 1 that is
maximal with respect to being abelian. Of course, %G(A) = A and
GjA is monomorphic to a subgroup of the abelian group Aut(A); A = <a>
is cyclic of order 2k. Suppose that A ^ G and consider an element b e G\A
such that b2 eA. Then [b, A~] 7M, since otherwise the conditions on A are
contradicted. Therefore there exists an integer / such that «tf'>:<?2» = 2.
However, this implies that <y> <6> is nonabelian with two cyclic maximal
subgroups and precisely one subgroup of order two. Hence <a!> <6> is the
quaternion group. Therefore / = 2k~2. So b2 = a2k~\ It is proved similarly
that (abJ = a2k~\ Combining the two results, one obtains b~lab = a~~x.
Moreover, this result holds for each element beG\A having b2 e A. If 0
and y are two automorphisms of order two having a& = ay, Va e A, then
a&2 = ay® _ a jmpijes that y = 0 = 0. Therefore, GjA has precisely
one subgroup of order two and it must be cyclic. If y e Aut(A), yA = t4,
ay = a\ and ay2 = a@ = a, then a'2 = a'1 implies that i2 = —1 mod 2k.
A contradiction arises. So (G:A) = 2 and G is the generalized quaternion
group.
The remarks preceding Theorem 7.5.1 justify the converse.
Theorem 7.5.3. A finite /?-group is a finite extension over a cyclic maximal
subgroup of prime power order pn~x iff the group is isomorphic to one of the
following groups G:
1. G is cyclic, n >l,
2. G = (ala*"-1 =1>® <b\bp =l>, n ^2,
3. p is odd and G = {a,b\apn~l = bp =1, b'^ab = a1+p">, n > 3,
4. p = 2, G = (a,b\a2n~i = b2 =1>, and
a) b~lab = a, « ^ 3, (the dihedral group D2n), or
b) b~1ab = a1 + 2n~\n > 4, or
c) Z?-1^ = a-1 + 2n-2, or
5. ,p = 2 and G = (a^la2" =1, Z?2 = a2", b'^b = a'1}, n ^ 3, (the
generalized quaternion group <22")-
Proo/. The second paragraph of this section verifies A) and B) whenever
G is abelian. So assume that G is nonabelian. First consider the case that
(ja\aPn~x = 1> is a maximal subgroup of G and that there exists an element
be G\A of order p. Then G = [<a>] <6>. By Theorem 7.4.4, b~xab = aJ for
/ =1 mod/?" and (p", j -1) =1, that is, j $\ mod/. One can

7.5 Finite Extensions over a Cyclic Maximal 93
assume / = 1 + tp" 2. There exists an integer k such that kt = 1 mod p. So
b~kabk = a1 +pn~2 whenever/? is odd. Then A + pn~2)p =1 and l + pn~2 #1
mod/?". Hence C) is obtained by replacing bk by b.
Suppose that p = 2. For n = 3, j2 =1 and 7 =? 1 mod 4 implies that
7 = — 1. However, for n > 3, and using again the fact that generators have
to go onto generators, there are solutions 7 = —1, j =1+ 2n~2, and
j = — 1 + 2n~2. The three possibilities lead to distinct extensions. Whenever
/ = — 1, G would contain precisely one cyclic subgroup of order 2", since
(ambJ = am6am6 = amb2 {b-lamb) = am62<Tm = A2 =1.
If 7= -1+ 2"-1, then |G'| = 2n > 2 since w > 3. If j = 1+ 2", then
\G'\ = 2. For either7 = 1 + 2"~2 or7 = -1+ 2n~2, G has at least two cyclic
maximal subgroups. This completes D).
Finally, E) is obtained from Theorem 7.5.2 when the case that G has
precisely one subgroup of order two is considered.
It is evident that groups of the form (l)-E) satisfy the hypothesis.
EXERCISES 7.5
1. Prove that if a finite /7-group has precisely one subgroup of order pm for
1 < m < n, then the group is cyclic. (Use Theorem 7.5.2.)
2. Prove that there are five types of groups of order p3. The abelian types are of
the form A,1, 1), B, 1), and C). For p = 2, there are the quaternion and the
dihedral groups. For/? ^ 3, there will be the types <tf, b\ ap2 = bp = 1, b~1 ab =
al+p} and <a, b, c | ap = bp = cp = 1, ac = ca, be = cb, b'1 ab = ac}. (Com-
(Compare methods with Exercise 5.1.6.)
3. Show that every subgroup of order p2 is cyclic in a finite ^-group if? there is
only one subgroup of order p.

8
THE THEORY OF THE TRANSFER
8.1 THE TRANSFER
In this chapter a group G will be examined as an extension by a subgroup of
the factor group B\B' for a subgroup B in G. The transfer relies on the coset
decomposition of G with respect to B, (G: B) being finite, and the fact that if
G -» A is a homomorphism a of G into an abelian group A, and (I is the
epimorphism of G -> G/G\ then there is a unique homomorphism 77 such that
the diagram here is commutative.
Consider a subgroup B in a group G such that (G:B) = n. Form the coset
decomposition G = axB + ••• + #„? and note that for a fixed element g g G
and for each / there are both a uniquely determined integer j and an element
b e B such that gat = a p. Suppose that gak = ajb1. Then ga^^1 = gap'1
and fl^1^ g B, that is, ^ and ak are in the same left coset. Therefore i = k.
Hence there exists a mapping G -* Sw, 5n the symmetric group of degree n,
dependent upon the coset representatives chosen. So denote gat = aipbt
for p e Sn and ip denoting the image of / under the mapping p. Suppose that
G = cxB + ••• + cnB is another coset decomposition and that for the same
element g there is a o g Sn for which gct = ciabf. By the nature of the coset
representatives, there must exist tg^ such that ct = aixbh for i = 1,...,/?.
Since
C(ixpz-1) = a{izpx-1)x ^(ixpx-1) = aixpb(ixpx-^
one obtains
gCi = gaixEi = aixpbifii = c(/TpT_i) B^.^b^Bi.
So b? = 5G^.1N^. Now consider B/B''. Note that
(n br) v = (n k^) (n **) (n 5,) *',

8.1 The Transfer 95
since BjB' is abelian. As / ranges over 1,... n, so does both hpx'1 and h.
Consequently,
Hence the mapping of G -* B\B' denned by
is independent of the choice of coset representatives and dependent only on
the fundamental properties of B\B' and the coset decomposition of G with
respect to B. It remains to determine whether or not this mapping is a homo-
morphism.
Suppose that g and h are two elements of G = axB + ••• + anB such
that gat = aipbi and hat = aiabf having p, aeSn under the mapping of
G -> Sn as determined in the preceding paragraph. Then ghat = gaiabf =
aiopbiob* and
(n biabt) b' = (ft k) (n **•) ^ = (n ^) ^ (n *?) ^.
So, if VG^B, or simply V whenever G and 5 are clearly understood,, is the
mapping defined by
then (jA)F = #F/zF. Hence V is a homomorphism. Summarizing:
Theorem 8.1.1. If B is a subgroup of finite index n in a group G, G = ax 5 +
... + anB is a coset decomposition of G with respect to B, and for each g e G
and all /, #<^ = o^-, then
1. there exists an element p&Sn such that ga^ = a^b^
2. there exists a homomorphism VG_+B of G -> B/IT defined by
and
3. P^_^B is independent of the system of coset representatives.
The homomorphism VG_+B given in Theorem 8.1.1 is called the transfer.
It defines an extension 1—> A-*-> G-&-* C—-*1, C c 5/F, such that
G' c Ker(j5). The nature of the epimorphism fi-.G-^G/G' implies the

96 The Theory of the Transfer 8.1
existence of a unique homomorphism Vg^B such that the diagram here is
commutative.
-> BJB'
G/G'
Unless B is abelian, we should not expect transitivity of the transfer since the
homomorphisms are from groups into factor groups. However, the next
result can be obtained.
Theorem 8.1.2. If B and C are subgroups of a group G such that C c B c G
and (G:C) is finite, then FG*_C = V^BV*^C.
Proof. Let n = (G:B) and m = (B:C). For G = axB + • • + aw?, 00, = 0^,
and B = bfC + ••• + Z?*C, there exist permutations (Ji^Sm such that
&.^* = b*aicifj, Vz and j.
Moreover,
G = J^afifC and #0^/ = aipbtbf = aipbfa.cifj.
Therefore,
*.«-((iW
-(
= (fi cu)c,
and since CjC is commutative,
1
n / m
n(r
ii \ i /
As in the case of the factor systems in extension theory, direct compu-
computation of the transfer is seldom used. But it does permit certain developments
that other methods apparently fail to obtain.
EXERCISES 8.1
1. Consider the symmetric group of degree three,
S3 = (a,b\a3 = b2 = 1, ZT1 ab = a2}.
Show that FS3_*<a> must send S3 -> 1.

8.2 Burnside's Theorem 97
2. For KG_>B, (G:B) = n and G = AB = BA, if JJ c #G(y4), show that for each
# e G,gVG->B = g"B'. Also prove that if B c Z(G), then #FG-».b = g'1.
3. Suppose that P is a proper abelian Sylow p-subgroup of a finite group G. Show
that G/nZ(G)nP = l.
4. For FG_*B and G = AB = BA, if A n 5 = 1, then prove that A c
8.2 BURNSIDE'S THEOREM
The next theorem is needed as a preliminary result in this section, but it
can be used to prove several of the preceding exercises. Whenever reference
is made to the transfer VG_B for a group G, it is assumed that (G:B) is finite.
Theorem 8.2.1. If FG_»B is the transfer of G into B, then there is a set of coset
representatives {st}k and a set {nt}k of positive integers with respect to each
g g G such that
2. ?if, = (G:B),
1
3. «f is the least positive integer for which sJ1gnisi&B.
Proof. Since the transfer is independent of the coset representatives, then,
for (G: B) = n, consider a coset decomposition of G with respect to B given
by G = atB + ••• + anB. Hence for each g e G there exists p e 5n such that
g#f = afpZ??. Assume p to be represented as a product of k disjoint cycles.
Consider a cycle (il9 i2,..., in) and the associated set of coset representatives
{atj}. Then
gah = ai2bh,gai2 = ahbh, ...,gaini = ahbini,
and
9 = ahbinfill = "' = 0*3**2**2 * = ai2bhah1'
Consequently,
There are /: cycles and n1 + ••• + nk = «. Denote by {^}&, for / =1,...,/:,
the subset of the coset representatives for which st = ah in each cycle. This
verifies A) and B). As for C), note that for t < nh g* = aitbit... b^^1.
If afl1gtail = af1iaitbit... bheB, then aJ^a^eB. Hence ah and ait belong
to the same left coset of B. A contradiction is reached.

98 The Theory of the Transfer 8.2
Theorem 8.2.2. If X and Y are normal subsets of a Sylow p-subgroup P in
a finite group G that are also conjugate in G, then there is an element
9 G ^g(P) such that X9 = Y.
Proof. Since there exists an element aeG such that Xa = Y, then Y is a normal
subset of Pa. Hence P and Pa are contained in the subgroup JfG(Y) and so
there is an element b e <sVG(Y) such that Pab = P, that is, ab e ^TG(P). There-
Therefore g = ab is the required element since Xab = Yb = Y.
One immediately notes that in a finite group G, Theorem 8.2.2 implies
that if two distinct elements in the center of a Sylow ^-subgroup P are
conjugate in G, then they are conjugate in jVg(P).
The next theorem is due to Burnside.
Theorem 8.2.3. If a Sylow /7-subgroup P in a finite group G is contained in
Z(/TG(P)), then G = [iV]P for some subgroup N^ G. (The condition is
equivalent to #G(P) = ^)
Proof. Since P is abelian, then T^p sends G into P and hence G = Ker(FG_F)P.
By Theorem 8.2.1, for a non-identity element h e P there is a subset of coset
representatives {st}k and a set of positive integers {nt}k such that
k k k
E«? = (G:P) = w, /z^G-p = [](^71/z^.)P'==[]4y-1/z»^, and jf^^eP.
i 11
Consequently, /z"? is conjugate in G to an element of P. Since P is abelian,
then hni and sj1h"isi are normal subsets in P. By Theorem 8.2.2, there exists
an element geJ^G(P) such that g'^g = sj1^^^ However, ^TG(P) =
#G(P) implies that sj1tilisi = hnt. Therefore,
since p is prime to n. Hence Ker(FG_>P) n P = 1.
If for a subgroup B of a group G there is a subgroup ^4 in G such that
G = [^4]?, then B is said to have a normal complement in G. If a Sylow
/7-subgroup B in G has a normal complement A, then ^4 is called a normal
p-complement of 2? and G is called p-nilpotent.
One application of Theorem 8.2.3 is the following theorem.
Theorem 8.2.4. If for the smallest prime p dividing \G\ of a finite group G,
a Sylow ^-subgroup P is cyclic, then G has a normal ^-complement, that is,
G is /7-nilpotent.
Proof. |Aut(P)| = pn - p"-1 = f~\p -1) whenever \P\ = pn and

8.2 Burnside's Theroem 99
P) is monomorphic to a subgroup of Aut(P). Since P is cyclic,
thenP c #G(P). This implies thatpf(jVG(P):<$G(P)) and so \jVg(P)I<$g(P)\
divides p — 1. A contradiction is reached unless ^VG(P) = ^G(P) since /? is
the smallest prime dividing |G|. By Theorem 8.2.3, P has a normals-comple-
normals-complement.
We are now able to identify the finite groups having all their Sylow sub-
subgroups cyclic.
Theorem 8.2.5. A finite group G having all of its Sylow subgroups cyclic is
metacyclic.
Proof. Use induction on \G\. For the smallest prime/? dividing |G|, a Sylow
subgroup P has a normals-complement, say N, such that G = [NJP. Hence
N is metacyclic and G is solvable. The factors of the consecutive terms of
the derived series are not only abelian but also cyclic, since they would be the
direct product of cyclic subgroups. By Theorem 7.4.5, this implies that
G" =1. Hence G is metacyclic.
Corollary 8.2.5. A finite group G has all of its Sylow subgroups cyclic iff G
satisfies the following conditions for some pair of generators a, b:
1. G = (a,b\an = bm=\y,
2. b~xab = aj, Uj</i-1,
3. (nj~l)=h
4. jm = 1 mod n, and
5. (m,n)=l.
Proof. By Theorem 8.2.5 G is metacyclic and by Theorem 7.4.4 G satisfies
conditions (l)-D). Since G = \A~\B for A = {a} and B = <6>, then n and
m are relatively prime, for otherwise not all of the Sylow subgroups are
cyclic. Conversely, by Theorem 7.4.4, conditions (l)-D) imply that G is
metacyclic and E) implies that all Sylow /^-subgroups are cyclic.
Theorem 8.2.6. If G is a finite nonabelian simple group, then
1. \G\ is divisible by 12 or
2. \G\ is divisible by p3 forp the smallest prime dividing |G|.
Proof Assume for the smallest primes dividing |G|, that a Sylows-subgroup
P has order less than/?3. By Theorem 8.2.4, \P\ =? p and if \P\ = p2, then P
cannot be cyclic. So consider P to be an elementary abelian group of order
p2. Then
| Aut(P)| = (p2 -1) (p2 -p)= p{p -1J (p +1).
By Theorem 8.2.3, JTG(P) # <gG(P). Therefore JrG{P)l<$G{P) is mono-

100 The Theory of the Transfer 8.3
morphic to a nontrivial subgroup of Aut(P) and |/c(Wg(^)I divides
(p —IJ (p +1). If p is odd and g is a prime greater than p, then q X
(p -IJ (p +1). So, suppose that/? = 2. Then |^"C(P)/*C(P)| must divide
3 and hence must be 3. Therefore, 12 divides |^TG(P)| and so 12 divides |G|.
One concludes that if 2| |G|, then |G| is divisible by 12 and if 2 X |G|, then
|G| must be divisible by the cube of the smallest prime dividing |G|.
EXERCISES 8.2
1. Prove that if P is a Sylow p-subgroup of a finite group G and JfG(P) = ^G(P),
thenG'nP = l.
2. Prove that if for a finite simple group G, \G\ = pk, (p, k) = 1, and P is a Sylow
^-subgroup of G, then ^G(P) is properly contained in J^G(P) and (jV%g(P):(^g(P))
divides Q? — 1).
3. Prove that all Sylow subgroups of a finite group G are cyclic iff G = [_A~\B for
A and B cyclic and G' = A.
4. Determine a counterexample to the converse of Theorem 8.2.5.
5. Prove that all finite groups of square-free order are supersolvable.
6. Prove that each subgroup and each homomorphic image of a finite /?-nilpotent
group is also /?-nilpotent.
7. If N is a minimal normal subgroup of a finite i?-nilpotent group G and p divides
\N\, then prove that N c Z(G).
8. If the finite group G isi?-nilpotent for all primes/? dividing |G|, then prove that
G is nilpotent.
8.3 THE THEOREMS OF GRUN
Theorem 8.3.1. For each prime p dividing the order of a finite group G, there
exists a unique normal subgroup G'(p), called the p-commutator subgroup,
such that
1. GjG'{p) is an abelian /?-group and
2. if G/A is an abelian ^-group, then Gf(p) c A.
Proof. It is enough to prove that if GjA and GjB are abelian /7-groups for a
given prime p, then GjA n J5 is an abelian /?-group. The details are left as an
exercise.
Theorem 8.3.2. For each Sylow /?-subgroup P in a finite group G
2.
3. Ker(FG_P|P) = PnG', and
4. V

8.3 The Theorems of Griin 101
Proof. Since Ker(FG_+P) contains all elements of order prime to /?, it is clear
that A) holds. G = G\p)P implies that GjG'(p) ^ P/Pn G'O). Since
G' c G\p\ then P n G' c G'O)- But PG'/G' is a Sylow ^-subgroup of GjG'
and so G\G' = (PGf/Gf) ® Gf(p)jGf. Therefore, PG' n G'Q?) <= G' and this
leads to PnG'lp) e G'. Hence, Pn G'O) = PnG' and this proves B).
As for C), first note that G'(jp) ? Ker(FG_P). By Theorem 8.2.1, there exists
a set of coset representatives {st}k and a set {«*}* of positive integers with
respect to each element g e P such that
and nt is the least positive integer for which sj1 gnist e P. Moreover,
= f] g^lsT1, gnil st) Pf = gn g P',
l
for some geG'. Therefore, gVo->p = P' implies that gneG' and (n,p) =1
implies that g e G'. Therefore, gePnG'. Hence KerCF^p \P) = P n G'.
D) is a result derived from A) and C).
Corollary 8.3.2. For a Sylow ^-subgroup P of a finite group G,
Ker(FG_P) = G'{p).
Proof. By Theorem 8.3.1, G'(p) c KerCFc^p). By B) and C),
GIG\p) s PIP nG' = P/Ker(FG_P \P) = PjP n
So
G/G'O?) s Ker(FG^p)P/Ker(FG^P) = G/Ker(FG^P).
Consequently, |Ker(FG_P)| = \G'(p)\. Hence G'(p) = Ker(FG^P).
In the next theorem it will be convenient to have available several ele-
elementary properties of double cosets. A double coset of a group G with respect
to subgroups A and B is a set AgB for some element geG. For another
double coset AhB, consider xeAhBnAgB. Then x = agb = alhb1, for
a, ax eA and b, bteB. Thus # = a~1a1hb1b~1. Consequently,
AgB = A(a-1a1hb1b~1)B = AhB.
Therefore, either AhB = AgB or the two double cosets are disjoint. In
particular, if A and B have finite index in G, then G has a double coset de-
decomposition of the form G = AB + AgtB + ••• of pairwise disjoint double
cosets.

102 The Theory of the Transfer 8.3
It is apparent that a double coset AgB is the set-theoretical disjoint
union of either right cosets with respect to A or left cosets with respect to B.
Assume that A and B have finite index in G. Let us then suppose that
agB = axgB, for a, axeA. But g~1a^iag eB implies that a^1 aegBg~1.
Hence af1 aeAnB9'1. So a and ax belong to the same left coset of
C = A n B9~\ Suppose that aC = atC. Then a = axc and c = gbg'1, for
some ceC and beB. Therefore agB = axcgB = a^gbg'1)gB = axgB.
One can now conclude that the number of left cosets of B in AgB is equal to
(A: A n B9'1). In a similar manner one can also conclude that the number of
right cosets of A in AgB would be (B:B n A9).
The next two theorems are generally referred to as the First and the Second
Theorems of Grun, respectively.
Theorem 8.3.3. For a Sylow /?-subgroup P in a finite group G and FG_»P,
PnG' = (Pn ^G(P)', {PnF9\ge
Proo/. Since P* = (Pn /G(P)', {P n P'» | # e G}> c p n G\ it remains to
show that every element ofPnG' is in P*. Induction will be used on the
orders of the elements of P n G' since le P n G' implies that le P*. Consider
an element ^ 1 in P n G' and assume that any element of order less than
g is in P*. By Theorem 8.2.1, there exists a set {^}fc of coset representatives
and a set {wjk of positive integers such that
Y^\nt = (G:P) = n, and «; is the least positive integer for which sj1 gnisteP.
Consider G = P + Ph±P + ••• + P/*rP, the set of st that belong to a par-
particular double coset PhP, and the factors (XL^T1 0"^) ^ associated with it.
Then st = x^jf for xt, yt e P. Since PjP' is abelian, then
for /= h~1(YlhxY1gHixi)h. By the remarks preceding the theorem, the
number of left cosets of P in PhP is (P:P n Ph~l) = pm, for m > 0. There-
Therefore, 2>; = ^.
First suppose that m = 0. Then PAP = ZzP implies that h e JVG(P).
Hence there is only the factor/= h~xgh. Since fg = h~1ghg~1 eJrG{P)r
and feP, then fg~xePn JTG{P)f c p*. Therefore feP*g = gP*.
Suppose that m > 0. Then hfh'1 = T\hX71gnixi and so hfh~1P' =
n^r1^"^'^' = 9pmP'- Without loss of generality, it can be assumed that
one of the si9 say sj9 is h. The integer n} in sj1gtljsj = h~1gnjh e P is a power
of/? since #eP. If m >1, then h~1gpmheP, that is, g^ehPh'1. Since

8.3 The Theorems of Griin 103
/eP, then (hfh~1y±g*mehPh~1. From the above, it follows that
(hfh'1y1gpmePf. Therefore (h/h'1)- 1gpmePt n P* and f-1h~1gpmhE
Pn{P'fczP*. Hence t//*" 1gpmh)~1 eP* and so fe(h'1gpmh)P*.
Since gePnG', then #fg-+p = P' implies that P' = (h~1gpTnh)vG-*p. Thus,
h~1gpmheKer(FG_P \P) = Pn G'. Since ra > 0, this element inPnG' has
order less than g. By induction, A" ^"Vz e P*. Also by induction, gpm e P*.
Consequently, /e (A" xgpmh)P* = gpmP*.
In both cases, /e#pmP*, for pm the number of left cosets of P in P#P.
By considering all the factors, it follows that g^G^p c gT'. Since gePnG\
then g^c-p = P' c P*. Thus 0*"g»p c #"P* n P* and therefore #"eP*. But
(«?^)z=l. Hence geP*.
Definition 8.3.4. A finite group G is p-normal if the center, Z(P), of a Sylow
/7-subgroup P in G is contained in the center of each Sylow /?-subgroup in G
that contains Z(P).
It is evident that a finite group having abelian Sylow /?-subgroups for a
given prime p is p-normal. The same is true for a group in which the pair-
wise intersection of the Sylow ^-subgroups is the identity for each prime p
dividing the group order.
Equivalently, a finite group G is p-normal if Z(P) of a Sylow p-subgroup
P in G is Z(Pt) for each Sylow p-subgroup Pt that contains Z(P). It will be
left as an exercise to prove that if a finite group G is p-normal and Z is the
center of a Sylow p-subgroup P, then Z is the center of all Sylow p-subgroups
in G lying in JfG(?) and only those in J\rG{Z).
Theorem 8.3.5. If Z is the center of a Sylow p-subgroup P in a finite p-normal
group G, then GjG'{p) s JfG(Z)\JfG(Z)\p).
Proof. By Theorem 8.3.2,
G\G\p) s PIP n Gf and ^G{Z)\JfG(Z)\p) s P\P n ^G(Z)'.
By Theorem 8.3.3,
PnGf = {Pn JfG(P)\ {PnPf9\ge
Since Pc\JfG(Z)' ^PnGf, it remains to show that PnG' ^ PnJTG(Z)f.
Clearly, JTG(P) ? ^TG(Z) and so P n ^G(P)' c P n ^rG(Z)'. It will now be
proved that C = P n P'9 ^ P n JTG(Z)' for each geG. Note first that
Z c JfG(C). Then Z*7 c ^TG(C) since Z9 permutes with each element of
P'9 ? P9. Furthermore, Z and Zg are contained in the centers of two Sylow
p-subgroups Pt and P2 of JfG(C), respectively, since G is p-normal. More-
Moreover, there exists an element x e JfG{C) such that Px = Px2. But this implies
that Z and Zgx are contained not only in Px but also in a Sylow p-subgroup

104 The Theory of the Transfer 8.3
Px* of G that contains P1. Since G is /^-normal, this means that Z and Z9X
are contained in Z(P*). Hence Z = Z9X and therefore gx = yeJfGB).
Since x e ^(C), then C = Cx = Px n P'9X = Px n P'y. But P c ./TG(Z)
implies that Pfy c ^(Z)'. SoCg ^G(Z)'. Therefore C <= P n JfG(Z)'
for each ^eG. Consequently, Pr\G' ^ Pr\ jVg(Z)'.
If P is abelian in Theorem 8.3.5, then Gvg-+p can be identified more pre-
precisely as it will be in the next corollary. The following theorem is an applica-
application of the transfer in order to obtain the structure of an abelian normal
Hall subgroup in a finite group.
Theorem 8.3.6. If N is an abelian normal Hall subgroup of a finite group G,
then
1. G^g^n = NnZ(G) and
2. N = (Nn G') ® (Nn Z(G)).
Proof. Set VG_>N=V. Since (G:Ker(F)) divides |JV|, it follows that
G = N Ker(F). Consequently, Gv = Nv. Nv is a subgroup of JV since N
is abelian. Moreover since iV<a G and V is independent of the coset repre-
representatives, then Gv is unchanged under Inn(G), that is, GF<3 G.
By Theorem 8.2.1, there exists a set of left coset representatives {sjfc
and a set of positive integers {nt}k for each g e G such that
and n{ is the least positive integer such that s~l1gnisieN. Consequently,
gv = gng* for #* e JV n G'. If g eN such that gveNn G', then gneNn G'.
It follows that geNnG' since (w, |iV|)=l. Hence gv =1. Therefore
(N n G')n Nv =1. Since the «th powers of the elements of N are in
(N n G') Nv, (n, \N\) = 1 implies that all the elements in N are in (JV n G') Nv.
So N = (Nn G') ® Nv. For each element heG and geNv, \_g,h] eNv
since Nv <a G. Of course lg,Ji]eN n G''. Hence [#, A] e (JV n G') n NF = 1.
One concludes not only that g e Z(G) but also that Nv = JV n Z(G). There-
Therefore A) is valid and B) follows because JV = (JV n G') ® NF.
Corollary 8.3.6. For an abelian Sylow /^-subgroup P of a finite group G,
o/. By Theorems 8.3.2 and 8.3.5, G^p ^ G/Gr(^) ^
However,
GVg-*p = PVg->p = j
By Theorem 8.3.6, applied to J^G(P), it follows that
= PnZ(jrG(p)).

8.4 Some Applications of the Theorems of Griin 105
EXERCISES 8.3
1. Complete the proof of Theorem 8.3.1.
2. Prove the statements in the paragraph following Definition 8.3.4.
3. Prove that a^-nilpotent group G is also ^-normal.
4. Prove that Definition 8.3.4 is equivalent to the following: The finite group G
is ^-normal if Z(P)9 = Z(P) for all g e G such that Z(P)g c p.
5. A subgroup N in a group G is Q-admissable, Q a set of mappings of N -> N,
if N* c N for all \j/ e Q. Consider a subset Q in Aut(G) for a finite group G.
If TV is an Q-admissable subgroup of G, then prove that Gvg-+n is Q-admissable.
8.4 SOME APPLICATIONS OF THE THEOREMS OF GRUN
Consider two subgroups T and N in a group G. The set of all elements of T
that normalize N form a subgroup of T. This set will be denoted by jV*t(N)
and it coincides with the previous use in which N was considered as a sub-
subgroup of T. This extended form will be used in the next theorem.
Theorem 8.4.1. (Burnside). If N is a normal subgroup of one Sylow p-sub-
group P in a finite group G for a prime p\ \G\ but N is not normal in another
Sylow /^-subgroup of G, then there exists an element g e G\J^G(N) of order
#s for a prime q ?= p such that
1. M = NN9 ... JV3" is a p-group for n = qs and
2.
Proof. Among the Sylow/^-subgroups of G in which iV is not normal, denote
by T one in which (T: rfT(Nj) is minimal. Set A = JVT{N). Of course
^4 = T n */TG(iV). Then P is a Sylow ^-subgroup of tyTG(iV), since iVo P,
and there exists a conjugate Pt of P in *WG{N) that contains A. Since ]V is
normal in Pl5 we can without loss of generality assume that P is Pt.
Clearly, i c T properly and consequently A a jVt(A) properly because
T is nilpotent. Since \A\ < \T\ = \P\, it also follows that A c Jff(A)
properly. Suppose that P2 is a Sylow ^-subgroup of G that contains JfF{A).
Then
J/pOl) <= P <= ^rG(N) and ^P(A) S jrPi(N) = P2n JTG(N).
However,
(P:A) = (T:A).
This contradicts the fact that (T:A) is minimal. So JV<i P2. This implies
that for any j^-group B of G that contains ^VP(A), N o S. Next note that
^4 <i {.yTp^), jV^A)} = C. If P3 is a Sylow ^-subgroup of C that contains
jV*p(A), then by the previous remark, No P3. Furthermore, ^4 <= jVt(A)

106 The Theory of the Transfer 8.4
properly implies that JfT(A) ? JTG(N). Otherwise, JTT(A) c JTT(N) = A.
So C ? j\rG{N). Therefore there is a Sylow ^-subgroup Q of C, for a prime
q ^ p, such that Q ? /G(iV). Consider an element g e Q, \g\ = qs, such that
g <fc <Wg(N). Then for all /, Ngl <i A01 = A, since JV <a yt <a C. Consequently,
M = N Ng ... N9"'1 exists as a/7-group for ^ = #, and M9 = M. But g$J^G(N)
since JV* ^ N.
Theorem 8.4.2 (N. ltd). If each proper subgroup in a finite group G is
/?-nilpotent, but G is not, then G contains a normal Sylow /7-subgroup.
Proof. First consider G to be /7-normal. For a Sylow /?-subgroup P of G,
if yf/G(Z(P)) c G properly, then «yf/G(Z(P)) is /?-nilpotent. Moreover, by
Theorem 8.3.5, GjG'{p) sz J\rG{Z(PJ)j JTG(Z(P))'(/?) ^1, and so G'(/?) c G
properly. This implies that G'Q?) is/7-nilpotent. However, since the normal
/7-complement N of G'{p) is characteristic in G'(p), then iV<i G and G/7V
is a /?-group. Hence G splits over N and therefore G is /7-nilpotent. The
contradiction implies that J^G(Z(P)) = G. Of course, if P is abelian, then
P <3 G and the conclusion is satisfied. So then suppose that P is nonabelian.
Consider G* = G\Z(P) and note that the proper subgroups of G* are
/?-nilpotent by Exercise 8.2.6. If G* is not/7-nilpotent, then P/Z(P) <i G\Z{P)
by induction. Therefore, P ^ G. If G* is /7-nilpotent, then there exists a
normal/7-complement N/Z(P) c G* properly since Z(P) c P properly. Hence
N is a/7-nilpotent proper subgroup of G. However, the normal/7-complement
Nt in JV is also normal in G (A^ is a Hall/?'-subgroup). This means that Nt
is a normal /7-complement in G. So G is /7-nilpotent. This contradiction im-
implies that the case of G* being/7-nilpotent whenever G = J^G(Z(P)) does not
arise. So, whenever G is /7-normal, then G contains a normal Sylow
/7-subgroup.
On the other hand, suppose that G is not /?-normal. By Exercise 8.3.4
one can conclude the existence of an element xeG such that Z(P)X =? Z(P)
whenever Z{Pf c p. Consider Z(Pf ^ P. Then P, P*'1 <= yTG(Z(P)).
By a Sylow theorem there exists an element ye JfG(Z(P)) such that
p* = px~\ But jxe^G(P) ? ^rG(Z(P)) implies that xeJfG(Z(P)\ that
is, Z(P)X = Z(P). So Z(P)*^ P and Z(P)^ Px~\ Note that Z(P) c PnP*
By application of Theorem 8.4.1 to Z(P), P, and Px~\ there is an element
g e G\J^G(Z(P)) of order q", p =? q, and a /7-group M such that g e ^i/G(M)\
^G(M). Then A = [M] <#> exists. If A c G properly, then ^4 is /7-nilpotent.
Thus <#> <i ^4. This implies that A = M ® <#>. Hence ^r e ^G(M). But this
is a contradiction. Therefore A = G and M is the Sylow /?-subgroup of G.
Hence, if G is not /?-normal, then G has a normal Sylow /7-subgroup.
Theorem 8.4.3 (Frobenius). If J^G(H)/^G(H) is a/7-group for each /7-subgroup
if of a finite group G, then G is /?-nilpotent.

8.4 Some Applications of the Theorems of Grun 107
Proof. Use induction on |G|. If K is a subgroup of G and H ^ 1 is a p-sub-
group of K, then
n K^G{H) nK^ (J^G(H) n X) <€ G{E)\<€ G{H)
is a subgroup of ,yG(H)l^G(H). So ^f K(H)/^K(H) is a ^-group and induc-
inductively iC is /?-nilpotent. Hence each subgroup of G is /?-nilpotent. If G is
not/?-nilpotent, then the Sylow ^-subgroup P of G is normal in G by Theorem
8.4.2. Therefore G = P%G(P). By Theorem 4.2.3, <6G(P) splits over
^g(p) nP = Z(P). Since %G(P) is /?-nilpotent, then Z(P) has a normal
/7-complement B in ^G(P). But 1? is characteristic in <^G(P)<a G, and so
B<z G. Appealing to the same theorem, G splits over B by a Sylow p-sub-
group of G. Therefore G is />-nilpotent.
As an exercise, it will be shown that if J^dH) is /?-nilpotent for each
/^-subgroup H of a finite group G, then G is /?-nilpotent.
The next theorem is also due to Frobenius and it gives further insight
into the divisors of the orders of simple groups.
Theorem 8.4.4. If \G\ is relatively prime to Oi (/?' — 1) for a finite group G
having a Sylow /^-subgroup of order pn, then G is />-nilpotent.
Proof. By Theorem 5.4.4 and by
it follows that ^KG(if)/^G(if) is a /?-group for each ^-subgroup H in G.
Hence G is ^-nilpotent by Theorem 8.4.3 and /?-normal by Exercise 8.3.3.
Suppose first that JfG(Z(P)) c G properly. Then inductively, JfG(Z{P))
is/Miilpotent. So, by Theorem 8.3.4, G/G'ip) s ^G(Z(P))/^G(Z(P))/(/?) # 1.
Again, induction leads to a normal p-complement N in Gr(/?) which of course
is characteristic in G'(p) and hence normal in G. Moreover, G splits over N.
This implies that G is/?-nilpotent. Next consider Z(P) <i GandG* = G/Z(P).
Inductively, G/Z(P) is j9-nilpotent and there exists a normal ^-complement
iV/Z(P) in G*. Since JTG(Z{P))j^G(Z{P)) must be a/?-group, it follows that
JV c <gG(Z(PJ), that is, Z(P) c Z(iV). Hence iV splits over Z{P). But since
N is /?-nilpotent, then iV = Z(P) ® ^4, for A characteristic in N. Therefore
G and G splits over A. Consequently G is /?-nilpotent.
Theorem 8.4.5. If \G\ ^2 for a finite simple group G of even order, then 12,
16, or 56 divides |G|.
Proo/. Suppose that |G| = 2nr, 2f r, r #1, and w > 0. By Theorem 8.2.3,
if n =1, then G has a normal 2-complement. If n ^ 4, then of course
16||G|. For /i = 2, Eii B'-1) = 3 and for /i = 3, 111 B'-1) = 21. By

108 The Theory of the Transfer 8.4
Theorem 8.4.4, r must be divisible by 3 in the first case and by either 3 or 7
in the second case. Hence, if \G\ is not divisible by 16 and \G\ # 2, then \G\
is divisible by 12 or 56.
The above theorem is clearly an improvement over Theorem 8.2.6, but
it took the heavier machinery of Griin's results to make it so. Several other
results of this nature will be found in the exercises.
The final theorem of this section is the result of a question that arises in
a natural way. The alternating group of degree five is simple and each proper
nonabelian subgroup is solvable. However, is it possible that there exist
simple groups in which each proper subgroup is nilpotent or even super-
solvable? The answer is due to B. Huppert.
Theorem 8.4.6. A finite group having all of its proper subgroups supersolv-
able is solvable.
Proof. Suppose that G denotes a finite group with the required properties.
If N is a proper normal subgroup of G, N # 1, then by induction G/N is
solvable. Since N is supersolvable, it follows that G is solvable.
So consider G to be a nonabelian simple group and let P be a Sylow
/7-subgroup for the smallest prime p dividing \G\. Of course jVg{Z{P)) ^ G
and it is supersolvable. Hence jVg(Z(P)) contains a normal subgroup of
index p. By Theorem 8.3.4, this implies that
GIG'{p) s JrG(Z{P))l^G{Z(P))'(p)*\
whenever G is ^-normal. The contradiction implies that G cannot be p-
normal. Hence, if G contains two Sylow /7-subgroups P and P9 such that
Z(P) c P9, thenZ(P) # Z(P9). However, if Z(P) <a P9, thenP,P9 czjfG(Z(P))
and there exists an element heJrG(Z{P)) such that P9 = Ph. This means
that g e jrG(Z{PJ). Therefore Z(P) = Z(P9) and a contradiction arises. So
Z{P)*fi P9. Therefore, by Theorem 8.4.1, there exists a/?-group M and an
element g e J^G(M)\^G(M) such that g # 1 has order qn for p ^ q. Since
JTG(M) 7^ G, then JfG(M) is supersolvable. JfG(M) contains a normal
^-complement N. But this implies that g e N since N contains all elements
of order prime to p. In turn, this implies that g e %>G(M) because [M, g~] ?
[M,N~\ cMniV=l. A contradiction arises. Since contradictions are
reached in both cases, then G cannot be simple.
These theorems only touch upon many useful results that stem from the
Theorems of Grim. However, it is hoped that the reader has gained some in-
insight into the type of situations that the transfer seems best at being able to
handle.

8.4 Some Applications of the Theorems of Griin 109
EXERCISES 8.4
1. Prove: If JfG(H) is p-nilpotent for each ^-subgroup H ^ 1 in a finite group G,
then G is p-nilpotent.
2. Prove that if the Sylow^-subgroups of a finite group G are bicyclic and \G\ is
relatively prime top2— 1, then G is p-nilpotent.
3. Suppose that the Sylow 2-subgroups of a finite group G are metacyclic. Prove
that if \G\ is prime to 3, then G is 2-nilpotent.
4. Prove that a simple group whose order is odd and smaller than 1000 is of prime
order. (Prove without using the known result that every group of odd order
is solvable.)
5. Let T and N be two subgroups of a group G. Prove that jVt{N) is a subgroup
of Tand that NJ^T(N) is a subgroup

9
FREE GROUPS AND COPRODUCTS
This material is independent of the preceding discussions. Since the converse
holds, it has been placed here at the end. The purpose of this chapter is to
establish the significance of the notions involving group generators, defining
relations, and a group being generated by a collection of its subgroups.
9.1 FREE GROUPS
Definition 9.1.1. A group G is generated by a set X of elements in G if each
element in G is expressible as a finite product of the elements and the inverses
of the elements in X. X is called a set of generators for G and the elements
in X are called generators.
Clearly, a set of generators for a group is never empty. Since G, consid-
considered as a set, is always a set of generators for the group G, then it is apparent
that we will be interested primarily in X being a proper subset of G. Also
note that a group G is generated by a set X iff the only subgroup of G that
contains X is G itself. One can readily verify that a subgroup S of a group
G is generated by a nonempty set X iff S is the intersection of all subgroups
in G that contain X.
Consider a nonempty set X and the set of all mappings from X -> G,
GeGrp, denoted by M(X,G). For M(X,G*), G*eGrp, a mapping
(j>*: M(X, G) -> M(X, G*) is defined as the homomorphism 0, if it exists,
of g -+ G* such that for each feM(X, G) there is an /* eM(X9 G*) that
makes the diagram commutative;
that is, <?*:/h->/* = /</>.
Actually, we have implicitly defined a two-variable function F from the
collection of groups, Grp, and their homomorphisms into a collection of
110

9.1
Free Groups 111
sets and the mappings between these sets as indicated by the diagram.
G—"--
Set 0F
M(X, -).
G*—L_> M(X, G*) = G*F
= F^ and denote the collection of sets M(X, G), VGeGrp, by
Definition 9.1.2. A free group with respect to a nonempty set X is a pair
[f; &(X)\ for &(X) e Grp and f e M(X, #XX)), such that for each G e Grp
and for each g e M(X9 G) there is precisely one element h e Hom(F(I), G)
that satisfies # = f Fh. #XX) will be called the free group object and it will
be denoted by 3F whenever X is clear from the context.
Hence the existence of a free group with respect to the set X means that
for any mapping g of a nonempty set M(X, G) there is a unique homo-
morphism h e Hom(J% G) such that the diagram is commutative.
Before showing the existence of 3F in Grp, one needs to observe the
following relationship between generators and elements in M(X, —). Con-
Consider/:Z -» G such that Xs generates G. Suppose that g:X -» G*. Then
there is at most one homomorphism h:G -+ G* that preserves commutativity
in the diagram.
G
(The reader should verify this.)
There may be none! Let G be the symmetric group of degree three, G*
be the cyclic group of order six, and X = {a, b \ a ^ b). Define / to have
af of order three and bf to have order two. Then {Xf} = G. For G*, sup-
suppose that a9 = 1G* and b9 to be of order three. Then clearly h does not
exist, since the only homomorphisms of G -» G* have kernels 1G, G, or the
normal subgroup of order three.

112 Free Groups and Coproducts 9.1
It seems quite reasonable to anticipate that we will be looking for a
universal object in Grp that is generated by an injective image of the non-
nonempty set X.
The next proof has been attributed to J. Tits.
Theorem 9.1.3. For a nonempty set X there exists a free group [f; #"] having
f injective and & = <Zf>.
Proof. The proof can be broken into two cases. Since the second is analo-
analogous to the first, only one case will be given in detail.
Case 1. \X\ is finite.
Each countable set C is bijective with the integers. So the set of group
structures, «s/, on C is not empty. For each aestf, denote by Aa the corres-
corresponding group and by Jia the set of all mappings m: X -> Aa. For each
meJia, index Aa by Aam and form the product object A = Y\stYljiaAam.
Define a mapping f:xi—> A by f:xi-^{xm}/*, i* = \j^Jia. Suppose that (j)
is a mapping of X -* G for G e Grp and without loss of generality
assume that <X*> = G. Since |G| < |C|, then the group G* = GIIZ,
Z the additive group of integers, has \G*\ = \C\. The injection
rj:G -> G*, defined by rj: g \-^ (g, 0), gives rise to the sequence X —^-» G -^—>
G* —^-> ^4a for some aestf and an isomorphism i/^. (The existence of i^
stems from \G*\ = |C|.) Consequently, y = (f)r\\l/EJia and so y induces a
mapping y':X -> ^4ay where ^4ay is now considered as a factor of A Let
y* = aO//*)/? for the projection a of A onto ^4fly, the projection ft of G*
onto G, and i^* the mapping of G* -> ^4ay defined by G* -^- ^4a ^ ^4ay.
Then the diagram here is commutative.
Let #" = <Xf> and h = y*|#". Then h makes the next diagram commuta-
commutative.
X
By the remarks preceding this theorem, there can exist at most one homo-

9.1 Free Groups 113
morphism from SF into G. Hence h is unique. Since f is an injection of
X-+A9 then [f: 9^\ satisfies Definition 9.1.2.
Case 2. \X\ is infinite.
Consider a set C such that \C\ = \X\. Note that if Zc is the set of integers
indexed by c e C, then
nzJ = \x\.
c I
Therefore the set of group structures on C is not empty; even if X is not
countable. Let
= cn(nzc).
By making these replacements for the C and the G* in Case 1 and proceeding
in a similar manner, one shows that the free group exists.
Corollary 9.1.3. Each group G is the image of a free group.
Proof. Consider G as a set and replace X in the Theorem by G. The result
follows by taking 0 as the identity mapping.
How unique is the group given in Theorem 9.1.3? The answer to this
really requires an answer to the uniqueness of a free group and that only
requires an application of Definition 9.1.2. The next theorem is the state-
statement of this result. The proof will be left to the reader.
Theorem 9.1.4. If [f;^PO] and [f*;#"*(X)] are free groups with respect
to the same set X, then there exists an isomorphism h e Hom(#XX), ^*(X))
such that f* = fFft.
In other words, the free group object for a given set X is unique up to
isomorphism.
Since for each nonempty set X, the free group 3F{X) = 3F exists, then
by Theorem 2.2.5, we can assume without loss of generality that X c= #\
Quite often, ^(X) is called the free group rather than [f ;^(X)]. Then X
will be referred to as the set of free generators of #" and the elements in 3F
are called words of X. If X and X* are sets having the same cardinality,
then of course #"(X) ^ #"(X*).
Suppose that X is a set of generators for the group G. Then the homo-
morphism h in the commutative diagram here is an epimorphism with
respect to the inclusion map /.

114 Free Groups and Coproducts 9.1
Consequently, <F contains a normal subgroup N such that SF\N ^ G. N
consists of the elements x = x\l o x322 o • • • o x3nn, for Xj eX, 8t = ± 1,
and xh = 1G. (Use (o) to denote the binary operation in 3F and in G use the
usual multiplicative notation.) The set of all expressions (xl1I1 (x322)h...
(xsnn)h, such that (x3^ o ••• o xsnn)h =1G, is called the set of defining relations
for G with respect to X.
Conversely, if X generates G, then the minimal normal subgroup N in
SF that contains the finite products of all words and their inverses that are the
identity in G, yields <F\N ^ G.
Summarizing: A group is completely determined by its defining relations
with respect to a given set of generators.
In fact, given a set X and a set X* of words of X, there exists a group
having the elements of X* as defining relations. However, please note! The
minimal normal subgroup in J* that contains X* may be $F itself.
Theorem 9.1.5. (von Dyck). If a group G is generated by a set X with a
system of defining relations D and a group H is also generated by X with a
system of defining relations D* that contains 1), then H is a homomorphic
image of G.
Proof Exercise.
EXERCISES 9.1
1. Prove that if X = {x}, then G = {x) is isomorphic to the additive group of
integers if there are no defining relations. On the other hand, if G has finite
order n, then xn = 1 is a defining relation in G. In this case, show that#XX)/iV = G
for TV generated by x o x o ••• o x (« factors), (o) the binary operation in !F(X).
2. Consider S3 = (a, b\ a3 = b2 = I, b'1 ab = a2}. Show that & ({a, b})/N ^ 53,
for ./V generated by the words a o a o a, bob, a o b o a o b and with (o) the
binary operation in#X{#, b}).
3. Prove Theorem 9.1.4.
4. Prove Theorem 9.1.5.
5. Prove that a subgroup S of a group G is generated by a nonempty set X if? 5
is the intersection of all subgroups in G that contain X.
6. Consider /e M(X, G) and # e M(X, G*) such that JfJ generates G. Prove that
there is at most one homomorphism h: G —* G* that preserves commutativity in
the diagram shown here,

9.2 Free Products and Coproducts in Grp 115
7. Prove case 2 in Theorem 9.1.3.
8. If the epimorphic image Gd of a group G is a free group, then prove that Ker@)
is complemented in G.
9. Prove that a finitely generated group is an epimorphic image of a finitely gen-
generated free group.
9.2 FREE PRODUCTS AND COPRODUCTS IN Grp
Definition 9.2.1. A group G is the free product of a collection {Gjj of its sub-
subgroups if
1. G = <{GJ,> and
2. each element g e G, g ^ 1, has a unique representation of the form g =
QiQi'-Qm f°r some integer n, such that for t =1, ...,/?, gt #1, gteGit,
and no two adjacent elements are in the same Git.
The free product will be denoted by G = fli* G(. If |/| = n is finite, then
11/ Gi will be expressed as Gx * G2 * ••• * Gn. The Gt are called the free
factors and the elements g e fl/* Gt, in the form given in B), are said to be
in reduced form.
If G = fir* G;> then it follows from the definition that Gt nGj = 1
whenever i # /. Also from the definition it is clear that if g{ and g} belong to
distinct Gt and Gy, respectively, then ^^ g^ ^ #; #*• So the free product has
elements of infinite order even if each Gt is finite.
Theorem 9.2.2. (van der Waerden). For a nonempty collection {G^j from
Grp, there exists a free product G* of a collection {Gf }7 of its subgroups
such that G* = Gh V/e/. Hence free products exist in Grp.
Proof Consider a collection {Gjx in Grp. Define a word as an ordered
ft-tuple g = (gl9 ...,#„), for n ^1, gteGit, t =1,...,«, such that & is not the
identity in Git and no two adjacent gt belong to the same Gt. For n = 0,
the 0-tuple will be called the empty word. Denote the set of all words formed
from {GJj by G. Two words are equal if they have the same length and they
coincide on each component. Suppose h = (hu ..., hr) is another word. The
product of the word g by his defined as gh = (gu ...,gn,hl9 ..., hr) whenever
gn and hx belong to different Gh gh = (gl9..., #„_;, hj9... hr) whenever gn =
hi1, ...,gn-J+1 = hj}x and gn_j9 hj belong to different Gi9 and gh =
@i, ~;9nhp ¦¦•> *r) whenever #„, A,, belong to the same Gt but #„ ?= hj1.
For gf define g'1 = (g'1, ...^f1) and note that gg and ^f^ are the
empty word.
Let PG denote the set of all bijections of G -> G. Note that for each non-
identity element x e G( the following bijection exists:

116 Free Groups and Coproducts 9.2
For each geG of the form (gug2, ..-,#„) and x ^\h 1; the identity
element of Gh define a mapping by
1- 0*-* (du ~-*9n*x) whenever gn and x belong to different Gh
2. g •—> (gt,..., gfnx) whenever #„ and x belong to the same Gt and #„* ^ lf,
and
3- g^(gi, ...,gn-i) whenever #n and x belong to the same Gt and gnx = lf.
Associate with x = lf the identity mapping of G -> G. The mapping induced
by the element xe Gt will be denoted by x*. As a result, if ye Gt is another
such mapping, then xy is also a mapping and (xy)* = x*y*. Moreover,
y = x~x implies that (xy)* =lt. Since the empty word must be mapped
onto x or y, then x* = j* iff x = y. Therefore this set of mappings G * is
bijective with Gt and furthermore it is a subgroup of PG. In PG, let G* =
<{G?}>. Note that if g = (gu ...,gn)e G, then g* ...g*eG* is the element
that takes the empty word onto g. Therefore each element of G* satisfies
B) of Definition 9.2.1. So G* is a free product of {Gf }7 and has Gf ^ G/5
V/e/. Of course, G is also a free product since G = G*. (The associativity
of the product in G can be proven directly but the isomorphism with G* is
sufficient.)
Without loss of generality, we can assume that Gt is contained in the
group G in the proof of Theorem 9.2.2 and replace the ordered ^-tuples by
the usual multiplicative forms. Further use of the expression in this dis-
discussion of the "free product of an arbitrary collection of groups" is meant in
this sense.
The relationship between free products and free groups is given in the
next theorem.
Theorem 9.2.3. Consider a nonempty set X, an index set / having \X\ = \I\,
a denoted bijection of / with X given by /1—> xh Vz e /, xt e X, and the infinite
cyclic groups Gt = ^(xt)9 for each iel. The mapping 6:^(X) -> n* Gt = G
defined by xt i—> xt is an isomorphism.
Proof. For a positive integer s and an element xt e X, let xt o xt o • • • o xt-
(s factors) be denoted by x/ and xj1oxj1o ...oxj1 (s factors) be
denoted by xjs. Hence, if s is a nonzero integer, then x* is defined in !F(X).
Moreover, an element x = xst\ o xst22 o ... o xstkk e&(X) that is not the identity
element in 3F(X) could not have an image under 9 that is the identity in G,
since G is a free product. Hence 9 is a monomorphism. It is clear that 9 is
a surjective mapping. So 9 is an isomorphism.
Let us examine the free product G = fl/* G/ in regard to the homo-
morphisms of the factors Gt into a group H, V/e /. From the definition it is
clear that there exists a monomorphism, namely the inclusion mapping de-

9.2 Free Products and Coproducts in Grp 117
noted by ij of each Gj -> G. Suppose that if is a group such that for each
jel, there is a homomorphism Xj'.Gj -» H. Define a mapping (j) of G -+ H
by (gh, ...,&„)* = g\[x g*l22 ... #fn'n and associate the empty word with lH. It
can be readily verified that ^ is a Homomorphism and that the accompanying
diagram is commutative for each jef.
Moreover, if 9 was another element belonging to Hom(G, if) that would
make this same diagram commutative for each jef, then 9 = 4>. Hence 4>
is uniquely determined. Furthermore, it is evident that for each element of
Xj Hom(Gh H) there is exactly one homomorphism of G -> H that makes
the above diagram commutative for each is I. Hence each element of
XjHom(G;, H) is obtainable from {tjj in precisely one way.
The property brought out in this last paragraph warrants further
examination. In order to view this property in its proper perspective, we will
introduce the notion of coproduct in Grp. The relationship between the
coproduct and the free product will be analogous to that between the product
and the direct product in Grp. Actually, the coproduct is the dual of the
product in the sense that the "arrows are reversed". Before defining the
coproduct, a few preliminary remarks are necessary.
Consider a collection {Gjx in Grp with respect to an index set I. Each
if e Grp can be associated with the set X7 Horn(Gh H). Suppose that
K e Grp and 0 e Hom(if, K). If Xt s Hom(Gi9 H), then it follows that
Aj0 e Hom(G;, K). Consequently, each </> e Hom(ff, K) induces a mapping
of the set X7Hom(G?, H) into the set XiHom(GI-, K). So we have a two-
variable function F that associates a group with a set and a homomorphism
between groups to a mapping between sets as indicated in the figure.
H-I—+ HF = X
K—?—> KF =
Set (j)F = F^ and denote the collection of sets Xj Hom(Gh H), VH e Grp
by XjHom(G;, —). (Any similarity either to the wording or to the notation

118 Free Groups and Coproducts 9.2
previously used in Sections 2.1 and 9.1 is intentional and the reasons for it
will be brought out in the Appendix.)
The question arises: Is it possible to find a group G for which each
element of X7 Hom(Gl5 K) can be obtained from some particular element
of Xj Hom(G?, G) in precisely one way for each K e Grp? Whenever we are
using finitary methods or when |/| is finite, the free product and its injections
{tt}j provide the answer. In this case, Yl* Gt is called the coproduct object
and the set {tj7 is called the coproduct element. Let us define these terms
more generally and then show that they are nonvacuous in Grp.
Definition 9.2.4. The coproduct for a collection of groups {G^T in Grp is the
couple [{eJj;G], for GeGrp and {ejj e Xj Hom(Gh G), satisfying the
property:
For each H e Grp and each element {Ajj e Xj Hom(G^, H), there is
precisely one homomorphism <fi e Hom(G, H) such that X{ = e$ for each
i g /, that is, the diagram is commutative for each /" e /.
The group G is the coproduct object and {ejj is the coproduct element. G
will be denoted by LL* Gt.
In a manner similar to the proof of Theorem 9.1.3, we proceed to show
the existence of the coproduct in Grp.
Theorem 9.2.5. Coproducts exist in Grp.
Proof. Denote by Xt the set Gt if \Gt\ is infinite or take Xt to be a countable
set if \Gt\ is finite. Without loss of generality we can assume that the col-
collection of sets {X^j has the property that Xt r\Xj = 0 whenever / # j.
Let Y denote a set having \Y\ equal to |U/^»I and let $? denote the set of
all group structures on Y. stf is not empty (see the proof of Theorem 9.1.3).
For each aestf, let jj?a denote the set X/ Hom(Gh Aa) where Aa is a group
structure on Y corresponding to a e stf. Index the Aa by the elements h e j^a
and form A = Y\^Y\^aAah. For each iel, define a mapping ^f:Gf -»y4
by ^i1-^!^?}/* where /* = U^^a- Each ^ is a homomorphism.
Next, consider a group GeGrp and the set of mappings t] = {^1^:
Gt -> G}7. Without loss of generality, it can be assumed that G is generated
by {G^}/- Since the elements in G are finite products of these images,
then |G| < |Y|. Form G* = GIKJIjZ;) for which Zj is a copy of

9.2 Free Products and Coproducts in Grp 119
the additive group of integers and J is an index set so chosen such that \G*\ =
\Y\. There exists a homomorphism cj)*:A-+G* and a projection pr(G):
G* -> G n (Hi <0/» that satisfy ^ = pr(G)<?*^, V/e J. Without loss of
generality it can be further assumed that G* ^ G = ^4a for some a e j/ and
that r] = h for some /z e jfa. Let pr(^4aft) denote the projection of A -> ^4a/j.
If J^ is the subgroup of A generated by the images under the \j/h Vz e J, then
pv(Aah) is the unique homomorphism for which r\x = pr(Aah) i/^ | J% V/ e /.
This completes the proof. (Diagramming is left to the reader).
As an exercise, one can verify that the coproduct object is unique up to
isomorphism. (Exercise 9.2.4).
For a collection {Gj7 of elements in Grp, Uz* Gt contains a subgroup
generated by nonidentity elements g of the form g = glg2 ... gn such that
gj t^Ij, n^l, and no two adjacent gj belong to the same G^. Denote this
subgroup by Uj Gt. Of course, if |/| is finite, then JJ/8 Gt = Ui G>
Theorem 9.2.6. For each collection {Gjj in Grp, U/ G* = Tlf Gt.
Proof. There is a homomorphism fliLL Gf -> Yl* &i by restriction of the
unique homomorphism 0:LJ/* G? -> n/ ^/ to LJ7 Gf. Since a homo-
homomorphism <^> is defined by x^ = x, Vx e Gt and V/ e /, then <^> is surjectiye.
Reasoning as in Theorem 9.2.3, one concludes that Ker@) is the identity.
The theorem states that the free product can be embedded in the co-
coproduct, or, in view of Theorem 2.2.5, the free product is a subgroup of the
coproduct.
For each group G that can be expressed as G = ({G^t) for some col-
collection {Gi}j of proper subgroups of G, there exists a homomorphism
^ 'Hi G( -> G. Then Ker(??) consists of those reduced forms in Uj G( that are
mapped onto the identity of G. Because the mapping is defined by gi h-> gh
gti^l, it follows that if g e Ker(rj), then gn = 1 in G. So Ker(??) becomes a
set of defining relations among the distinct G/5 V/e/.
EXAMPLE. Consider 53 = (a, b \ a3 = b2 = baba = 1>. Then there
exists a homomorphism tj: <a | a3 = 1> * (b | b2 = 1> into G such that
KerO?) = {baba).
By the universality of the coproduct in the mapping ^LI/ Gt -+ 11/* Gt.
Ker(^) has a different content whenever Ker(^) is not the identity element,
It must consist of the identity elements and those elements in LJ/ Gt that
cannot be represented as a finite product of elements g = gxg2 ••• gn m re~
duced form. This suggests that ]_!/ Gt = [Ker(^)] (Ij/ Gt).
Theorem 9.2.7. For a collection {Gijj in Grp, there exists an epimorphism
0»rii* Gt -* rii ^i having Ker@) generated by commutators \_gh gj] for each
pair ijel, i * j, and ty, e G,, V^,- e G;. fb Gt s (Tl/* G,)/Ker@).

120 Free Groups and Coproducts 9.3
Proof. Exercise.
Note the diagram shown here, where a, /?, y, 5, and \j/ exist uniquely by
definition.
What can be said about Ker(^), whenever / is not finite?
Several other properties of free products will be found in the exercises.
These properties will be prerequisite for completing the details in examples
that describe some of the peculiarities of Sylow ^-subgroups in infinite
groups.
EXERCISES 9.2
1. Prove that the set of group structures on Y in Theorem 9.2.5 is not empty.
2. Prove Theorem 9.2.7.
3. Formulate a statement for coproducts similar to Theorem 9.1.4. Then prove it.
4. Show that the center of a free product of two or more groups, each of order
greater than one, is the identity element.
5. Let G = YI* Gt. Prove that each element of finite order in G must be conjugate
to an element of finite order in some Gt.
6. Let G = A * B, \A\ > 1, and A be a periodic group. Prove that A is a maximal
periodic subgroup.
The next exercises will use Exercises 5 and 6.
7. Let G = A * B such that A is cyclic of order p and B is cyclic of order p2 for a
prime integer/?. Show that A and B are Sylow/7-subgroups of G. Hence conclude
that all the Sylow /?-subgroups of G are not conjugate to one another.
8. Let G = A * B such that A and B are cyclic groups of order p for a prime in-
integer p. Clearly A = B. However, show that there are two conjugate classes of
Sylow /7-subgroups.
9. Let G = A * B such that A is a cyclic group of order p for some prime integer p
and B is isomorphic to the additive group of integers. Show that all Sylow
/7-subgroups of G are conjugate to A and that there are an infinite number of
subgroups in the conjugate class of A.
9.3 COPRODUCTS IN Ab
A few remarks are in order for the collection Ab of all abelian groups and
the set of all homomorphisms between pairs of them. The product of a
collection {A^j in Ab exists by a proof identical to that in the second para-

9.3 Coproducts in Ab 121
graph in Section 2.1; commutativity is stressed in the groups involved. The
coproduct also exists in Ab and it is usually called the direct sum.
Theorem 9.3.1. Coproducts exist in Ab.
Proof. For a collection {A^j in Ab consider A = fli A-v For each At there
exists an injection et: At-> A defined by at \-> {xj}j such that xt = a{ and
Xj = oh whenever / # j. Clearly, et e Hom(Ah A). Consider {^r}7 e
Xj Hom(^, B) for B g Ab. Define a mapping 0 of A into B by <f>: {xt}j ->
{*/'}/. One can easily verify that 0 e Hom(/1, B). Moreover, x*' = xei(p
for each iel and xg^. By its formation, 4> is clearly unique.
Corollary 9.3.1. For each collection {A^j in Ab, U7* ^ = 1^/ ^/-
In Section 9.1, replace Grp by Ab in Definition 9.1.2 and in the com-
comments immediately preceding and succeeding this definition. The free group
obtained is called the free abelian group. The free abelian group object is
denoted by #"Ab(X) with respect to the set X.
Theorem 9.3.2. For a nonempty set X there exists a free abelian group
[f;#Ab] having f injective and #"Ab = <Xf>.
Proof Denote the additive group of integers by Z and denote by Zx the set
of all mappings /: X -> Z for which at most a finite number of elements x
have xf ^ 0. Using addition of mappings for an operation, the set Zx be-
becomes an abelian group denoted by J*Ab(X) or simply ^Ah. The set of map-
mappings X* = {/*}/ defined by yfx = 1 if y = x and yfx = 0 if y =? x is bijective
with X via the mapping f:X->X* defined by f:xi->/x. Suppose that
G e &rAb sucn tnat x9 = k anc*./ = 0 whenever x # j. Then g =fx + ••> +fx
(k summands) = kfx, whenever k # 0. If fc = 0, then g is taken as the zero
mapping 0 of #"Ab. Hence fx0 = 0. Consequently, each element h e #"Ab has
the unique form h= Zkfx. Therefore<Xf> =<X*> = ^Ah. It remains to show
that #"Ab is the required free abelian group object. Consider an element
GcAb and a mapping ^eM(I} G). With respect to the bijection f given
above and the inclusion mapping t of X* -> #"Ab> there exists a mapping
<?*: #"Ab -> G defined by x^ = (V)^* that makes the diagram here commuta-
commutative.
X
Since <X*> = #"Ab, then (/>* can be extended to a homomorphism of
JrAb -> G. It is uniquely determined since x and xf must have the same image
in G. This completes the proof.

122 Free Groups and Coproducts 9.3
Corollary 9.3.2. Each abelian group is the image of a free abelian group.
Since the free abelian group object is unique up to isomorphism, it should
be noted that #"Ab is isomorphic to Y\x%x where Zx ^ Z for each xeX.
EXERCISES 9.3
1. Fill in the details in the paragraph following Corollary 9.3.1.
2. Show the respect in which the coproduct in Ab is unique. Repeat for the free
abelian group.
3. Let G be a free group with respect to a nonempty set X. Prove that G/[G, G]
is free abelian on a set Y such that \Y\ = \X\.
4. Reformulate this section with respect to modules over the ring of integers.

APPENDIX
SOME ELEMENTS OF CATEGORY THEORY
Category theory evolved from the theory of groups. Certain patterns of
behavior were recognized to occur often enough that it warranted abstracting
them. Moreover, the results that are obtained are applicable to other alge-
algebraic systems. This brought unity to concepts common to most of the known
algebraic systems. Category theory is now studied as an entity in itself. For
this reason, the discussion that follows is not meant to be comprehensive in
its scope. On the other hand, it will be enough in order that if it had been
presented earlier in the text, then several repetitious arguments could have
been eliminated. In general, application of category theory, its methods,
and its terminology are not yet widely used in the current studies and litera-
literature on nonabelian groups. Consequently it has been placed here in the Ap-
Appendix and Chapters 2 through 8 were developed more or less along the
classical approach.
A.I CATEGORIES AND FUNCTORS
Definition A.I.I. A category ^ is a collection of objects, Ob(^), together with
the sets Mor(^4,B) for each ordered pair A,BeOb(^), called morphisms
of A into B, and a mapping Mor(^4, B) x Mor(?, C) -» Mor(v4, C), for each
ordered triple A, B, C e Ob(^), fog denoting the image of (f,g) for
/eMor(i,B) and g e Mor(B, C), subject to the following conditions:
1. Unless A = C and B = D, Mor(A, B) n Mor(C, D) = 0.
2. There exists a morphism iAeMor(A,A) for each AeOb(^) such that
tAof = f and g o iA = g, V/eMor(,4, B) and Vg e Mor(?, A).
3. Whenever the mapping exists it is associative, that is, (fog)oh =
/o (g o h), for /e Mor(^, B), g e Mor(?, C), and h e Mor(C, D).
A is called the domain and B the codomain of Mor(v4, B). An element
fe Mor(^4, B) is also expressed as /: A -> B.
REMARK. Right-hand notation for morphisms is being used. This is
123

124 Some Elements of Category Theory A.I
consistent with the notation used for mappings throughout the preceding
chapters.
A morphism/eMor(;4, B) is an isomorphism if there exists an element
g e Mor(?, A) such that/o g = iA and gof= iB. For A = B, an isomorphism
is called an automorphism. Each element of Mor(v4, A) is called an endo-
morphism and frequently Mor(A, A) is expressed by End(v4).
The collection of all automorphisms of A e Ob(^) forms a group and it
is usually denoted by Aut(A). End (,4) forms a monoid, that is, an algebraic
system having an identity element with respect to an associative binary
operation.
The nature of a category is barely touched upon in the following
examples. The verification that each is an example of a category will be left
as an exercise.
Examples A.I.2
1. Consider the one-element sets A, B, C, and D, and the functions defined
by the commutative diagram here.
B
C -k >D
Together with the identity mappings on each element, the four elements and
the indicated functions form a category.
2. The collection of all sets with respect to a given universe together with
the collection of all sets of mappings between each ordered pair of sets is a
category denoted by $f.
3. The collection of all groups together with the collection of all sets of homo-
morphisms between each ordered pair of groups A and B, Hom(,4, B) is a
category denoted by Grp.
4. Restrict the objects in Grp to abelian groups. This category is denoted
by Ab.
5. The set of all subgroups of a given group and the collection of the sets of
homomorphisms between each ordered pair of subgroups form a category.
6. The collection of all rings together with the collection of the sets of ring
homomorphisms between each ordered pair of rings form a category denoted
by Rng.
7. In 6, replace rings by K-modules with respect to a ring R and replace ring
homomorphism by i^-homomorphisms. This category is denoted by ModR.

A.I Categories and Functors 125
8. For an element A of a category <g, denote by %>A the category having as its
objects the elements of Mor(A, B), for each BeOb(%) for which Mor(?) ^ 0.
If f:A -> B and g:A -> C, then define a morphism h*:fv-±g in ^ by the
morphism h:B-*C in ^, if A exists, that makes the diagram here com-
commutative;
A
that is, h*:/1->/oh = g.
9. In a manner similar to 8, one defines the category ^A having as its objects
the elements of Mor(?, A), for each B e Ob(^) for which Mor(?, A) ^ 0.
If /: B -> A and g: C -^A, then define a morphism h*:g »-»/ in <&A by the
morphism h:B-*C in #, if /z exists, that makes the diagram here com-
commutative;
A
that is, h*:g\-+hog =/.
Definition A.1.3. A covariant functor F of a category # into a category ^*
is a mapping of <& -* ^* such that each A e Ob(^) has a unique image
^FGOb(^*) and each /e Mor(^, B) has a unique image fF e Mor(iF, BF)
subject to the conditions that
1. for each A €<?, (la)f = iAf and
2. for each/e Mov(A, B) and flf e Mor(?, C), (/ ^f)F =fFogF.
A contravariant functor F of a category # into a category ^* is a map-
mapping of # -> ^* such that each ^ e Ob(^) has a unique image AF e %* and
each fe Mor(^4, B) has a unique image fF e Mor(,BF, ^4F) subject to the
conditions that
1. for each ^4e^, (t^)F = i(Af) and
2. for fe Mor(A, B) and g e MorE, C), (/ o g)F = gF o/F.
Covariant functors are usually referred to as functors. The contra-
variant functor is a "dual" to the functor in the sense that the arrows are
reversed. For example, if ^ and ^* are categories and F is a functor from

126 Some Elements of Category Theory A.I
# to <?*, then the diagrams here are commutative for A, B, CeOb(^) and
for / g Mor(,4, B), g e Mor(?, C).
If F is a contravariant functor, then the right-hand diagram is replaced by the
one shown here.
4F A
CF
Examples A.1.4
1. Let AeOb(%) for a category ^. Consider a mapping F(A):% -*?,&>
the category of sets (Example A. 1.2.2), defined by BF(A) = Moi(A,B) for
each ?eObG9*). For geMov(B,C) set fid(FA)) = f o g for each /e
Mor(A9B)9 that is, ^F(il):MorD, ?) -> Mor(,4, C). Then FD) is a
covariant functor of C -» ^ F(v4) is called the /zom functor and it is often
denoted by hom(A, —).
(Note how this differs from Example A. 1.2.8.)
2. Replace # by Grp in A) and observe how this gives a functor from Grp
into ?f. Note that Mor(^4, B) is never empty in Grp.
3. This is similar to A). Let AeOb(^) and consider a mapping ^ -> ^
defined by ?F*(^ = Mor(B, A) for each 5 e Ob(#). For ^f e Mor(?, C),
set f(aFHA)) = gof for each /eMor(B,4 that is, ^F*(^: Mor(?, A) -+
Mor(C, A). Then F*(A) is a contravariant functor of ^ -> ^. This functor
is called a contravariant hom functor and it is often denoted by hom( — , A).
(Note how this differs from Example A. 1.2.9.)
4. Replace V by Grp in C).
5. Another co variant functor from a category ^ to Sf that will be used in
the next section is described as follows: For fixed A, BeOb(^), associate
the set Mor(,4, C) x Mor(?, C) with each C e Ob(#). Then for,/eMor(.4, C)
and #eMor(?, C), associate to each AeMor(D, C) the morphism (f oh) x
(goh). This functor is denoted by hom(^4, —) x hom(B, —).
6. A contravariant functor from a category # to «9^, that will also be used
in the next section, is described as in E). with the exception that each

A.2
Products and Coproducts 127
heMor(D, C) is associated with (ho /) x (hog).
hom(—,A) x
It is denoted by
This section, albeit too brief, should give some insight to the notions of
a category and a functor. In the next section, a concept associated with the
functor is introduced. It is regarded, by some, as one of the most important
concepts in the area of algebra. However, the proof of that conclusion will
be omitted.
EXERCISES A.1
1. Verify the examples in this section.
2. Discover and identify the categories and the functions found in Chapters 2-9.
A.2 PRODUCTS AND COPRODUCTS
Definition A.2.1. Consider a functor F from a category ^ to the category
of sets Sf, A universal element of F, if it exists, is an ordered couple [#;^4]
for A e Ob(X) and a e AF subject to the condition that for each B e Ob(^)
and for each /? e BF there is precisely one element / e Mor(A, B) such that
a('F) = p.
Let Fbea contravariant functor from a category %> to the category of
sets Sf. A universal element of F, if it exists, is an ordered couple [ot;A~] for
A eOb(^) and oceAF subject to the condition that for each BeOb(^) and
for each fieBF there is precisely one element / e Mov(B, A) such that
a^ = p.
In both cases, the a in the universal element [a;^4] is frequently referred
to as the universal element and A is called the universal object.
The definitions suggest the following heuristic figures:
1. Universal element of a functor.
A *. > AF
B
BF
aeAF

128 Some Elements of Category Theory
A.2
2. Universal element of a contravariant functor.
A JL—y AF ae/
B *
fF
There is no implication that universal elements must exist for each
category. However, when they do exist, they will be unique in the sense
given by the next theorem.
Theorem A.2.2. If [a;A] and [a*; A*] are universal elements for the func-
functor F from the category # to ?f, then there is an isomorphism h e Mor(^4,^4*)
such that a(/jF) = a.
Proof. From the definition there is a morphism / e Mor(^4, ^4*) such that
a(/F) = a* and a morphism g e Mor(,4*, A) such that a*(^F) = a. So
a = a*(/F) = a(fF°9F) = a(foe)Ft However a = oA Hence iA=fog. In
like manner, one obtains iA* = g of Hence / is an isomorphism.
A similar theorem exists for universal elements of contravariant functors
and it has the same conclusion. In both cases the universal objects are unique
up to isomorphism.
If the universal element [a;A] exists for a functor F from a category
# to 6?, then the images of the objects in ^F have a representation with
respect to bijections as indicated next.
Theorem A.2.3. Suppose that [a;A] is a universal element for a functor F
from a category # to Sf. Then, for each J3eOb(X), BF is bijective with
, B).
Proof Since F is a functor, then a(fF) e BF for each/ e Mor(A, B). Moreover
each element in BF is obtained from a by fF for precisely one morphism
/ e Mor(y4, B). Hence there is a bijection between the elements in BF and the
morphisms/ e Mor(A, B).
As a result of this theorem, if A is a universal object for a functor F
from a category # into Sf, then the set ^F is bijective with the set

A.2 Products and Coproducts 129
Once again, a similar result exists for a universal element of a con-
travariant functor. In this case BF is bijective with Mor(B,A) for each
B e Ob(X). Hence the set %F is bijective with the set
{Mor(B,A)\BeOb(%)}.
In the examples of universal elements that are to follow, note that the
first two examples have obvious analogues in both ring theory and module
theory.
Example A.2 A
1. Consider the category Grp, a group G, and a subgroup N <3 G. To each
group H associate the set of all homomorphisms / e Hom(G, H) such that
Nf =lH,lH the identity of H. Denote this set by g(G9 H). If g e Uom(H,K)
for a group K, then g can be associated with a mapping of <f (G, H) -»
#(G, K) by /1->/ o g, V/ e <f (G, #). Hence <f (G, -) defines a functor from
Grp into ?f. A universal element of this functor is the pair [$; G\N~\ where
0 is the epimorphism of G -> GjN. If / e<f (G, H), then there is precisely
one homomorphism # e Hom(G/iV, if) that makes the diagram shown here
commutative.
-> H
G/N
2. For a homomorphism f:G -+ H in Grp, Ker(/) is the set of all x e G
such that j*/ = 1H. Ker(/) = X can be thought of either as a subgroup of
G or as an element in Grp for which there exists an inclusion mapping
(f>:K-^G9 that is, <f> = tG | K. The functor Hom( —, G) is a contra-
variant functor from Grp to Sf. The restriction of Hom(y4, G) to the set of
all g eHom(^4, G) such that A9°f =lH also gives rise to a contravariant
functor from Grp to S? that will be denoted by $*( —, G). A universal ele-
element of ?*(-, G) is [0;X]. The verification of the details is left to the
reader. However, it should be noted that the diagram here is commutative
and that the indicated homomorphism h always exists uniquely.
K

130 Some Elements of Category Theory A.2
3. Consider a group G and a functor from Ab to ?f defined by Hom(G, —).
A universal object of this functor is the quotient group G\\G, G] and the
corresponding universal element is the epimorphism G -» Gj{G,G~\.
Definition A.2.5. The product of two objects A and B in a category # is the
universal element, if it exists, of hom( —, A) x hom( — ,B).
The coproduct of two objects A and B in a category # is the universal
element, if it exists, of hom(yl, —) x hom(B, —).
It is explicit from the definitions that the existence of these universal
elements must be proven.
The product of two objects may be thought of in terms of ordered triples,
for convenience. For brevity, (/, g) e hom(C, A) x hom(C, B) can be
expressed as [/, g\C\. Then a morphism 0 from [/, g\C~\ to [/', g'\C~\ is
a mapping 0:C -> C, if it exists, such that both of the diagrams shown
here are commutative.
> C
A universal element, if it exists, is the triple [pr(^4), pr(B);P] for which
there is a unique morphism \j/ e Mor(C, P) for each [/, g; C] such that
commutativity is preserved in the diagram here.
Vr(A) A pr(B)
P is the product object, usually expressed as P = A U B, and pr(^4), pr(B)
are the projections from P to A and P to B, respectively.
Example A.2.6. Consider nonempty sets A, B e Ob(^), ?f the category
of sets. Then the cartesian product P = A x B together with the projections
pr(v4): (a,b)\-^a and pr(?): (a,b)\-^b, a,beA x B, form a product of A
and B in ^.
Coproducts can be approached in the same manner. For the triple
[f,g;C], CeOb(?),/eMor(i,C), and g eMor(B, C), a mapping 0 to

A.2 Products and Coproducts 131
[/', g'\C~\ is defined as the mapping </> from C to C, if it exists, such that
both of the diagrams here are commutative.
g' = g°<t>
A universal element, if it exists, would be the triple [inj(^), inj(B);P*].
There would exist a unique morphism \j/ e Mor(P*, C) for each [/, g; C] such
that the diagram shown here is commutative.
P* is the coproduct object, usually expressed as P* = A*B, and
inj(B) are the injections from P* to A and B respectively.
Example A.2.7. Consider nonempty sets A,BeOb(?f) and the set-
theoretical disjoint union P* = A1kjB1 formed by the sets Ax and Bx for
which At is bijective with A and Bx is bijective with B. Then P*, together with
the bijections m){A):A -* Ax and inj(B):B -> Bl9 is the coproduct of A and B
in Sf.
Examination of Section 2.1 and Chapter 9 provides additional examples
of both products and coproducts in Grp. Since in most of these examples
reference is made with regard to arbitrary index sets, one needs to reformu-
reformulate the previous definitions.
Definition A.2.5'. The product of a collection of objects {Gjf, / an index set,
in a category ^ is a universal element, if it exists, of Xjhom( —, Gt). The
product object P is denoted by Yli Q and the product element by {pr(Gt)}j
with pr(Gf) called the projection of P onto Gt.
The coproduct of a collection of objects {G^, I an index set, in a cate-
category # is the universal element, if it exists, of Xj hom(G?, —). The coproduct
object P* is denoted by LL Gt and the coproduct object by {inj (Gf)}j with
inj(Gf) called the injection of Gt into P*.
The last two examples in Sf can obviously be extended to an arbitrary
index set and the reader is encouraged to do this.

132 Some Elements of Category Theory A.2
EXERCISES A.2
1. State and prove the theorem for contravariant funtors analogous to Theorem
A.2.2.
2. State and prove the theorem for contravariant functors analogous to Theorem
A.2.3.
3. Provide the details in the examples.
4. In Section 2.1 and in Chapter 9, identify each use of the product and the co-
product. In each use, identify the categories involved, the functors, and the uni-
universal elements and objects.

BIBLIOGRAPHY AND
INDEX OF SPECIAL SYMBOLS

BIBLIOGRAPHY
This is not meant to be a comprehensive list of textbooks and reference books on
the various aspects of group theory. However, it does provide a source for supple-
supplemental reading. The more recent reference books have excellent bibliographies
which facilitate the identification of the journal articles in which results have first
appeared. The book by H. Wussing is unique in that it is the only one of a histori-
historical nature.
Baumslag, B. and Chandler, B., Group Theory, McGraw-Hill, New York A968).
Boerner, H., Darstellungen von Gruppen, Springer, Berlin A955).
Boerner, H., Representation of Groups with Special Consideration for the Needs of
Modern Physics, Wiley, New York A969).
Burnside, W., The Theory of Groups of Finite Order, Cambridge A911); Dover,
New York A955).
Carmichael, R. D., Groups of Finite Order, Ginn, Boston A937); Dover, New
York A956).
Coxeter, H. and Moser, W., Generators and Relations for Discrete Groups, Springer,
Berlin A957).
Curtis, C. and Reiner, L, Representation Theory of Finite Groups and Associative
Algebras, Interscience, New York A962).
Dickson, L.E., Linear Groups with an Exposition of the Galois Field Theory, Teub-
ner, Leipzig A901); Dover, New York A958).
Dieudonne, J., La Geometrie Des Groupes Classiques, Springer, Berlin A955).
Dieudonne, J., Sur Les Groupes Classiques, Herman, Paris A958).
Dixon, J., Problems in Group Theory, Blaisdell, Waltham, Mass. A967).
Feit, W., Characters of Finite Groups, Benjamin, New York A967).
Fuchs, L., Abelian Groups, Pergamon Press, New York A960).
Gorenstein, D., Finite Groups, Harper and Row, New York A968).
Gruenberg, K., "Cohomological Topics in Group Theory," Lecture Notes in
Mathematics, Vol. 143, Springer-Verlag, Berlin A970).
Hall, M., The Theory of Groups, Macmillan, New York A959).
Hall, M. and Senior, J., The Groups of Order 2n (n < 6), Macmillan, New York
A964).
Huppert, B., Endliche Gruppen I, Springer, Berlin A967).
Kaplansky, I., Infinite Abelian Groups, University of Michigan Press, Ann Arbor
A954).
Kochendorffer, R., Lehrbuch der Gruppentheorie unter besonderer Berucksichti-
gung der endlicher Gruppen, Akademische Verlagsgesellschaft, Leipzig A966).
Kurosh, A., Theory of Groups, trans, K. A. Hirsch, Chelsea, New York A955).
Lang, S., Rapport sur la Cohomologie des Groupes, Benjamin, New York A966).

136 Bibliography
Ledermann, W., Introduction to the Theory of Finite Groups, Oliver and Boyd,
London A957).
Macdonald, I., The Theory of Groups, Clarendon Press, Oxford A968).
Miller, G. A., Blichfeldt, H. F. and Dickson, L. E., Theory and Application of
Finite Groups, Wiley, New York A916); Dover, New York A961).
Neumann, H., Varieties of Groups, Springer, Berlin A967).
Polites, G., An Introduction to the Theory of Groups, International Textbook,
Scranton A968).
Robinson, G., Representation Theory of the Symmetric Groups, University of Toronto
Press, Toronto A961).
Rotman, J., The Theory of Groups: An Introduction, Allyn and Bacon, Boston
A965).
Schenkman, E., Group Theory, Van Nostrand, Princeton A965).
Schmidt, O. U., Abstract Theory of Groups, Moscow A933); Freeman, San Fran-
Francisco A966).
Scorza, G., Gruppi Astratti, Perrella, Rome A942).
Scott, W. R., Group Theory, Prentice-Hall, Englewood Cliffs A964).
Smirnov, V. I., Linear Algebra and Group Theory, McGraw-Hill, New York A961).
Specht, W., Gruppentheorie, Springer, Berlin A956).
Speiser, A., Die Theorie der Gruppen von endlicher Ordnung, Springer, Berlin A937);
Dover, New York A945); Birkhauser, Basel A956).
Suzuki, M., Structure of a Group and the Structure of Its Lattice of Subgroups,
Springer, Berlin A956).
Van der Waerden, B. L., Gruppen von Linear en Transformationen, Chelsea,
New York A948).
Wielandt, H., Finite Permutation Groups, Academic Press, New York A964).
Wussing, H., Die Genesis des Abstrakten Gruppen Begrijfes, VEB, Deutscher Verlag,
Berlin A969).
Zassenhaus, H., The Theory of Groups, Chelsea, New York A958)

INDEX OF SPECIAL SYMBOLS
This list contains some of the symbols used throughout the text. Except for
the first few, they are listed in the order of first appearance. The number
refers to the page on which they are first found.
iff if and only if
V for all
g belongs to
n, u set-theoretical union, intersection
m\n the integer m divides the integer n
A x B cartesian product of two sets A and B, 1
a:A -> B mapping a of a set A into a set B, 1
Aa image of a set A under the mapping a, 1
a:xHj = xa mapping of an element xeA onto an element
yeB under the mapping a:A -» B, 1
iA identity mapping of a set A, 1
pr(v4) projection of A x B onto A, 2
\A\ cardinality of a set A, 2
B\A elements of a set B that are not in A, 2
A a B A is a subset of B and B\A is not empty, 2
A c B A is a subset of ?, 2
0 empty set, 2
(G:v4) index of the subgroup A in a group G, 2
<M> subgroup generated by the set M, 2
^ = g~xAg the conjugate of a set /I by an element g in a
group G, 3
.ArG{A) normalizer of a set ^4 in a group G, 3
^G(A) centralizer of a set A in a group G, 3
Z(G) center of a group G, 3
i<G A is a normal subgroup of a group G, 3
GjA quotient group, 3
G ^ H groups G and H are isomorphic, 3
Hom(G, H) set of all homomorphisms of a group G into a
group H, 3
137

138 Index of Special Symbols
Ker(a)
Tm(a)
Aut(G)
End(G)
Inn(G)
A ® B
Z
Q
Grp
*EL c,
®iGt
YLGi
AUB
(AB)C
y\A
GWtH,GwtH,G\H
VG^H
YL*Gt
A*B
LL* G,
kernel of a homomorphism a, 4
image under a homomorphism a, 4
set of all automorphisms of a group G, 4
set of all endomorphisms of a group G, 4
set of all inner automorphisms of a group G, 4
a~1b~1ab (unless otherwise stated) for elements
a, b in a group G, 4
commutator subgroup of a group G, 5
y4 is subnormal in the group G, 5
direct product, 5
set of integers, 7
set of rationals, 7
collection of sets indexed by the set /, 8
collection of sets with n = 0 (or 1), ...,/?, 8
collection of all groups with the collection of all
homomorphisms between pairs of groups, 8
product object of a collection of groups {Gj/, 8
direct product of a collection of subgroups
in a group, 10
a special subgroup of *ri/ Gh 11
product object of groups A and B, 11
direct product with amalgamated subgroup, 13
the restriction of a mapping to the set A, 13
semidirect product, 28
split product, 28
wreath products, 30
transfer, 95
free product of a collection of subgroups {G^j, 1
free product of subgroups A and 5,115
coproduct of a collection of groups {Gijj, 119
a special subgroup of LL* Gh 119
15

INDEX

INDEX
abelian extension, 87
alternating group, 6
ascending central series, 49, 50
ascending chain condition, 71
automorphism, 4
inner, 4
outer, 4
automorphism group, 4
of cyclic group, 6
of elementary abelian /?-group, 6
Baer, 38
Basis Theorem, 63
bicyclic, 90
Burnside, 63, 98, 105
Burnside Basis Theorem, 63
group, 7, 23
properties of, 23
category, 123
center, 3
central extension, 87
centralizer, 3
characteristic series, 49, 61
characteristic subgroup, 4
class equation, 3
commutator identities, 53, 55
commutator subgroup, 5
complement, 35
normal, 98
P-, 98
composition series, 5
conjugate, 3
conjugate subgroup, 3
coproduct
category, 130, 131
group, 117
coproduct element
category, 131
group, 118
coproduct object
category, 131
group, 118
core, 71
coset decomposition, 1
cyclic extension, 87
defining relations, 114
descending central series, 50
dihedral group, 6, 28
direct factor, 5, 10
direct product, 5, 10
properties of, 11, 12, 16, 55
with amalgamated subgroup, 13
direct sum, 10
direct summands, 10
Dixon, 35-38
double coset, 101, 102
elementary abelian, 6, 46, 48
elementary commutator, 66
endomorphism, 3
epimorphism, 3
equivalent extension systems, 81
equivalent extensions, 24, 25, 80-83, 87
141

142
Index
exact sequence, 23
extension system, 80
extensions, 23, 76
abelian, 87
of abelian groups, 84
central, 87
cyclic, 87
splitting, 26, 83, 88
factor set, 76
factorizable, 22
Fitting subgroup, 67
Frattini subgroup, 56, 67
free abelian group, 121
free factors, 115
free generators, 113
free group, 110, 116
free group object, 111
free product, 115
Frobenius, 106, 107
fully invariant subgroup, 4
functor, 125
contravariant, 125
contra variant hom, 126
covariant, 125
hom, 125
Gaschutz, 38, 39, 40, 64
generalized quaternion group, 91
generating set, 2, 110
minimal, 63
generator, 56, 110
group
alternating, 6
bicyclic, 9
C,., 7, 23
dihedral, 6
elementary abelian, 6, 46, 48
free, 110, 116
free abelian, 121
generalized quaternion, 91
metacyclic, 89, 90, 99
nilpotent, 5, 50, 53, 54, 74
/?-, 5, 26, 51, 52, 56, 64, 84, 90, 91, 92,
93
p-nilpotent, 98, 100, 105-107, 109
p-normal, 103, 105
7T-, 35
periodic, 5
quaternion, 6, 66, 75, 80
simple, 5, 99, 107, 109
square-free order, 42, 100
supersolvable, 69, 108
symmetric, 6
symmetric of degree four, 22, 42, 46,
66
torsion-free, 5
Grim, 102, 103
theorems of, 100
Hall, P., 44, 45, 48, 58
Hall Ti-subgroup, 43
Hall subgroup, 43, 104
properties of, 43
hereditary property, 65
Higman, D. G., 41
holomorph, 28
homomorphism, 3
Huppert, 75
hypercenter, 49, 50, 55
hypercommutator, 49, 50, 55
index relations, 21, 22
injections, 131
invariant series, 5, 49, 70, 72
invariant subgroup, 4
isomorphism, 4
isomorphism theorems, 4
Ito, 55, 106
kernel, 4
kth roots, 47
lower central chain, 49

Index
143
lower central series, 50
length of, 50
mappings, 1
bijective, 1
cartesian product of, 1
injective, 1
projection, 1
surjective, 1
Maschke's Theorem, 47, 48
maximal condition, 71
metacyclic group, 89, 90, 99, 109
minimal generating set, 63
modular identity, 21
monomorphism, 3
morphism, 123
nilpotent, 5, 50, 53, 54, 74
nongenerator, 56
normal complement, 98
normal series, 5, 49
normal subgroup, 3
normalizer, 3
order, 2
element, 2
group, 2
Ore, 39
^-commutator subgroup, 100
^-complement, 98
normal, 98
/?-group, 5, 26, 51, 52, 56, 64, 84, 90,
91, 92, 93
i?-nilpotent, 98, 100, 105-107, 109
p-normal, 103, 105
Ti-number, 35
7r'-number, 35, 43
7c-group, 35
properties of, 35
Ti-subgroup, 56
periodic group, 5, 39
permutable subgroup, 20
permutably decomposable, 22
product
in category, 130, 131
of groups, 8
of subgroups, 21
product element, 9
product object, 9, 130, 131
projections, 8, 130
quaternion group, 6, 66, 75, 80
quotient group, 3
reduced product, 41, 64
Sah, 54
Schreier, 79
Schur, 38
semidirect product, 28
representable as, 28
series
characteristic, 49, 61
composition, 5
descending central, 50
invariant, 5, 49, 70, 72
lower central, 50
normal, 5, 49
upper central, 49
short exact sequence, 23
simple group, 5, 99, 107, 109
sockel, 69
solvable group, 5, 45, 46
splits, 28, 37-41, 66, 98
splitting extension, 26, 83, 88
subdirect product, 16
properties of, 18, 19
representation as, 16

144
Index
subgroups
center, 3
characteristic, 4
commutator, 5
conjugate, 3
fully invariant, 4
Hall, 43, 104
Hall 7T-, 43
invariant, 4
normal, 3
/7-commutator, 100
7T-, 56
permutable, 20
subnormal, 5
Sylow p-, 5, 23, 26, 32, 34, 67
Sylow 7i-, 39
system normalizer, 46
subnormal subgroup, 5
subnormal 7C-subgroup, 56
supersolvable groups, 69, 108
Sylow basis, 45
Sylow /?-subgroup, 5, 23, 26, 32, 34, 67
properties of, 5, 23, 26
of symmetric group, 32, 34
Sylow 7r-subgroup, 39
properties of, 39, 46
Sylow theorems, 5
symmetric group, 6
system normalizer, 46
torsion-free group, 5
transfer, 94
universal element, 127
universal object, 127
upper central chain, 49
upper central series, 49
length of, 49
van der Waerden, 115
von Dyck, 114
words, 113, 115
wreath product, 30
properties of, 30
restricted, 30
standard restricted, 30
standard unrestricted, 30
unrestricted, 30
Zassenhaus, 39, 43
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