Exercises in
Group Theory
E. S. Lyapin, A. Ya. Aizenshtat,
and M. M. Lesokhin
Translated by David E. Zitarelli
Temple University
Philadelphia, Pennsylvania
Plenum Press. New York
Wolters-Noordhoff Publishing. Groningen
1972

The original Russian text, published by Nauka Press in Moscow in 1967,
has been corrected by the authors for this edition. The English translation
is published under an agreement with Mezhdunarodnaya Kniga, the Soviet
book export agency.
Library of Congress Catalog Card Number 78-141243
Plenum Press ISBN 0-306-30505-4
Wolters-Noordhoff Publishing ISBN 90 01561004
@ 1972 Plenum Press, New York
A Division of Plenum Publishing Corporation
227 West 17th Street, New York, N. Y. 10011
and Wolters-Noordho1f Publishing, Groningen
All rights reserved
No part of this publication may be reproduced in any form without
written permission from the publisher
Printed in the United States of America

TRANSLATOR'S PREFACE
The present book is a translation of E. S. Lyapin, A. Ya. Aizenshtat, and
M. M. Lesokhin's Uprazhneniya po teorU grupp. I have departed somewhat
from the original text in the following respects.
1) I have used Roman letters to indicate sets and their elements, and
Greek letters to indicate mappings of sets. The Russian text frequently
adopts the opposite usage.
2) I have changed some of the terminology slightly in or<i:er to conform
with present English usage (e.g., "inverses" instead of "regular conjugates").
3) I have correcteq,  n um ber of misprin ts which appeared in the original
in addition to those corrections supplied by Professor Lesokhin.
4) The bibliography has been adapted for readers of English.
5) An index of all defined terms has been compiled (by Anita Zitarelli).
6) I have included a multiplication table for the symmetric group on
four elements, which is a frequent source of examples anGcounterexHmples
both in this book and in all of group theory.
I would like to take this opportunity to thank the authors for their
permission to publish this translation. Special thanks are extended to
Professor Lesokhin for his errata list and for writing the Foreword to the
English Edition. I am particularly indebted to Leo F. Boron, who read the
entire manuscript and offered many valuable comments. Finally, to my
unerring typists Sandra Rossman and Anita Zitarelli, I am sincerely grateful.
Philadelphia, Pa., 1971
David E. Zitarelli
v

FOREWORD TO THE ENGLISH EDITION
The two years which have passed since this book came off the press have fully
confirmed the author's belief concerning the expediency of actively studying
the fundamentals of modern algebra. By discovering the answers to specially
posed problems and gradually overcoming the increasing difficulties the
beginning student of group theory is led step-by-step to certain fundamental
concepts, where he encounters important results and becomes proficient in
methods of reasoning.
Just as we assumed, the book has proved to be suitable both for in-
dependent study and as a supplementary textbook for classwork and for
seminars.
We base our conclusion both on the authors' use of the book and on
the actual experience of a number of mathematicians whose opinions have
been communicated to us.
It was with great satisfaction that the authors became aware of the
intentions to publish the book in English. This will extend considerably the
domain of its use. We will be interested in determining to what extent the
indicated ideas concerning the teaching of modern algebra will be confirmed
by work being carried out in other countries under different conditions.
The authors have taken this opportunity to correct a number of typo-
graphical errors which occurred in the first (Soviet) edition of this book.
Leningrad, 1971
M. Lesokhin
E. S. Lyapin
A. Aizhenshtat
vii

FOREWORD
The aim of the present book is to promote the study of the basic methods,
results, and points of departure of modern algebra. Group theory is unques-
tionably the most developed of a number of algebraic disciplines which
comprise what is often called general, or modern, algebra (and which
properly speaking should be called general theory of algebraic operations).
Thus it is natural to begin the study of modern algebra with group theory.
Furthermore, one should take into account that at the present time group
theory touches upon nearly all of the other algebraic disciplines.
The points of departure of group theory itself can be learned most
naturally as they arise in connection with the ideas of a general character
which go beyond the bounds of modern group theory. It is this consideration
which determines the scope of the material in this book.
We here consider those parts of group theory which form the basis for
the most general concepts. These sections illustrate the foundations of group
theory and serve as a suitable vehicle for studying other algebraic disciplines.
The important role that algebra plays in all mathematics has been
evident for a long time. Various ideas, concepts, and methods are often
developed in algebra and later spread to other mathematical domains.
Therefore an acquaintance with the rudiments of algebra is necessary for
mathematicians in various special fields.
In this connection it is desirable to introduce this material as soon as
possible in the first courses taken at the undergraduate level. However,
in trying to achieve this, one encounters considerable difficulties. For the
abundance, complexity, and complete generality ("abstractness") of the
concepts impede a mastery of this material by those students just beginning
their study of higher mathematics. The most successful way of surmounting
this difficulty is to illustrate the newly introduced concepts by a large number
of concrete examples which show how such concepts arise in various cases.
In addition it is desirable for the student to work out these examples in-
dependently, rather than merely having them displayed for him by the
instructor or the author of a book. An active role by the student will
ix

x
Foreword
guarantee him a complete and effective understanding of the material. It
is to this end that we have written this book.
A brief introduction to new, basic concepts .is given at the beginning
of each section. Examples are then cited in order to make these concepts
concrete. Next, by means of a sequence of exercises, the reader himself is led
to prove various properties of the given concepts. These properties fall into
three categories: important basic theoretical results, less significant but
useful auxiliary results, and, finally, simply practice exercises.
The authors have found that a beginning student of mathematics who
works through this book (either in its entirety or in part) should be able not
only to learn and remember some results and methods of group theory but
also to master the basic concepts creatively. After this he can continue his
study of group theory and also become acquainted with other directions of
modern general algebra.
It is clear that a student's chances of success with this book will be
enhanced if, while studying it, he can attend classes and obtain guidance from
his instructor or if he consults other texts (a list of appropriate works is cited
in the Bibliography). This is all the more important since this book does not
provide extensive explanations or meanings of introduced concepts and
obtained results, and does not give the history of various questions or the
origins of the concepts encountered. Thus, to become acquainted with these
facets of group theory, and, perhaps, later to extend the study of the material
itself, it will be necessary for the reader to consult specialized monographs or
appropriate lecture notes.
Taking into account that different readers come from quite different
backgrounds, the present book was written so that it could be used without
any further sources. Thus the book is self-contained, which naturally
increases the number of ways in which it can be used. Of course the reader
who is already familiar with some of the material from another source can
simply omit the corresponding exercises recommended in the book.
Answers to all of the problems are given at the end of the book. Short
hints are supplied, sketching solutions to the more difficult ones. The letter
H after a problem number indicates that a hint to the solution is' found at
the rear of the book. The letter T means that the result obtained bears
significant theoretical interest.
Some problems are followed by remarks. The purpose of these is to focus
attention on some feature in the solution or a meaning of the obtained result
which deserves attention but might otherwise go unnoticed.
The material is divided into chapters and sections. The problems are
enumerated separately in each section, preceded by the number of the
chapter and section. For example 2.3.13 denotes the thirteenth problem in
Chapter 2, Section 3.

CONTENTS
Chapter 1
Sets
1. Basic Concepts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1
2. Mappings of Sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 7
3. Binary Relations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 11
4. Multiplication of Binary Relations. . . . . . . . . . . . . . . . . . . . . . .. 16
Chapter 2
Algebraic Operations of a General Type
I. The Concept of an Algebraic Operation. . . . . . . . . . . . . . . . . . . . 21
2. Basic Properties of Operations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3. Multiplication of Subsets of a Multiplicative Set. . . . . . . . . . . . . 31
4. Homomorphisms....................................... 33
5. Semigroups............................................ 38
6. Elementary Concepts of the Theory of Groups. . . . . . . . . . . . . . 42
Chapter 3
Compositions of Transformations
1. General Properties of the Composition of Transformations. . . 51
2. Invertible Transformations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3. Invertible Transformations of Finite Sets. . . . . . . . . . . . . . . . . .. 62
4. Endomorphisms........................................ 65
5. Groups of Isometries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 70
6. Partial Transformations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 74
Chapter 4
Groups and Their Subgroups
I. Decomposition of a Group by a Subgroup. . . . . . . . . . . . . . . . .. 79
2. Conjugate Classes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 83
3. Normal Subgroups and Factor Groups. . . . . . . . . . . . . . . . . . .. 86
4. Subgroups of Finite Groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 90
xi

xii
Contents
j 5. Commutators and the Commutator Subgroup. . . . . . . . . . . . .. 91
. . Solvable Groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 94
. Nilpotent Groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 96
8. Automorphisms of Groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 99
9. Transitive Groups of Transformations. . . . . . . . . . . . . . . . . . . . .102
Chapter 5
Defining Sets of Relations
1. Defining Sets of Relations on Semigroups. . . . . . . . . . . . . . . . . . 107
2. Defining Sets of Relations on Groups. . . . . . . . . . . . . . . . . . . . . . 112
3. Free Groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
4. Groups Defined by Sets of Relations. . . . . . . . . . . . . . . . . . . . . . . 121
5 . Free Prod ucts 0 f Gro u ps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6. The Direct Product of Groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
Chapter 6
Abelian Groups
I. Elementary Properties of Abelian Groups. . . . . . . . . . . . . . . . . . 133
j2. Finite Abelian Groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
. Finitely Generated Abelian Groups. . . . . . . . . . . . . . . . . . . . . . . . 139
4. Infinite Abelian Groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
Chapter 7
Group Representations
1. Representations of a General Type. . . . . . . . . . . . . . . . . . . . . . . . 145
2. Representations of Groups by Transformations. . . . . . . . . . . . . 148
3. Representations of Groups by Matrices. . . . . . . . . . . . . . . . . . . . 152
4. Groups of Homomorphisms of Abelian Groups. . . . . . . . . . . . . 156
5. Characters of Groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
Chapter 8
Topological and Ordered Groups
I. Metric Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
2. Groups of Continuous Transformations of a Metric Space. . . . 166
3 . To po I 0 gi cal Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 70
4. Topological Groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
5. Ordered Groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
Hints
Chapter I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
Chapter 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
Chapter 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

Contents
xiii
Chapter 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
Chapter 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
Chapter 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
Answers
Cha pter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
Cha pter 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
Chapter 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
Chapter 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
Chapter 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
Cha p ter 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
Chapter 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
Appendix. Multiplication Table for S4' . . . . . . . . . . . . . . . . . . . . . . . . . . 231
Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

Chapter 1
SETS
1. Basic Concepts
Let us define a property such that for each object which can be considered
in any mathematical theory, one can determine, at least in principle, whether
or not this object has the given property. We can then consider the collection
I
of all objects having this property as a ne mathematical object, which is
called a set. The objects are called elements of the given set.
Thus in order to describe a set one can either state the property, so that
if a mathematical object has the property it becomes an element of the set,
or list all of its elements. (Properly speaking, the fact that a given set contains
elements can also be considered as such a property. Thus these two approaches
differ only in a minor way.)
The fact that x is an element of a set M is denoted by x E M. In this case
we also say that x belongs to M, or that x is contained in M.
Two sets are equal, i.e., they coincide, if they consist of the same elements.
Therefore the proof that two sets M and N are equal usually consists of two
parts. First one proves that for each x EMit follows that x EN; second that
YEN implies Y E M.
The terms "collection," "totality," and "class" are used in the same
sense as the term "set." We should point out, however, that a distinction
is made in some of the deeper axiomatic theories between the concepts of
"set" and "class." Since this is not necessary within the limits of this book,
we will make no such distinction.
We will use braces to denote that a set M consists of the elements
x, Y, . . . , t, . . . . Thus
M = {x, y, ..., t, ...}
If the elements of a set are denoted by means of indices, for example,
Xa, xp, . . . , xc;, . . . , then we also write
M={xe} or M={xe}eEr
where r is a set of indices, r = {ex, f3, . . . , , . . .}.
1

2
Sets
It will not be necessary in this book to distinguish between an object x
and the set {x} consisting of the one element x (a unit set), although such a
distinction is necessary in some other theories.
A set can consist of a finite number of elements, or it can be infinite.
The so-called empty set is the set which does not have any elements. We will
denote the empty set by 0.
If all elements of a set M are also elements of another set N, then we say
that M is a subset of N (or that M is contained in N). We write MeN or
N :::> M.
In particular, the set N itself and the empty set are always subsets of N.
These subsets are called improper. All other subsets are called proper subsets
of N.
It is clear that two sets M and N are equal, M = N, if and only if the
conditions MeN and N c M are satisfied simultaneously.
The union (or set-theoretic sum) of two sets M and M' is the set consisting
of all elements which belong to at least one of the sets M or M'. The union is
denoted by MUM'.
The intersection of M and M' is the set of all elements which belong
simultaneously to M and M'. The intersection is denoted by M n M'.
If M [) M' = 0, then we say that M and M' are disjoint.
Let there be given a collection of sets:
{ M tit M t ..., Me, ...} = {M e } e E r' r = {a, , ..., ;, ...}
The union of all sets Me;' denoted by
MtI U M U ... U Me U
or by
U Me;
e;er
is the set of all elements which are contained in at least one of the sets Me;'
The intersection, denoted by
M« n M n ...
or by
U Ai-
E r 
is the set of all elements which belong to everyone of the sets Me;'

Basic Concepts
3
If a set M has subsets Ma, Mp,.. . such that their union equals M,
M=Ma U M,,! U...
and no two of these subsets have any element in common (they are mutually
disjoint), then we say that M is the disjoint union of the sets Ma,Mp,....
We also say that this defines a partition of M into classes Ma, Mp,.... If in
addition all of the sets Ma, Mp,.. . are nonempty, and there is more than one
of them, then we say that the partition is proper.
For two sets M and N, we denote by M"N the set of all elements in M
which are not contained in N.
Let M be any set, and let r be a set of indices (i.e., elements which are
considered only as distinctive symbols)
r = {a, , ...}
To each of the indices  E r there corresponds an element Xc; in M. In this
connection one element in M can correspond to distinct indices. We then
say that (x a , x p ,. . .) = (xc;)c;er is a system of elements in M. For example,
an infinite sequence of numbers is a system in which the set of indices consists
of all natural numbers: 1, 2, 3, . . . .
It is possible to consider a pair of elements (x, y) in a set M, taken in a
definite order, as a system of elements in M with set of indices {I, 2} (i.e., x
is the first entry in this pair, corresponding to the index 1, and y the second,
corresponding to the index 2).
Suppose that to each index  in a set of indices r = {lX, p, . . .} there
corresponds a set (where it is assumed that some of the sets M a' M p, . . . , or
even all of them can coincide). The Cartesian product of the sets Ma, Mp,...
is the set of all elements x of the following form:
To each index  E r there is associated an arbitrary element Xc; E Mc;.
Then x is defined by
x= (xE)E E r, x= (x«, x, ...)
( A"« E M«, x E M, ...)
We can say that x is a system of elements of the set U Mc; with set of
c;er
indices r (obviously not just any system, but one in which Xc; E M c; for each
 E r). The Cartesian product of the sets M a' M p, . . . is denoted by
or by
M« X M X ... X M; ... (e E .r)
n X M
Er

4
Sets
When the set of indices r of a Cartesian product is finite and consists
of n elements, we usually regard this set as being the integers 1,2,..., n.
The Cartesian product is then written MIX M 2 X ... x M n' and the
elements are denoted by finite sequences
(XI' XI' ... ,x n ) (where Xt E Mt, XI E MI' ..., X n EMil)
In particular, if M i = M for i = 1, 2, . . . , n, then we have the Cartesian
product
M X M X ... X M
..
y
n times
1.1.1. Determine which of the following numbers are elements in the sets
given below:
0, I, : ' V 2 ,
I
- 2, 1t, 7' i,
1 1
-i, V2 + Y2 i, 2+i
Sets:
M 1 = set of all natural numbers;
M 2 = set of all integers;
M 3 = set of all rational numbers;
M 4 = set of all positive real numbers;
M 5 = set of all complex numbers z with the property Z2 = z:
M 6 = set of all numbers z such that Z2 = z.
1.1.2. Which of the matrices
x=(_: ), y=( :), z= ( - )
u= ( ), v=( :)
are elements of the set of matrices whose determinant equals one?
1.1.3. How many elements are contained in the set of all roots of the equation
x 5 -2x 3 +x=O
1.1.4. Determine which of the sets M l' M 2, M 3' M 4 , M 5 of Problem 1.1.1
are subsets of the others.

Basic Concepts
5
1.1.5. Let f(x) and g(x) be two algebraic polynomials. Prove that the set of
roots of the polynomial F(x) = f(x)g(x) is the union of the set of roots of
f(x) and the set of roots of g(x).
1.1.6. Find the intersection of the set of all nonnegative integers and the
set of all nonpositive integers.
1.1.7. For each real number lX let Ma denote the set of all real numbers
greater than lX. Find (1) U Ma and (2), n Nta (the union and intersection taken
a a
over all real numbers lX).
1.1.8. Let M n be the set of all natural numbers which are divisible by the
natural numbet n. Find:
1) U M n (the union taken over all n);
n
2) n M n (the intersection taken over all n);
n
3) M n n M m ;
4) LJ M p (the union taken over all primes p = 2, 3, 5, 7, . . .).
1.1.9. Prove that the following equalities hold for any sets M, N, and S:
MUM=M, MnM=M
MUN=NUM, MnN=Nn M
MU(NU S)=(M U N)U S
M n(NnS)=(M n N)nS
M n (NUS)=(M n N)U (M n S)
M U (Nn S)=(M UN) n (M US)
1.1.10. F!nd (1) M U 0 and (2) M n 0 for an arbitrary set M.
1.1.11. Let M be any set. We will employ the notation A = M"A for each
subset A c: M. For A, B c: M, find:
1) A; 2) AU A; 3) AnA; 4) ( A U B)
 _ N
5) (A n B); 6) (A n B) U (A n B); 7) A U (A n B)
8) (A n B) U (A n B) U (11 n B); 9) M; 10)(j)
1.1.12. Find the number of subsets of a set with four elements. Find the
number of distinct proper partitions of this set.
1.1.13. Form as many partitions as possible of the set of all integers from the
sets given below:
Mo = {O}
M 1 ={1}
M 2 = set of all positive integers

6
Sets
M 3 = set of all negative integers
M 4 = set of all even integers
M 5 = set of all odd integers
M 6 = set of all primes
M 7 = set of all composite numbers
1.1.14. Suppose that
M = V Ae'
..
M = U B).
A
are two partitions of a set M. Will the collection of all nonempty sets of the
form A n B;. be a partition of M?
1.1.15. For each element x of a set M, let
Nx=M"{x}
Find
n N x
x E Ai
1.1.16. Denote the number of elements of a finite set X by m(X). Prove that
the following equalities hold for any sets M, N, and S:
1) m(M U N)= m(M) +nz (N)- m(M n N)
2) In (M U NUS) = m (M) + m (N) + m (S) - nz (Ai n N) -
- m (M n S) - m (N n S) + m (M n N n S)
State and prove analogous formulas for an arbitrary number of finite sets.
1.1.17. How many systems of elements is it possible to construct from the
set { ::-1,0, I} with set of indices {I, 2, 3,4, 5}?
1.1.18. Write out all elements of the Cartesian product of the three sets:
M 1 = { - 1, I}, M 2 = {a, b, c}, M3 = {a}
1.1.19. How many elements are there in the Cartesian product of three
finite sets consisting of k 1 , k 2 , and k3 elements?
1.1.20. What form must a partition of a finite set M into two nonempty
classes take,
M=M 1 UM g

Mappings of Sets
7
in order that the Cartesian product M 1 X M 2 have a maximum number of
elements?
2. Mappings of Sets
Let A and B be two given sets. A mapping cp of A into B is a rule which
associates with each element a of A a unique element of B, which in this case
is denoted by cp(a) or cpa, and is called the image of the element a under the
mapping cp. If in addition for any two distinct elements a, a' E A their images
cp(a) and cp(a') are also distinct, then cp is called a one-to-one mapping.
If for each bE B there exists an element a E A (perhaps more than one)
such that cp(a) = b, then we say that cp is a mapping of A onto B. The student
should always be on the alert to distinguish between "onto" and "into"
mappings, as the former is a special case of the latter.
For A' c: A, by cp(A') (also cpA') we mean the set of all images cp(x) of
elements x E A'. The set cp(A') is called the image of the subset A' under the
mapping cpo Consequently, cp is a mapping of A onto B if and only if cp(A) = B.
Let B' c: B. The set of all a E A such that cp(a) E B' is called the inverse
image (or complete inverse image) of the set B'.
Let cp be a one-to-one mapping of a set A into a set B. Then we can define
in a natural way a mapping of the set cp(A) onto the set A. We call this mapping
the inverse of cp, and denote it by cp - t. Since cp is one-to-one, for each
Z E cp(A) there exists a unique a E A such that
 (a) = z
Therefore we can define
-1 (z) = a
In the discussion above, it is possible for A and B to have elements in common,
and in particular for A and B to coincide.
An element a E A is called a fixed point of the mapping cp if cp(a) = a.
Let CPt be a mapping of a set A 1 into a set B l , and CP2 a mapping of A 2
into B 2 , where At c: A 2 and B 1 c: B 2 . The mapping CP2 is called an extension
of the mapping CPt if CPt and CP2 coincide on A l , i.e., if CPl(a) = CP2(a) for all
a E Al.
Two sets A and B are said to be equivalent (or have the same cardinality)
if there exists a one-to-one mapping of A onto B (and by the same token, as
follows from 1.2.9, also a one-to-one mapping of B onto A).
We say that the cardinality of a set A is less than the cardinality of a set
B if A and B are not equivalent but A is equivalent to a subset of B.

8
Sets
By the cardinality of a set M we mean the symbol associated with the
class of all sets which are equivalent to M (see 1.2.15). The cardinality is also
called the cardinal number.
If a set consists of n elements, then we say that it has cardinality n.
For a set with an infinite number of elements, we say that its cardinality is
infini tee
A set which is equivalent to the set of all natural numbers is called
countable. Thus a set is countable if its elements can be enumerated by means
of the set of natural numbers.
The following basic theorem is well-known. However, we shall not have
occasion to make use of it .
For any two sets A and B, precisely one of the following three possibilities
must hold :
1) A is equivalent to B,
2) the cardinality of A is less than the cardinality of B,
3) the cardinality of B is less than the cardinality of A.
It is necessary to point out that in some mathematical works the image
of an elemen t a under a mapping cp is denoted by acp or a" (instead of cpa = cp( a),
which we have adopted in this book, and which is used more frequently).
1.2.1. Let A and B be sets consisting of nand m elements, respectively.
Determine the number of distinct mappings of A into B. What conditions
will insure the existence of ' mappings of A onto B, one-to-one mappings of
A into B, and one-to-one mappings of A onto B? How many of these last
mappings are there?
t.2.2.T. Prove that every infinite set has a counta'ble subset.
Remark. This result shows that the cardinality of a countable set is less
than or equal to the cardinality of every other infinite set.
1.2.3.8. Let A be the set of integers, B the set of natural numbers, and C
the set of primes. Do there exist one-to-one mappings cp of C onto A and C
onto B such that
cp (PI)  cp (P'J)
for all PI' P 2 E C, P 1  P 2 ?
1.2.4. How many mappings are there of the set M = {a, b, c, d} into itself
which have fixed points?
1.2.5. Let n be a fixed natural number, and define a mapping cp of the set of
all natural numbers M = {I, 2, 3,. . .} into itself by
{ n-k (k<n)
({I(k)= n+k (kn)

Mappings of Sets
9
Is qJ one-to-one? What is the image qJ(M)?
1.2.6. Let N be the set of all continuous real functions defined on the real line.
Let qJ be the mapping of N into itself which takes the function f(x) onto
(Xi - 1) f(x)
Is qJ one-to-one? Is qJ onto?
1.2.7. Lef S be the set of all real functions defined on the real line, and let
h(x) be a fixed function in S. Let.ph be the mapping of S into itself which takes
f(x) E S onto
h (x) f(x)
What condition must the function hex) satisfy for .ph to be one-to-one?
Remark. Compare this result with the preceding problem.
1.2.8. Let T be a triangle whose sides have length a, b, and c. Is the mapping
which associates with T the triangle T' whose sides have length
a+b
2 '
b+c
2 '
c+a
2
a one-to-one mapping of the set of all triangles into itself? Is it onto? Which
triangles are fixed points of this mapping?
1.2.9. Let qJ be a one-to-one mapping of a set A into a set B. Prove that the
inverse qJ - 1 is a one-to-one mapping of qJ(A) onto A.
1.2.10. Let qJ be a one-to-one mapping of a set A into a set B. Prove that
(cp-t)-t = cp
1.2.11. Let M be a set which is written as a disjoint union of subsets which
have either one or two elements. Let qJ be a mapping of Minto M defined by
qJ(x) = x if x constitutes one of the components of the given partition, and
qJ(x) = y if x is contained in a component consisting of two elements, where
y is the second element in this component. Prove that qJ is a one-to-one
mapping, and that qJ - 1 = qJ.
1.2.12. Let qJ be a one-to-one mapping of a set M into itself such that
qJ - 1 = qJ. Prove that qJ can be obtained by the method described in the
preceding problem.
1.2.13.T. Let A, B, and C be sets such that the cardinality of A is less than or
equal to the cardinality of B, and the cardinality of B is less than or equal to

10
Sets
the cardinality of C. Prove that the cardinality of A is less than or equal to the
cardinali ty of C.
1.2.14. Prove that the cardinality of any finite set is less than the cardinality
of any infinite set.
1.2.15. Let A, B, and C be sets such that A is equivalent to Band B is
equivalent to C. Prove that A is equiva1nt to C.
1.2.16. Prove that every infinite subset of a countable set is countable.
1.2.17. Given a countable collection of finite sets M l' M 2" . . , prove that
U M k is either finite or countable.
k
1.2.18.T.H. Given a finite or countable collection of sets M k (k = 1,2,.. .),
where each M k is either finite or countable, prove that U M k is finite or
k
countable. When is this union finite? When is it countable?
1.2.19.T.H. Prove that the set of all rational numbers is countable.
1.2.20.H. Let M be an infinite set, and let N be a set which is either finite
or countable. Prove that M and M U N have the same cardinality.
1.2.21.H. Prove that the set of all polynomials with rational coefficients is
countable.
1.2.22. Prove that the set of all matrices with rational entries is countable.
1.2.23.T.H. Prove that the set of all real numbers r such that 0  r  1 is
uncountable.
1.2.24.T.H. Prove that the set of all real numbers is uncountable.
Remark. The cardinality of the set of all real numbers is called the
continuum. A set which is equivalent to the set of all real numbers is called
continual, or is said to have continual cardinality.
1.2.25.T.H. A complex number is called algebraic if it is the root of some
polynomial (different from zero) in one variable with integral coefficients.
A number which is not algebraic is called transcendental.
Prove that the set of all algebraic numbers is countable, the set of all
transcendental numbers is uncountable, and the set of all real transcendental
numbers is uncountable.
1.2.26.H. Prove that the following sets of points on a straight line are
equivalent:
1) the set of all points on the line;
2) the set of all points on an arbitrary open interval of the line;
3) the set of all points on an arbitrary closed interval of the line;
4) the set of all points on the line which are not contained in a given
interval (either open or closed).
1.2.27.H. Let A and B be any sets, where B contains more than one element.
Let M be the set of all mappings of A into B. Prove that the cardinality of M
is greater than the cardinality of A (A =1= 0).

Binary Relations
11
1.2.28.8. Let PM be the set of all subsets of a set M. Prove that the cardinality
of PM is greater than the cardinality of M (M =1= 0).
1.2.29. Suppose a mapping ({J2 is an extension of a mapping ({JI. If ({J2 is
one-to-one, then so is ({J I. Prove.
1.2.30. Let ({J1, ({J2, and ({J3 be mappings such that ({J3 is an extension of
({J2, and ({J2 is an extension of ({J1' Prove that ({J3 is an extension of ({JI.
1.2.31. Let N be the set of all natural numbers, N' the set of all even natural
numbers, and ({J a mapping of N' into N. When does there exist a one-to-one
mapping of N' into N which is an extension of ({J?
3. Binary Relations
Let X 1 and X 2 be arbitrary sets. The Cartesian product of X I and X 2'
i.e., the set of all pairs of the form (a, b), where a E X I' be X 2  is called the
universal binary relation on elements of X I and X 2 (the order in which the
sets X I and X 2 are given is essential). A binary relation on elements of X I
and X 2 is any subset of their Cartesian product, p c: X 1 X X 2'
Let p be a binary relation on elements of the sets X I and X 2 . We say
that the elements a E X I and b E X 2 belong to this relation if (a, b) E p.
In addition to this notation, we will also write
a"' (p)
F or certain binary relations we use special symbols in place of "'. If no
confusion can arise as to which binary relation is being considered, then we
shall omit the term in parantheses. Let us point out that the condition
(a, b) E P is also written in the form apb.
Let p be a binary relation on elements of the sets X 1 and X 2' Denote by
pr 1 p the collection of all elements a E X 1 for which there exists b E X 2 such
that a '" b(p). Define pr2P c: X 2 in an analogous manner.
Let p be a binary relation on elements of X 1 and X 2 such that pr IP = X 1
and pr 2 P = X 2 , where a1 '" b(p), a2 '" b(p) implies a l = a2 and a '" b 1 (p),
a '" b 2 (p) implies b l = b 2 . In this case we say that p is a one-to-one corres-
pondence between the sets X 1 and X 2' In order to visualize this we often
write
ae ... be,
a1J .... b l1 , ...
where a '" b(p), a" '" b,,(p),. . .. In view of these properties of p, each
element a in X 1 appears in exactly one of these pairs (a  b) and each
element bPI in X 2 also appears in exactly one pair (a"  bPI).
Given a binary relation p on elements of X I and X 2  we define a binary
relation p* on elements of X 2 and X I by
b '" a(p*) if and only if a '" b(p)
(a E X I' be X 2)

12
Sets
The relation p* is called the conjugate of p (or the inverse of p, in which case
we write p - 1 instead of p*).
We also define another binary relation p on elements of X I and X 2
by a '" b(p) if and only if a '" b(p) does not hold, where a E Xl' b E X 2 .
Since binary relations are subsets of the universal binary relation
Xl x X 2 , it is clear what is meant by the expressions PI C P2, PI U P2,
PI n P2'
In what follows we will be interested in binary relations on one set X,
i.e., subsets of the Cartesian product X x X.
The universal binary relation X x X will often be denoted by (JJx, or
simply by (JJ if no confusion can arise. The empty binary relation, i.e., the
binary relation corresponding to the empty subset of X I x X 2' will also be
denoted by 0.
The diagonal of a set X is the binary relation  on X defined by a '" b()
if and only if a = b.
Let P be a binary relation on a set X. Then
P is reflexive if a '" a(p) for all a EX,
P is transitive if a '" b(p) and b '" c(p) imply a '" c(p),
p is symmetric if a '" b(p) implies b '" a(p),
p is antisymmetric if a '" b(p) and b '" a(p) cannot both hold for a =1= b.
A binary relation on X which is reflexive, transitive, and symmetric is
called an equivalence relation, or simply an equivalence, on X.
When p is an equivalence relation the notation a = b(p) is often used
in the literature in place of the notation a '" b(p).
Let p be an equivalence on X. If x '" y(p) (and therefore y '" x(p»,
then x and y are said to be equivalent with respect to p, or p-equivalent.
A nonempty subset K of X is called a p-class if any two elements in K are
p-equivalent and no element in K is p-equivalent to any element in X"K.
The set of all p-classes is usually denoted by X / p.
A binary relation on X which is reflexive, transitive, and antisymmetric
is called an ordering relation, or simply an ordering. The terms order, partial
order, and partial ordering are also used.
An ordering p on X is called linear if for all a, b EX, either a '" b(p) or
b '" a(p). A linear ordering of a set is sometimes called a chain.
A linear ordering p is called complete (or well-ordered) if for every non..
empty subset X' c X there exists an element a o E X' such that a o '" a(p)
for all a EX'.
The symbol  (or <) is often used to denote an ordering instead of "',
In addition, a  b has the same meaning as b  a. If a  b(p), then we say
that a "precedes" b with respect to the ordering p.
If a set has an ordering defined on it, then we say that it is an ordered set.
If the ordering is linear or complete, then we call the set linearly ordered or
completely ordered (well ordered).

Binary Relations
13
An element a of an ordered set X is called maximal if a  b cannot hold
for any b E X distinct from a, and minimal if c  a cannot hold for any C E X
distinct from a. An element a is called universally maximal if x  a holds for
all x E X, and universally minimal if a  x for all x EX.
1.3.1. Let X I and X 2 be any sets, and let P and (f be binary relations on
elements of X I and X 2. Prove that the following conditions are equivalent:
1) a '" b(p) implies a '" b((f); 2) P C (f; 3) P U (f = (f;
4) P n (f = P; 5) P n a = 0; 6) a C p.
1.3.2. Let P be a binary relation on elements of the sets X I and X 2' Show
that
pUp = 00,
pnp=(/)
Con versel y, if
p U a = 00,
pn a =(/)
for some binary relation (f on elements of X 1 and X 2' then (f = p. Prove.
1.3.3. Define the binary relations Pk (k = 0, 1,2, 3,. . .) on the set N of all
natural numbers by n '" m(Pk) if n - m = k. Find: (1) pr I (Pk); (2) pr 2(Pk);
(3) (f = U Pk; (4) (f* ; (5) 't = npk; and (6) 't*.
k k
1.3.4. Let P, PI , and P2 be binary relations on elements of the sets X I and X 2'
Prove that: (1) pUp = P; (2) P n P = P; (3) (p*)* = P; (4) (P I U P2)* =
pT U p!; (5) (PI n P2)* = pT n p! ; (6) p* = (p)* ; (7) P = P; (8) P t U P2 =
Pt n P2; and (9) PI n P2 = PI U P2'
1.3.5. Let R be the set of all real numbers, and let r be the Cartesian product
r=RXRX...R
...
..
n times
Let Pij(i,j = 1, 2, . . . , n) be binary relations on r defined by
(ab ag, ..., an) r"J (bb bi' ..., b n ) (Pi})
if a l + a2 + . . . + a i = b l + b 2 + . .. + b j . Find each of the following
binary relations:
1) 'tt = Pit n P21 n · · · n Pnb
2) 'ti = Ptt n P,g n · · · n Pnn'
3) 't3= n Pi)
 it}

14
Sets
1.3.6. What basic properties (reflexivity, transitivity, symmetry, anti-
symmetry) do each of the following binary relations on the set of all natural
numbers satisfy:
1) n '" m(Pl) if nand m are relatively prime;
2) n '" m(P2) if n divides m;
3) n '" m(p 3) if n = m 2 ;
4) n '" m(p 4) if n < m;
5) n '" m(ps) if n  m;
6) n '" m(u k ) if m - n = k (k = 0, 1,2,. . .)?
1.3.7. Let X' c X, and let p be a binary relation on X. Define a binary
relation p' on X' by a '" b(p/) if a '" b(p) (a, be X'). Show that if p has any
of the basic properties (reflexivity, transitivity, symmetry, antisymmetry) on
X, then p' also has that property on X'. If p is an equivalence relation, then
so is p'. Similar assertions can be made if p is an ordered or linearly ordered
relation. Verify.
1.3.8. Suppose that a partition is given on a set M. Define a binary relation
p on M by x '" yep) if x and y lie in the same class of the partition. Prove that
p is an equivalence relation.
1.3.9.T. Let p be an equivalence relation on a set M. Prove that any two
p-classes either coincide or are disjoint, and that the collection of all p-classes
forms a partition of M. The equivalence corresponding to this partition is,
by 1.3.8, the original equivalence p.
Remark. Comparing 1.3.8 and 1.3.9 we see that being given a partition
of a set is the same as being given the equivalence relation corresponding to
this partition, and vice versa. We do not usually distinguish between
equivalence relations and partitions (without empty classes), so that if p
is an equivalence then the corresponding partition is also denoted by p.
1.3.10.T. Let p be a binary relation on a set X. Define a binary relation p'
in the following way: a '" b(p/) if there exist C 1 , C 2 ,. . . , C n E X such that
c 1 = a, c 1 '" C 2 (p), C2 '" C 3 (p),. · · , C n - 1 '" Cn(p), C n = b. Prove that:
1) pcp';
2) p' is transitive;
3) p' is the intersection of the set of all transitive binary relations u on X
which contain p.
Remark. The relation p' is the universally minimal element (under
inclusion) of the set of all transitive binary relations on X .containing p.
We call p' the transitive closure of p.
1.3.11. Let 8 be any fixed positive real number. Define a binary relation p
on the set R of all real numbers by a '" b(p) if 0  b - a < 8. Determine
the transitive closure of p and the transitive closure of p U p* (cf. 1.3.10).

Binary Relations
IS
1.3.12. Consider the binary relations Pt and P2 on the set K of complex
numbers, where Zt '" Z2(Pt) if IZtl  I Z 21, and Zt '" Z2(P2) if arg Zl = arg Z2
for all Z t , Z2 E K. Find: (1) Pt n pT; (2) Pt U pT; (3) Pt n pT n P2 ; and (4)
the transitive closure of (Pt n p!) U P2 (cf. 1.3.10).
1.3.13. Given a binary relation P on a set X, find the binary relation P'
which is the universally minimal element in the ordered set (by inclusion)
of all symmetric binary relations (J on X such that P c: (J (the symmetric
closure of p).
1.3.14. Given a binary relation P on a set X, find the binary relation P'
which is the universally minimal element in the ordered set (by inclusion)
of all reflexive binary relations containing P (the reflexive closure of p).
1.3.15. Determine which of the following binary relations on the set of all
continuous real-valued functions defined on the interval [a, b] are orderings:
1) f(x) '" g(x)(Pt) if f(c)  g(c) for all c E [a, b];
2) f(x) '" g(X)(P2) if f(cI)  g(C2), where f(cI) is the maximum value of
f(x) in [a, b] and g(C2) is the maximum value of g(x) on [a, b];
3) f(x) '" g(X)(P3) if S: [f(x) - g(x)] dx  0;
4) f(x) '" g(x) (P4) if there exist C t , C2 E [a, b] such thatf(ct)  g(C2);
5) f(x) '" g(x)(ps) if f(a)  g(a) and f(b)  g(b).
1.3.16. Which of the following binary relations are equivalences (orderings)
on the set of all infinite sequences of real numbers:
1) (at, a2, . . .) '" (b t , b 2 , . . .)(Pt) if a k  b k for all k = 1, 2, 3, · · · ;
2) (at, a 2 , . . .) '" (b l , b 2 , . . .)(P2) if there exists a number n such that
a k = b k for k = n, n + 1,...;
3) (at, a2, . . .) '" (b t , b 2 , . . ,)(P3) if there exists a number n such that
ak  b k for k = n, n + 1,...;
4) (at, a 2 ,...) '" (b 1 , b 2 ,.. .)(P4) if either a k = b k for all k = 1,2,... or
there exists a number n such that an < b n and a k  b k for k = n + 1,
n + 2,. . . ?
1.3.17. Prove that every finite ordered set has both minimal elements and
maximal elements.
1.3. 18.T.H. Prove that for each ordering P on a finite set X there exists a
linear ordering P' for which P c: p'.
Remark. This statement is also valid for infinite sets.
1.3.19. Let (P)er be a collection of orderings on a set X. Prove that the
binary relat'ion
a = n P E
 Er
is also an ordering on X.

16
Sets
1.3.20. Let X be a set with the property that whenever a  b there can be at
most a finite number of elements c for which a  c  b. Find all linear
orderings on X.
1.3.21. Which of the following ordered sets of rational numbers (with the
usual ordering) are well ordered:
1) the set of all integers;
2) the set of all positive integers;
3) the set of all negative integers;
4) the set of all rational numbers;
5) the set of all numbers of the form (i)ft (n = 1, 2, 3, . . .);
6) the set of all numbers of the form (!)ft (n = 1, 2, 3, . . .);
7) the set of all numbers of the form l/n (n = 1, 2, 3, . . .)?
1.3.22. What can be said about a set M which has a linear ordering p such
that both p and p* are well-ordered?
1.3.23. Suppose that there is defined a well-ordering on the classes of a
partition {N c;} c;er of a set M, and in turn a well-ordering on each of these
classes. Define a binary relation p on M by x  y(p) if either (1) x E NIX'
YEN fJ' (X =1= {3, and N IX  N fJ with respect to the ordering of the classes of the
partition, or (2) x, YEN c; and x precedes Y in the ordering of N C;' Prove that
p is a well-ordered relation on M.
4. Multiplication of Binary Relations
Let p and a be binary relations on a set X. Their product is the binary
relation! = pa on X defined by a  b(!) if and only if there exists c in X such
that a  c(p) and c  b(p).
In particular, it is possible to consider the product of a binary relation p
with itself, where we use the notation pp = p2. Since this multiplication is
associative (see 1.4.4) one can write p3, keeping in mind that p3 = (pp)p =
p(pp), etc.
Let X' c X, and let p be a binary relation on X. We define the subset
pX' as the set of all a E X for which there exists bE X' such that a  b(p).
Similarly X' p denotes the set of all a E X for which there exists b E X' such
that b  alp).
If, in particular, X' consists of one element, X' = {a} for some a EX,
then the sets pa and ap are sometimes called the right and left cuts (or sections)
of the binary relation p by the element a.
1.4.1. Define the binary relations p, a, !, A. k (k = 1,2, 3, . . .) on the set N of
natural numbers by
n  m(p) if n divides m;
n  m(a) ifn < m;

Multiplication of Binary Relations
17
n  met) if nand m are relatively prime;
n  m(A k ) if 1m - nl = k, for k = 1,2, 3,. . . .
Find the following products:
p, pa, ai, g, PAk' AkP, OAk' Aka, AkAZ
Remark. Note those cases which show that multiplication of binary
relations is not commutative.
1.4.2. Let P and (1 be binary relations on a set X. Prove that
pc, (po) C pCt p,
prg (po) C pri CJ
1.4.3. For a binary relation P on a set X, prove that
prlP = pX
and
pr2P = Xp
1.4.4. Let PI , P2, and P3 be binary relations on a set X. Prove that
(PI p) P3 = PI (PiPs)
(the associative law of multiplication of binary relations).
1.4.5. Let X' c X, and let PI and P2 be binary relations on X. Prove that
(PIP2)X' = PI(P2 X ') and X'(PIP2) = (X'Pl)P2
1.4.6. Let Z be the set of all complex numbers. Define the binary relations
P, (1, and t on Z by
ZI "" Z2 (p) if I Z. I = I Zg I
Zl "'" Z2 (a) if arg ZI = arg z,
Ztl'"'.J Zg (-t) if ZI = Zi
Let R be the set of all real numbers, C thet of complex numbers z for which
Izl  1, and P = {i}. Find:
pR, Rp, aR, R't, pC, Co, C't, pP, Po, P't
1.4.7. Let PI' P2 and (1 be binary relations on a set X, with PI c P2. Prove
that
apt C ap2,
Pia C Pia

18
Sets
1.4.8. Show that the following hold for any binary relations !, Pa., pp,. . .
on a set X:
1: (Pm UP U · · .) = 1:pm U 1:pU · · ·
(p« U P U · · .) 1: = Pa. 1: U P 1: U · · ·
1: (Pa. n P n · · .) C 1:pm n 1:pn · · ·
(p« n P n · · .) 1: C Pm 1: n P  1: n · · ·
1.4.9. Define the binary relations P1 and P2 on the set {a, b} by
PI = {(a, a), (b, a), (b, b)}, Pi = {(a, b), (b, a)}
Find and compare
W (PI n P, WPt n wPi
Remark. Compare this result with 1.4.8.
1.4.10. Find the following products of binary relations on a set X:
w t , w  , 4 w, 4 '
1.4.11. Define the binary relations Pk (k = 1,2,..., n) on the set of all
sequences of n elements from a given set M by
(ab a ..., an)  (bb b" ..., b n ) (p",> if ak = b k
Find PiPj.
1.4.12. Consider the binary relation P of inclusion on the set PM of all
subsets of a given set M (i.e., P '" Q(p) if P c: Q, where P, Q c: M). Find
i - - * - *
p, pp, pp, pp , PP
1.4.13. Let P be an arbitrary binary relation on a set X. Determine the
products P, P, rop, pro.
1.4.14. Which binary relations P on a set X satisfy the following equalities:
(1) .rop = ro; (2) pro = ro; (3) prop = ro; (4) ropro = ro; (5) pp* = ro?
1.4.15. Define the binary relations (11' (12' and (13 on the set of all real
continuous functions defined on the closed interval E = [a, b] by
f(x) '" g(X)((11) if fee)  gee) for all e E E
f(x) '" g(X)((12) if f(a) = g(a), f(b) = g(b)
f(x) '" g(X)((13) if fee) :1= gee) for all e E E

Multiplication of Binary Relations
19
Find
222
01' Og, 0 3 , O.Og, Og03' 03a
1.4.16.T. Let P be an arbitrary binary relation on a set X. Prove that in
order for P to be transitive it is necessary and sufficient that p2 c p.
1.4.17.T. Let PI and P2 be binary relations on a set X. Prove that
(PIP,)* = P:P
1.4.18.T. Let P be an arbitrary binary relation on a set X. Prove that pp* is
symmetric.
1.4.19.T. Prove that a binary relation P on a set X is an ordering if and only
if it satisfies the two conditions
"
p'=p, p n p*=4
1.4.20. If P I and P2 are reflexive (transitive, symmetric, antisymmetric)
binary relations on a set X, does it necessarily follow that PIP2 is a reflexive
(transitive, symmetric, anjisymmetric) binary relation on X?
1.4.21. Let PI and P2 be equivalence relations on a set X such that PIP2 =
P2Pl' Prove that PIP2 is also an equivalence relation on X.
1-.4.22. Let PI and P2 be linear orderings on a set X. Under what conditions
will PIP2 be a linear ordering on X?
1.4.23. Let PI and P2 be symmetric binary relations on a set X, with PIP2 c
P2PI' Show that PIP2 = P2PI-
1.4.24. Let P be an equivalence and (J a linear ordering on a set X. Which of
the basic properties (reflexivity, transitivity, symmetry, antisymmetry) do
each of the following products satisfy:
939 a
p, p, 0, 0, po, op, opa, pap
1.4.25. Let P be an arbitrary binary relation on a set X. Prove that
pi n p = <])
if and only if a  b(p) and b  c(p) implies a  c(p).

Chapter 2
ALGEBRAIC OPERATIONS
OF A GENERAL TYPE
1. The Concept of an Algebraic Operation
We say that an algebraic operation, or simply an operation, is defined on
a set M if there is a rule which to certain ordered pairs of elements of M
associates another element of M. Thus an operation is a mapping from some
subset of the Cartesian ptoduct M x Minto M.
The term "composition" is also used in place of the term "operation."
Sometimes we use the more precise terminology of "binary operation." This
is necessary since one can consider rules which associate an element of M
with each triple of elements of M (a ternary operation), or even rules which
associate an element with each sequence of n elements of M (an n-ary
operation). A set which is being considered with respect to an operation
defined on it is sometimes called a groupoid, or operative, although the term
groupoid is often used in a narrower sense.
The element z which corresponds to the ordered pair (x, y) (x, y E M)
under a given operation is called the result of this operation performed on the
elements x and y. Various notations are employed to indicate the result of an
operation. For example, having fixed a letter, say f, we can write z = f(x, y).
However, we often use the following forms of notation instead of the functional
one given above. We write the elements on which the operation is performed
side by side, putting some special symbol between them. Some well-known
symbols from elementary mathematics are +, -, x, and +. These will be
used to denote the usual operations of addition, subtraction, multiplication,
and division of numbers, as well as for other operations which bear no
resemblance whatsoever to these usual operations. Other symbols which
we shall have occasion to use are 0, ., and *.
We shall make use of the multiplicative notation and terminology
extensively. When an operation is performed on elements we write these
elements side by side with either a dot between them or without any symbol
separating them whatsoever:
xy = z, x. y = z
21

22
Algebraic Operations of a General Type
In this case the elements x and yare called factors (left and right, respectively),
and z is their product. Moreover, this operation may have no connection
whatsoever with ordinary multiplication.
The additive notation and terminology are sometimes used. The result
of the operation on x and y is then denoted by x + y, and corresponding
terminology is employed: sum, summand, etc.
When the name of a specific operation and its corresponding symbols are
firmly established (for example, addition and multiplication) on a concrete
set (the integers, functions, matrices), they will be used without express
mention. In general the multiplicative terminology is employed, and we too
will usually adhere to this.
The actual description of an operation on a set M can be carried out by
various methods. If the elements of M are of a concrete nature, it is possible
to state a law (in particular, a formula) which distinguishes those pairs for
which the result of the operation is defined. The element so constructed is
then the result of the operation for each such pair.
It is also possible to list directly all results of the operation. This can be
accomplished by means of the so-called Cayley table (also called a multi-
plication table). All elements of the set are written on the left and above the
square table. At the intersection of the row corresponding to the element x
and the column corresponding to the element y, we write the result of the
operation by the ordered pair (x, y), or by a dash if the result of the operation
is not defined.
Although a Cayley table of this form can only be constructed for a finite
set M, indeed for a set with not too many elements, one can theoretically
consider a Cayley table for any set M, either finite or infinite.
If an operation is defined on a set M, then for each subset M' c M the
operation on M induces, in a natural way, an oeration on M'. Namely, for
x, y EM', the result of the operation on M' is defined and equal to Z EM'
if for x, y considered as elements of M, the result is defined and equal to
Z EM'. (It is very important to remember that when considering an operation
on a set we always require the results of the operation to be elements of the set.
We will have no occasion to consider "external operations," where the
results are regarded as not being elements in the given set.) Generally speaking,
we call this new operation the restriction of the original operation to M'..
Let M 1 and M 2 be sets, and let 0 and 0 be operations defined on M 1
1 2
and M 2' respectively. Then the sets M 1 and M 2 are called isomorphic relative
to these operations if it is possible to establish a one-to-one correspondence
between the elements of M 1 and M 2'
X 1 '----.X:z, Y1'----+Y1. ('.\'It YIE M 1 ; X:!t Y:.tE 12)

The Concept of an Algebraic Operation
23
such that Xl 0 YI is defined in M I if and only if X2 0 Y2 is defined in M 2 ,
1 2
and the element Xl 0 YI corresponds to the element X2 0 Y2' i.e.,
I 2
Xl 0 YI XJ 0 YI
1 9
The significance of the concept of an isomorphism lies in the fact that
isomorphic sets with operations are absolutely identical with respect to
these operations. If the elements on the border of the Cayley table of one
of these sets are placed in a certain order, and the elements in the second
set are arranged correspondingly, then the Cayley tables of the two sets
coincide (up to the way in which the elements are denoted). This indicates
that operations on isomorphic sets are essentially identical.
2.1.1. Consider the set of intergers 1, 2, 3,4, 5, 6, 7, 8, 9, 10 relative to the
operation of addition. Determine for how many pairs of integers the operation
is defined.
Remark. Remember that certain results of the operation (e.g., that on
the integers 6 and 7) are regarded as undefined since they (e.g., 13) are not
contained in the set being considered.
2.1.2. Construct the Cayley table for the set {I, 2, 3,4, 5} relative to the
operation of subtraction. Take into account the Remark following problem
2.1.1.
2.1.3. Consider the set consisting of the elements Xl' X2, X3, X4, Xs, X6'
where the operation Xi 0 x j is given by the Cayley table
Xl Xs Xa x. XI) Xe
Xl Xs X 4 XI) Xe
Xs XI XI XI) XI Xl
Xa Xa Xa X 8 Xa
X. XI x- XI X 5 XI XI
D
XI) X 5 XI) XI Xa XI
X 8 Xe X 8 XI XI
Determine for which Xi the results (Xi 0 Xi) 0 Xi and Xi 0 (Xi 0 Xi) are defined.
2.1.4. Is it possible to construct a Cayley table for the set {z l' Z2, Z3, Z4}
such that Zi 0 Zi is defined for all Zi except Zl, where Zl 0 Zl is undefined,
and neither (Zi 0 Zj) 0 Zk nor Zi 0 (Zj 0 Zk) is defined for any triple Zi, Zj' Zk
(i,j, k = 1, 2, 3, 4)? If so, find the general form of such a table.

24
Algebraic Operations of a General Type
2.1.5. Each of the following formulas defines an operation which is defined
for all pairs of elements in the set M of all rational numbers.
Ob - a+b
a - 2
b - a(a+l)+b(b+l)
aO - 2
aob=a' - 2ab +b'J
Consider these operations on the set N of all natural numbers. For
which pairs of elements of N are the results of these operations defined in N?
2.1.6. Prove that every set with an operation is isomorphic to itself relative
to this operation.
2.1.7. Consider the set of all positive rational numbers relative to the opera-
tion of ordinary multiplication. Let A be the subset of all numbers greater
than one and B the subset of all numbers less than one. Prove that A and B
are isomorphic relative to multiplication.
2.1.8. Let M be the set of all natural numbers, M 1 the subset of odd integers
and M 2 the subset of even integers. Are any of the sets M, M l' and M 2
isomorphic relative to the operation of multiplication?
2.1.9. Prove that the set of all positive real numbers and the set of all negative
real numbers are isomorphic relative to ordinary addition.
2.1.10. Consider the set G = {2, 3,4, 5} under addition, and H = {2, 4,5, 10}
under multiplication. Prove that G and H are isomorphic relative to the
stated operations.
2.1.11.". Let G be the set of all positive real numbers under ordinary
multiplication and H the set of all real numbers under ordinary addition.
Prove that G and H are isomorphic relative to these operations.
2.1.12. How many nonisomorphic sets having n elements (n  3) are there
whose operation is given by a Cayley table in the following form? One entry
is a dash (the result of the operation is undefined), while the other n 2 - 1
entries consist of a specific element.
2.1.13. Define an operation on the set M 1 = {a, b, c, d, e} by the Cayley table
a b c d e
a a a a a a
b abcae
c acbbe
d ddddd
e e e e e e

Basic Properties of Operations
2S
Let M 2 be the set consisting of - 1, 0, 1 relative to ordinary multiplication.
Is M 2 isomorphic to any of the subsets of M 1 relative to the operation
defined by the table above? How many are there?
2.1.14. Consider the set M of all matrices of the form
( ;)
where x is any real number, relative to the operation of multiplication of
matrices. Prove that M is isomorphic to the set of all real numbers under the
operation of addition.
2.1.15. Define an operation 0 on the set M = {x 1 , X2 , X 3 , . . .} by
XIOXj=XI
(i, j = 1, 2, ...)
Determine which of the subsets of M (relative to the operation on M) are
isomorphic to M.
2.1.16. Suppose that M 1 , M 2' and M 3 are sets with operations such that
M 1 is isomorphic to M 2 and M 2 is isomorphic to M 3 relative to these opera-
tions. Prove that M 1 is isomorphic to M 3'
Remark. Combining this result with 2.1.6 and the fact that the relation
of isomorphism is symmetric, we can conclude that the relation of isomorphism
is an equivalence relation on the class of all sets which have operations.
2.1.17.T. Define an operation on a set X such that for any a, b, C E X,
1) aOa=a;
2) if a 0 b is defined, then a 0 b E {a, b};
3) if a 0 b is defined, then bOa is also defined and a 0 b = bOa;
4) if a 0 b = a and b 0 C = b, then a 0 C is defined and a 0 c = a.
Define a binary relation on X by a I"'tJ b(p) if a 0 b is defined and a 0 b = a.
Prove that p is an ordering relation.
2.1.18.T. Let p be an ordering relation on a set X. Define an operation on
X by a 0 b = bOa = a if a I"'tJ b(p) (a, b E X). Prove that this operation
satisfies each of the four conditions of the preceding problem.
Remark. Compare this result with 2.1.17.
2. Basic Properties of Operations
The general theory of algebraic operations is broken down into a number
of algebraic disciplines. Each of these disciplines is concerned with sets
having one or more operations, which must satisfy certain conditions. In
addition there are conditions which relate these operations. We shall
restrict ourselves in this book to the study of sets with one operation. (Of

26
Algebraic Operations of a General Type
course, other operations can be defined rather naturally on each of these sets,
but we shall only be interested in one operation at a time.)
The theory of sets with one operation is composed of a number of
particular theories which differ only in the basic properties which they satisfy.
Which properties are regarded as basic and, by the same token, how the
general theory is composed of course depends, to a considerable degree, on
the actual state of science at the given moment. The boundaries of these
disciplines are currently changing.
At the present time we regard the properties stated below as being basic.
These have been chosen because they fall into the separate divisions into
which the general theory of operations has been classified.
Let M be a set with an operation, which we shall denote by 0 (although
we have not used multiplicative notation, we shall usually do so in that which
follows).
(1) The operation is called closed if the result of the operation x 0 y
is defined for all elements x, y EM.
(2) The operation is called commutative if for each pair x, Y E M for
which the result x 0 y is defined, yO x is also defined and satisfies
XOY=YOX
(3) The operation is called associative if for each triple x, y, Z E M for
which the products x 0 y and (x 0 y) 0 z are defined, the products y q z and
x 0 (y 0 z) are also defined, and vice versa, and satisfy
(xQy) Qz=xQ (yQz)
(4) The operation is called left invertible if for each pair u, v E M there
exists an element x E M such that x 0 u is defined, and satisfies
xOU=fJ
(5) The operation is called right invertible if for each pair u, v E M there
exists an element y E M such that u 0 y is defined, and satisfies
UQY=fJ
(6) The operation is called left canceUative if whenever x 0 u and
x 0 v are defined (x, u, v E M) and satisfy
xQU=xQfJ
it follows that u = v.

Basic Properties of Operations
27
(7) The operation is called right canceUative if whenever u 0 x and
v 0 x are defined (x, u, v E M) and satisfy
UOx=VOx
it follows that u = v.
The following concepts are closely related to the properties stated above
for an arbitrary operation.
An element e is called a left identity of M if for each x EM, eO x is
defined and satisfies
eOx=x
An element e is called a right identity of M if for each x E M, x 0 e is defined
and satisfies
xOe=x
An element is called a two-sided identity of M, or simply an identity, if e is
both a left and a right identity of M.
The terms "identity element" and "neutral element" are also used instead
of "identity. "
If a set M has an identity e, then the elements x and x' are called inverses,
and each one is called an inverse of the other, if both x 0 x' and x' 0 x are
defined and satisfy
x 0 x' = x' 0 x = e
In the multiplicative notation the element x', the inverse of x, is usually
denoted by the exponent -1,
x' = X-I
An element z is called a left zero of M if for each x E M, z 0 x is defined
and satisfies
zOx = z
An element z is called a right zero of M if for each x EM, x 0 z is defined
and satisfies
xOz=z
An element is called a two-sided zero, or simply a zero, of M ifit is both
a left and a right zero of M. Other terms used synonymously with "zero"
are "zero element" and "annihilating element. "

28
Algebraic Operations of a General Type
An element x is called an idempotent if x 0 x is defined, and x 0 x = x.
All of the foregoing definitions are simplified if the operation is closed.
The reader should keep these simplifications in mind.
For the remainder of this section, M will denote a set with an operation,
denoted by O.
Recall from Chapter 2.1 that for each M' c M we consider the operation
on M' as the same operation on M, in the sense stated in Chapter 2.1.
2.2.1. Determine which of the basic properties hold, and find all identity
and zero elements, for the set of all real numbers relative to each of the
following operations:
1) addition;
2) multiplication;
3) subtraction;
4) division.
2.2.2. Determine the same questions as in 2.2.1 for each of the following
sets relative to each of the operations above:
1) the set of all natural numbers;
2) the set of all integers;
3) the set of all rational numbers;
4) the set of all positive numbers;
5) the set of all negative numbers.
2.2.3. Let M be the set of all continuous functions defined on the closed
interval [a, b]. Determine which of the basic properties are satisfied when
considering M relative to each of the operations below. Also find all identity
and zero elements of M.
1) addition of functions;
2) multiplication of functions;
3) subtraction of functions;
4) division of functions.
2.2.4. Prove that if a set M has both a left identity and a right identity, then
M has a two-sided identity e, in which case there are no left identities or right
identities other than e.
Remark. In particular, a set M can have at most one identity.
2.2.5. Prove that if a set M has both a left zero and a right zero, then M has
a two-sided zero 0, and there are no other left zeros or right zeros except O.
Remark. Hence it follows that a set M can have at most one zero.
2.2.6. Let N denote the set of all natural numbers. For each of the operations
stated below, determine which of the basic properties hold. Are there any

Basic Properties of Operations
29
identities or zeros (left, right, two-sided)? Which of the elements are indem-
potents?
1) a 0 b = c, where c is the greatest common divisor of a and b;
2) a 0 b = c, where c is the least common multiple of a and b;
3) a 0 b = a b ;
4) a 0 b = (a/b) + (b/a)
2.2.7. Consider the set of all complex square matrices of order n relative to
multiplication of matrices. Determine which of the basic properties are
satisfied. Is there a zero or an identity? Which elements have inverses?
2.2.8. Let p, q, and r be any fixed real numbers. Consider the set of all real
numbers relative to the operation defined by
aob=pa+qb+r
for all integers a and b. For what values p, q, r does this operation satisfy the
basic properties?
2.2.9.8. Consider the set Rx of all binary relations on a set X relative to
multiplication of binary relations. Which of the basic properties hold? Is there
a zero or an identity?
2.2.10. Let R denote the set of all positive real numbers relative to the
operation of finding the geometric mean, i.e., a 0 b = .J(a. b) for all a, bE R.
What basic properties hold? Does R have a zero or an identity? Which
elements of Rare idempotents?
2.2.11. Let M be the set of all points in the plane. Let rand s be real numbers
such that 0 < r < 2n and s > O. Define an operation 0 on M by a 0 b = c,
where c is the point such that the distance from a to c is equal to s times the
distance from a to b and, for a #= b, the angle between the line through a and b
and the line through a and c is equal to r (measured in a counterclockwise
direction). Determine which of the basic properties are satisfied.
2.2.12. Let M be the set of all directed line segments in the plane, i.e., all
pairs of numbers (a, b). Define an operation on M in the following manner.
If (a, b), (c, d) E M then (a, b) 0 (c, d) is defined if the terminal end b of the first
segment coincides with the initial end c of the second, and
(a, b)O(b, d)=(a, d)
Determine the geometrical meaning of this operation. Which of the basic
properties hold?

30
Algebraic Operations of a General Type
2.2.13. Suppose an operation is defined on a set by a Cayley table. How is it
possible to tell directly from the table whether the operation satisfies each
of the following properties:
1) closure;
2) commutativity;
3) left invertibility;
4) right invertibility;
5) left cancellation;
6) right cancellation?
2.2.14. For which of the basic properties of an operation IS Jl l'ossibl to
assert that if this property holds in M, then it also holds for each N c M
relative to this same operation?
2.2.15. Consider the sequence of sets
M.CMiC ... CM"CMn+tC ...
where each M n has an operation 0 (n = 1, 2, 3, . . .). For x, y e M n' define
n
xOy = z(m  n)ifandonlyifxOyisdefinedinMnandxOy = z,zeM n .
m n n
Define an operation 0 on the set M = U M n by x 0 y = z (x, y, z e M) if
n
there exists an n such that x, y, z t:: M n and x 0 y = z.
n
Prove that such a rule d()J, in fact, define an operation on M.
Prove that if all the operations 0 (n = 1, 2, 3, . . .) satisfy one of the basic
n
properties, then 0 also satisfies this property on M.
2.2.16.T. Let Px denote the collection of all subsets of a nonempty set X
(in particular, the empty set is included in P x), Consider P x relative to the
operation of union and the operation of intersection, and determine which
of the basic properties are ..satisfied for each.
2.2.17. Define an operation 0 on the set of all sequences of n real numbers by
(ab ai' ..., an)o(bb bi' ..., b,,)=(al +bb at +ai+ b l +b i , ....
al+ a i+ ... +a,,+b 1 +b i + ... +b n )
Which of the basic properties does this operation satisfy?
2.2.18.T. Given a system of sets M(, e r), each of which has an operation
denoted by 0, we define an operation on the Cartesian product of these sets
(cf. Chapter 1.1)
M = n X Me
EEE

Multiplication of Subsets of a Multiplicative Set
31
in the following way. The result on (x)er and (y )er in M is defined if
x 0 y is defined in each M. In such a case set
(x e»)] 0 (ve»)] = (x e 0 Y e»)]
Prove that if each of the M  satisfies one of the basic properties, then M
also satisfies this property.
3. Multiplication of Subsets of a ultiplicative Set
If M is a set which has a closed operation defined on it then M, when
considered relative to this operation, is called a multiplicative set. We are,
of course, using multiplicative terminology here.
Let A and B be subsets of a multiplicative set M. The product AB is
defined as the set of all Z E M which can be written in the form z = xy, where
x E A, y E B.
We say that a nonempty subset A c M is closed with respect to multi-
plication (the term "stationary" is also used) if AA c A.
A is called a left ideal of M if MAc A;
A is called a right ideal of M if AM c A;
A is called a two-sided ideal of M if it is both a left ideal and a rigt ideal
of M.
If A is either a left ideal or a right ideal (in particular, a two-sided ideal)
then A is called an ideal.
However, one should keep in mind that in the literature the term ideal
sometimes refers only to two-sided ideals. If the multiplication is commu-
tative, then the concepts of left ideal, right ideal, two-sided ideal, and ideal
are obviously equivalent.
2.3.1. Let R be the multiplicative set of all rational numbers, Z the set of all
integers, P the set of all positive rational numbers, and N the set of all
negative numbers. Find the following products:
RR, RZ, ZR, ZZ, PP, PN, NN
2.3.2. Define an operation on a multiplicative set M by the multiplication
table
Jab c
a a a a
b a a a
ebb b
Find (M M)M and M(M M).

32
Algebraic Operations of a General Type
2.3.3. Let K l' K 2 , K 3 , K ' 1 , and K be subsets of a multiplicative set M,
where K ' 1 c K 1 and K c K 2 . Show that the following relations hold:
1) Kt(KiUK3)=KIK,.UKtKa
2) Kt (K, n K 3 ) C KtK, n KtKa
3) K;K C KtK,.
2.3.4. Let M be a multiplicative set and PM the collection of all subsets of M.
Then PM is also a multiplicative set relative to multiplication of subsets.
Prove that if M satisfies any of the basic properties-associativity, com-
mutativity, left identities, right identities, left zeros, right zeros-then PM
also satisfies this property.
2.3.5. Let M be the multiplicative set of all positive rational numbers and
PM the multiplicative set of nonempty subsets of M (see 2.3.4). Show that
PM does not satisfy any of the following properties: left cancellation, right
cancellation, left invertibility, right invertibility.
Remark. Note that M satisfies all of the stated properties. Compare this
result with the results of 2.3.4.
2.3.6. Prove that every ideal of a multiplica.tive set is closed with respect to
multiplication.
2.3.7. Let N be the multiplicative set of all natural numbers, A the set of all
even natural numbers, B the set of all odd natural numbers, C n the set of all
natural numbers less than or equal to n (n = 1,2, 3,. . .), and Dn the set of all
natural numbers greater than or equal to n.
Determine which of the sets
A, B, Cn' Dn' C n n Dm (n, m = 1, 2, 3, ...)
are closed with respect to multiplication, and which are ideals of N.
2.3.8. Consider the set of all real numbers and the set of all pure imaginary
numbers (i.e., numbers of the form bi, where b is a real number different from
0) as subsets of the multiplicative set of all complex numbers. Determine if.
either of these subsets is closed and if either is an ideal.
2.3.9. Consider the multiplicative set M of all real square matrices of order n.
Is the set of all nonsingular matrices, or the set of all singular matrices, either
closed or an ideal of M?
2.3.10. Prove that every nonempty intersection of closed subsets of a
multiplicative set is closed.
2.3.11. Prove that every nonempty intersection of left ideals is a left ideal.
The same holds for right ideals and two-sided ideals.

Homomorphisms
33
2.3.12.H. Let L be a left ideal and R a right ideal of a multiplicative set.
Prove that L n R is always nonempty.
2.3. 13.H. Prove that the intersection of a finite number of two-sided ideals
is a two-sided ideal.
2.3.14. Prove that the union of any class of left ideals of a multiplicative set
is a left ideal. The-same holds for right ideals and two-sided ideals.
2.3.15. Let Tl and T 2 be two-sided ideals of a multiplicative set M. Prove
that Tl T 2 is closed with respect to multiplication.
2.3.16. Define an operation on the multiplicative set M consisting of the
five elements a, b, c 1 , C2 , d by means of the multiplication table
I a b C 1 C 1 d
a a a a a b
b a a a a a
c. aaaaa
C 2 a a a a a
d b a a a a
Determine which of the sets T 1 = {a, b, c 1 }, T 2 = {a, b, c 2 }, and Tl T 2 are
ideals of M.
2.3.17. Let M be a multiplicative set, zeM. When is the set {z} closed?
When is {z} a left ideal, right ideal, or two-sided ideal of M?
2.3.18.H. How many ideals are there of the multiplicative set of all complex
square matrices of order n?
4. Homomorphisms
A mapping cp of a multiplicative set M 1 into a multiplicative set M 2 is
called.a homomorphism if for any x, y e M 1 the equality
cp (xy) = cp (x) cp (y)
holds in M 2 .
A one-to-one mapping cp which is a homomorphism is called an
isomorphism.
Note that for a homomorphism (and in particular for an isomorphism)
cp(M 1 ) does not have to coincide with M 2 . Indeed, cp(M 1 ) is in general a
proper subset of M 2'
If cp(M 1) = M 2 then the homomorphism cp is sometimes called an
epimorphism. If cp is a homomorphism which is one-to-one, i.e., cp(x) =F cp(y)
if x =F y, then it is sometimes called a monomorphism.

34
Algebraic Operations of a General Type
To every homomorphism cp of a multiplicative set M 1 into a multi-
plicative set M 2 there corresponds a binary relation Plp on M 1 defined by
x  Y(Plp) if and only if cp(x) = cp(y).
It is easy to see that Plp is an equivalence relation. As we already know
(cf. Chapter 1.3), this means that to each homomorphism cp there corresponds
a partition of the set MI' Each class of this partition consists of the set of all
elements of M 1 which are mapped by cp onto a particular element in M 2'
An equivalence relation (f on a multiplicative set M (and also the
corresponding partition) is called left compatible (or left stable or left regular)
if x  Y«(f) (x, Y E M) implies zx  zY((f) for all Z E M. An equivalence (f is
right compatible if x  Y((f) implies xz  YZ((f) for all Z E M. If (f is both
left and right compatible, then it is called a congruence (or is two-sided
compatible).
Let M be a multiplicative set and let t be a partition of M having the
property that for any two t-classes K and K' there exists at-class K" (which
is obviously unique) such that KK'  K". (The product KK' is obtained by
set multiplication; see Chapter 2.3.) It is then possible to introduce an
operation of multiplication on the set of all t-classes, i.e., on M/t, by setting
KK' = K". In such a case M/t becomes a multiplicative set relative to this
operation. It is called the multiplicative jactor-set(or quotient-set) of the multi-
plicative set M by the partition (or the equivalence) t.
It is necessary to keep in mind that the product of t-classes as elements
of the factor set M/t does not coincide with the product as subsets of the
multiplicative set M, since one can evidently have KK' c;: K" and KK' #= K"
in M.
2.4.1. Let C be the multiplicative set of all complex numbers and R the
multiplicative set of all real numbers. Which of the following mappings of C
into R are homomorphisms?
CPt (z) = I z I
cP (z) = I z I + 1
CPs(z)=O
cP" (z) = 2
2.4.2. Let R be the multiplicative set of all rational numbers. Find all
homomorphisms cp of R into R for which cp(R) c { - 2, -1,0, 1, 2}.
2.4.3. Let M be the multiplicative set of all complex matrices of order n > 1,
and C the multiplicative set of all complex numbers. Define the mappings
l{JI, l{J2, and CP3 of Minto C by

Homomorphisms
3S
CPt (a) = det a
CPi (a) = all
3(a)= 1
where det a is the determinant of the matrix a and all is the entry in the first
row and first column of a. Determine which of these mappings are homo-
morphisms.
2.4.4.T. Let cp be an isomorphism of a multiplicative set M 1 onto a multi-
plicative set M 2' Prove that M I and M 2 are isomorphic in the sense of
Chapter 2.1.
2.4.5.T. Prove that if two multiplicative sets are isomorphic in the sense of
Chapter 2.1, then there exists an isomorphism on M I onto M 2 .
Remark. Compare the results of 2.4.4 and 2.4.5.
2.4.6. Prove that if cp is an isomorphism of a multiplicative set M 1 onto a
multiplicative set M 2' then the inverse mapping cp - I (cf Chapter 1.2) is
an isomorphism of M 2 onto MI'
2.4.7. Find all homomorphisms of the multiplicative set of integers of the
form 5 n (n = 1,2, 3,. . .) into itself. Which of these homomorphisms are iso-
morphisms?
2.4.8. Let M 1 and M 2 be multiphcative sets and cp a homomorphism of M 1
onto M 2' Show that if M I has any of the properties-associativity, com-
mutativity, left invertibility, right invertibility, left identities, right identities-
then M 2 will also have the corresponding property.
2.4.9. Let M 1 be the multiplicative set of all positive integers and M 2 the
multiplicative set {O, I}. Prove that the mapping cp of M 1 onto M 2 defined
by
<p(t)=t,
<p(n)=O
(n > 1)
is a homomorphism.
Remark. Note that even though the commutative operation on M 1
is cancellative, this property does not hold for M 2' Compare this result with
2.4.8.
2.4.10. What condition must be placed on a multiplicative set M 0 such that
for every multiplicative set M there exists a homomorphism of Minto M 0 ?
2.4.11. Suppose a is an equivalence relation on a multiplictive set M.
Prove that in order for a to be a congruence on M it is necessary and sufficient
that Xl  Yl(a) andX 2  Y2(a)implyx l x 2  YIY2(a)forallxl,X2'Yl'Y2EM.

36
Algebraic Operations of a General Type
2.4.12. Let M be a multiplicative set whose operation is defined by the table
I a b c
a a b c
b bee
c c c c
Define a binary relation a on M by
a rv b (0), b rv C (0),
a rv a (0), b I"J b (0), C rvC (0)
Prove that if x  y(a) then xz  yz(a) and zx  zy(a) for all x, y, z E M.
Find elements Xl' X 2 , YI, Y2 EM such that Xl  YI(a) and X 2  Y2(a), but
that X I X 2  YlY2(a) does not hold.
Remark. Compare this result with 2.4.11.
2.4.13. Let M be the set of all nonzero complex p-olynomials relative to the
usual multiplication of polynomials, and let C be the multiplicative set of all
complex numbers. Determine which of the following mappings of Minto C
are homomorphisms and describe the partition of M which corresponds to
each homomorphism.
F = aoxn + at Xn - 1 + ... + an_tX + an (ao::j::. 0)
1) Cft (F) = ao
2) Cf2 (F) = lio (where ao is the conjugate of a o )
3) 3(F)=aO+al + ... +an-t +a n
4) 1>,(F)=ao+a n
5) Cfs (F) = I an I
6) 6 (F) = c ( where c is any real number different from zero)
2.4.14.T. Let 't be a partition of a multiplicative set M. Prove that in order
for M/t to be a multiplicative factor-set (i.e., for t to satisfy the property
stated in the introduction) it is necessary and sufficient that 't be a congruence
on M.
2.4.15. Partition the set Q of all rational numbers into three classes: Q +
consists of all positive numbers, Q - all negative numbers, and QO the number
O. Prove that this partition p is a congruence on Q. Construct the multiplica-
tion table for the multiplicative factor-set Q/ p.
2.4.16.T. Suppose T is a two-sided ideal of a multiplicative set M. Denote
by PT the partition of M in which T is one PT-class and all other classes of

Homomorphisms
37
PT consist of single elements which do not belong to 1: Prove that PT is a
congruence on M.
Remark. A partition (and the corresponding congruence) of this type is
usually called ideal. * The corresponding factor set is often denoted by MIT
instead of M I PT' It is called an ideal factor set.
2.4.17. Let M denote the multiplicative set of real numbers of the form
2 ft (n = 0, 1, 2, . . .), and  the subset of M consisting of all numbers 2 ft where
n  k (k = 0, 1, 2, . . .). Prove that each  is an ideal of M. Construct the
multiplication table for the multiplicative factor-sets MIT 1 , MIT 2 , and
MI T 3.
2.4.18.T. Let p be a congruence on a multiplicative set M. Define the
mapping cp of Minto MI P by letting cp(x) be the p-class of MI p which contains
the element x for each x E M. Prove that cp is a homomorphism and that the
equivalence Plp which corresponds to this homomorphism coincides with p:
Pf=P
For which cases is cp an isomorphism?
Remark. The homomorphism cp is called the canonical homomorphism,
or natural homomorphism, of the multiplicative set M corresponding to the
congruence p.
2.4.19.T. Let cp be a homomorphism of a multiplicative set M 1 into a
multiplicative set M 2. Let Plp be the equivalence relation on M 1 correspond-
ing to cp. Prove that Plp is a congruence.
2.4.20.T. Let M 1 and M 2 be multiplicative sets, cp a homomorphism of M 1
onto M 2' and Plp the congruence on M 1 corresponding to cpo Prove that the
multiplicative sets Mil Ptp and M 2 = cp(M 1) are isomorphic.
Remark. The results of 2.4.18, 2.4.19, and 2.4.20 as a whole show that
if we do not distinguish between isomorphic multiplicative sets, then all
houl0morphisms of one multiplicative set M onto another can be viewed as
canonical homomorphisms corresponding to distinct congruences on M
(cf. 2.4.18).
2.4.21. Find all congruences on the multiplicative set whose operation is
given by the table I a bed
a b a b 1J
b bbbb
ebb b b
d bbbb
How many congruences are there?
* Another name for this congruence which is used more often in English is the Rees congruence
of M modulo T. [Trans.]

38
Algebraic Operations of a General Type
2.4.22.8. How many nonisomorphic multiplicative sets can be mapped by a
homomorphism onto the multiplicative set of problem 2.4.21?
5. Semigroups
A set which has a closed and associative operation defined on it is called
a semigroup with respect to this operation. Hence in the usual multiplicative
terminology a semigroup is a multiplicative set with an associative multiplica-
tion.
Let K be a subset of a semigroup S. A finite sequence of elements of K
written as a row of symbols without any marks separating them is called a
word over K :
XIX! ... X"_IX"
(XI' XI' ... , x,._1t X,. E l()
By multiplying pairs of adjacent elements of K we can transform the word
into one element belonging to S (of course it need not belong to K). This
transformation is accomplished in a finite number of steps by reducing the
length of the word (i.e., the number n). The element thus obtained in S is called
the value of the given word in S. In an arbitrary multiplicative set, the value
of a word depends on the order in which the operation is performed on each
of the pairs of elements, since different processes can lead to different results.
Thus a word in the form
XtXsXa
where associativity does not hold, can have two different values, namely,
(X 1 X 2 ) Xa,
Xl (X 2 X S )
In a semigroup every word has a unique value (cf. 2.5.1). Therefore when a
word is written
X t X 2 ... Xn-tXn
without any further information being given, then we mean the value of the
word in S. An element of the form
XX ... xx
"-..-'
n
is denoted briefly by x n .
For K c S, the set of all elements which are values of different words over
K is denoted by [K]s, and often simply by [K]. If K = {x, y, z, . . .} then

Semigroups
39
instead of writing [{x, y, z,.. .}]s, we will simply write [x, y, z,.. .]s. The set
K is called the generating set for [K] with respect to the operation on S.
A particular case occurs when the set generated by K is equal to the semi-
group, [K]s = S. If, in addition, no proper subset K' c K (K' =1= K) is a
generating set for S, i.e., [K']s =1= S, then K is called an irreducible generating
set of S.
A semigroup which has a one-element generating set is called cyclic, or
monogenic.
A nonempty subset of a semigroup S which is closed relative to the
operation on S is called a subsemigroup of S. A subsemigroup is obviously a
semigroup relative to the operation on S. Conversely, every subset of S
which is a semigroup is a subsemigroup of S.
An element of a semigroup S is called reular if there exists an element
XES such that
axa = a
If all elements of a semigroup S are regular, then the semigroup S is said to be
regular.
A nonempty subset K of a semigroup S is called a normal complex,
or a normal subset, if for all k.. k' E K, a, be S, akb E K implies ak'b E K,
ak E K implies ak' E K, and kb E K implies k'b E K.
2.5.1.T.H. Prove that every word in a semigroup S has a unique value (i.e.,
if we multiply pairs of adjacent elements of a given word in any order
whatsoever, the same element is always produced).
2.5.2. Prove that for any element a in a semigroup S
(an)m = a nm , anam=a n + m
for all natural numbers m and n.
2.5.3. For the multiplicative semigroup of all natural numbers, find the
generating set which is contained in every other generating set for this
semigroup (i.e., the universally minimal generating set).
2.5.4. In the multiplicative semigroup of all 2 x 2 square matrices over the
integers, find [x], [x, t], [y, z], and [y, t], where
x=( ), y=( ), z=( ), t=()
2.5.5. Prove that for any nonempty subset K of a semigroup S the set [K]s
is a subsemigroup of S.

40
Algebraic Operations of a General Type
2.5.6. Prove that [K]s is the intersection of all subsemigroups of S which
contain K.
2.5.7. Prove that for any K, K' c A,
[KU [K']s]s=[KU K']s
2.5.8. Show that in an infinite cyclic semigroup S = [x]s, if m =1= n then
x m =1= x n .
2.5.9. Prove that any two infinite cyclic semigroups are isomorphic.
2.5.10. Suppose that in the cyclic semigroup S = [x]s' x r = X S for some
natural numbers rand s (r =1= s). Prove that:
1) S is finite;
2) there exist natural numbers hand d such that Xh+d = Xh, and
S = {x,x 2 ,...,X h + d - 1 }, where all of the elements in S are distinct;
3) the pair of numbers (h, d) defined in part 2 is unique for a given
semlgroup ;
4) the equality xn+m = x n holds if and only if n  hand m is divisible
by d.
Remark. The pair of numbers (h, d) is called the type of the cyclic
semigroup [x]s, as well as the type of the element x. If S = [x]s is infinite,
then we say that the type of S, as well as the type of x, is infinite.
2.5.11. Prove that two finite cyclic semigroups are isomorphic if and only
if they have the same types.
2.5.12. Determine the number of all subsemigroups and ideals of the cyclic
semigroup of type (5, 3).
2.5.13. Find all possible types of those finite cyclic semigroups which have
h + d - 2 generating sets, each consisting of one element.
2.5.14. Prove that if Tl and T 2 are two-sided ideals of a semigroup S, then
Tl T 2 is a two-sided ideal of S.
Remark. Compare this result with 2.3.16.
2.5.15. Define an operation on the set S consisting of the elements
0, a h a, ..., an' bIb b 12' ..., b i}' ..., b nn
by
ala} = b i }
and xy = 0 for all other cases.
1) Prove that S is a semigroup.
2) Find the types of all cyclic subsemigroups of S.
3) Which subsets of S are left, right, and two-sided ideals?

Semigroups
41
2.5.16.8. Prove that for each regular element a of a semigroup S there exists
an element a such that
aiJa = a, llall = II
Remark. The elements a and a which satisfy the above equalities are
called inverses, * or regular conjugates.
2.5.17. Define an operation on the set of all sequences of four integers by
(Pb Pi' Pa, pi) (qb q'1.' qa, q,) = (Ph Pi' qa, q,J
Prove that this operation is associative. Determine which elements in this
semigroup are regular, and find all inverses of each regular element.
2.5. 18.T. Let lp be a homomorphism of a semigroup S into a semigroup T,
and let t be an arbitrary element of T. Denote by Kt the set of all elements
XES such that lpX = t. Prove that if Kt is nonempty then it is a normal
complex of S.
2.5. 19.T.H. Let K be a normal complex of a semigroup S. Prove that there
exists a semigroup T and a homomorphism lp of S onto T such that K
consists of all XES for which lpX = t for some t E T.
2.5.20. Let S be any semigroup, T a semigroup with an idempotent e, lp a
homomorphism of S onto T, and K the set of all XES such that lpX = e.
What kind of subset is K?
2.5.21. Find all normal complexes of a cyclic semigroup of type (3, 2).
2.5.22. Let S be a commutative semigroup containing at least one idempotent.
Prove that the set of all idempotents in S is a subsemigroup of S.
2.5.23.H. Prove that every finite semigroup has an idempotent.
2.5.24. Prove that every finite commutative semigroup has a unique
partition in which each component is a subsemigroup containing exactly one
idempotent. Show that each component of this partition is a normal complex.
2.5.25. Every regular semigroup has an idempotent. A regular semigroup
has exactly one idempotent if and only if it is two-sided invertible. Prove.
2.5.26. If a semigroup S has a left identity and is left invertible, then S is
two-sided invertible. A similar result holds for a right invertible semigroup
having a right identity. Prove these assertions.
2.5.27.T. If an infinite semigroup has a finite or countable generating set,
then it is countable. Prove.
*We are following here the .terminology of Clifford and Preston, The Algebraic Theory of
Semigroups. The terms "generalized inverses," "reciprocal elements," and "relative inverses"
are also used for this concept. [Trans.]

42
Algebraic Operations of a General Type
6. Elementary Concepts of the Theory of Groups
An algebraic group, or simply a group, is a set with an operation defined
on it which satisfies the properties of
1) closure,
2) associativity,
3) two-sided invertibility.
Thus a group is a semigroup which is two-sided invertible.
According to what has previously been said, we will almost always
adopt multiplicative notation and terminology, usually without express
mention. However, some authors use additive notation, especially when
considering commutative groups.
In exercise 2.6.1 it will be shown that every group has an identity, which
we will denote by e G or e (the identity is of times denoted by 1). Moreover, the
identity is unique by 2.2.4. Each element x of a group G has an inverse, which
will be denoted by x-to The inverse element of each x EGis unique by
2.6.1. We will use the notation x n for all elements x in a group just as we did
for semigroups. In addition, in the group case we will write
.{"
(x- 1 )n = x- n
Further, by XO we will always mean the identity of the group.
If a group satisfies the commutativity property it is called commutative
or abelian (the latter term being used more often).
Although in general a group G need not be commutative, it is possible
for G to have an element z which commutes with every element x in G:
xz = zx
The set Z of all such elements z is called the center of the group G. If the center
of G contains more than one element (cf. 2.6.32), then we say that the group G
has nontrivial center.
A nonempty subset H of a group G is called a subgroup of G if H is closed
relative to the operation on G and if the inverse of each element of H also
belongs to H. (This latter condition can be stated by saying that H is closed
relative to the operation of taking inverse elements.) If H =1= G and H =1= e G ,
then H is called a proper subgroup of G.
A subgroup H of a group G is called normal (or a normal divisor of G) if
a- 1 xaEH
for all a E G and x E H. The importance of normal subgroups will become
manifest in Chapter 4.

Elementary Concepts of the Theory of Groups
43
If a subgroup H is finite, then the order of H is defined to be the number
of elements in H. If H is infinite, we say that it has infinite order. Since every
group is obviously a subgroup of itself then, according to the above, the order
of a finite group is defined to be the number of elements in it. If the group is
infinite it is said to have infinite order.
Let K be an arbitrary subset of a group G. The set of all elements in G
which can be represented in the form
X t X 2 ... X n
(n = 1, 2, 3, ...)
where each Xi (i = 1, 2, . . . , n) either belongs to K or is the inverse of some
element in K, is called the set generated bv K in the sense of the theory of
groups. It is denoted by [K] g W e call K the ge ;;e rating set for [K]g. If[K]g = G
then K is called t he generating set of..tbe S!roup G. ILin..a ddition, no proper
subset of K generates G, then K is called an irreducible gener::ltin set. A
group which has a finite generating set is calle dl nitel y f! enerated. We should
point put here that the term "system of generators" is often used in place of
the term "generating set" in the literature on group theory.
One should note that when considering a group as a semigroup (every
group is obviously a semigroup), it is altogether possible to have
[K]g =f:- [K]s
It is immediately clear, however, that
[K]g = [K U K']s
always holds, where K' is the set of all elements of G which are inverses of
elements of K.
Where no confusion can arise as to the difference between group-
theoretic and semigroup-theoretic generators, we will omit the expression
"in the sense of the theory of groups." We will also omit the subscript g, and
write [K] instead of [K]g. Those chapters which are entirely devoted to
groups will be treated similarly.
Let us point out that in the literature the notation {K} is often used
instead of [K]. We have not adopted this notation here, even though it is
customary in group theory, because such braces are used in the general
theory of sets in a way that may lead to confusion.
If 11 is a subgroup of a group G for which there exists an element x E H
such that H  [x]g, then H is called the cyclic subgroup of G generated by the
element x. If H = G then G is called a cyclic group.
Let x be an arbitrary element in a group G. If there exists a positive
integer n such that x n = e G , then the smallest such integer is called the
order of the element x. If no such integer exists we say that X has infinite order.

44
Algebraic Operations of a General Type
A group G is called periodic if the order of every element of G is finite.
The relation between the order of an element and the order of a subgroup
will be established below (2.6.9).
2.6.1.T. Prove that a set G with an operation is a group if and only if the
operation satisfies the following properties:
1) closure;
2) associativity;
3) the existence of a two-sided identity e;
4) the existence for each element x E G of an inverse element (i.e., an
element X-I such that xx- I = X-IX = e).
Moreover, the inverse of each element is unique. Prove.
Remark. This system of four conditions is often taken as the definition
of a group. Many other equivalent definitions exist (see, for example, 2.5.25
and 2.5.26).
2.6.2.T. Let lp be a homomorphism ofa group G onto a multiplicative set M.
Prove that M is a group, lpe G is the identity of the group M = lpG, and for
any x E G the element lp(x - I) is the inverse of lpX [i.e., (lpx) - I = lp(x - I)].
2.6.3. In the multiplicative set M of all n x n complex matrices, determine
which of the subsets consisting of the following matrices are groups:
1) the real matrices in M;
2) the nonsingular matrices in M;
3) the real nonsingular matrices in M with positive entries;
4) the nonsingular diagonal matrices in M;
5) matrices of the form
. . . . . .
au a12 ... at. n-l aln
o a22. .. a2, n-l a2n
o 0 . .. aa, n-l a3n (atl, a2i, ... J ann -::j:. 0)
o 0 . .. 0 a ,ln
6) matrices of the form
au a12 0 ... 0
a21 a22 0 ... 0
o 0 O. .. 0 (al1ai - ali a il =j:. 0)
. . . . .
000 0

Elementary Concepts of the Theory of Groups
4S
7) matrices of the form
o atg al3 ... aln
a2t 0 a23. .. a2n
a31 a3 0 ... a3n
. . . . . . .
anI a n 2 a n 3 ... 0
2.6.4.- In the multiplicative semigroup of all real 3 x 3 matrices, find the
maximal (with respect to inclusion) subset M such that M is a group and
contains the matrix
( 0 0 0 )
010
000
2.6.5.T. Prove that for any integers nand m, and for every element x in a
group,
(x")m = x"m'
2.6.6. In the multiplicative group of all nonsingular complex 2 x 2 matrices,
find the order of the following elements:
( 1  ) ( 1 1 ) ( I 0 ) ( -2+31 -2+21 ) ( 2 1 )
o - 1 J 0 l' 0 -I ' 1 -I 3 - 2i' 1 1
2.6.7.T. Let x be an element of finite order n in a group G. Prove that all of
the elements
)CO = e,
i n-1
X, .x, ..., x
are distinct, and that
[ ] { j X n - 1 }
X g = e, x, x, ...,
For Xk (0  k < n) show that
(Xk)-t = xn-k
Elements in the group [x]g, when written as powers of x, can be multiplied
according to the formula
{ .xk+l (if k + 1< n)
x"x l =
xk+l-n (if k + I;::: n)
where 0  k, I < n (note that in the second case, obviously 0  k + I - n < n).

46
Algebraic Operations of a General Type
2.6.8. Let x be an element of infinite order in some group. Prove that for any
integers n =1= m, we have x n =1= x m .
2.6.9. Let x be an element of a group. Prove that [x]g is a subgroup (and, by
definition, cyclic) and that the order of [x]g is equal to the order of the
elelnent x.
2.6.10. Prove that two cyclic groups are isomorphic if and only if they have
the same order.
2.6.11. Let G be a cyclic group of order 15. Find the number of elements
x E G such that
[X]g= a
2.6.12. In a cyclic group of order 20, find the number of distinct irreducible
generating sets consisting of two elements each.
2.6.13. Suppose that for the three elements x, u, v of a group G,
X=ll'V='VU, uP=e, 'Uq=e
where p and q are relatively prime integers. Prove that for some relatively
prime integers P' and q' we have
II = x P ',
'U = x ql
2.6.14. Let the order of an element x of a group G be pq, where p and q
are relatively prime. Prove that there exist elements u and v in G such that
X=UV=Vll, uP=e, vq=e
2.6.15. Assume that the four elements Ut, VI , U2, V2 of a group satisfy the
equations
lltVt = 'Utili = 112'UCJ = 'U211,
U p - U p - "..q - "..q - e
1- 2-"'1-"'2-
where P and q are relatively prime. Prove that
lit = 1'2,
'Ut = 'Ui
2.6.16.H. Suppose the order of an element x of a group G is equal to
««2 « h d '" P h h
n = Pt 1 P2 ... Pm m , were PI' P2,. . . , Pm are Istlnct prImes. rove t at t ere
exists a unique set of m elements YI, Y2,' . . , Ym such that
X=YIYi ...Ym, YiYj=YiYi(l=l, 2, ... , m)
«1 (X2 (Xm
Y PI - y P2 - - y Pm - e
1 - 2 -...- m -

Elementary Concepts of the Theory of Groups
47
2.6.17. In every group G both G itself and the subset consisting of the
identity element e G are normal subgroups of G. Prove.
2.6. 18.T. Prove that a nonempty subset H of a group G is a group with
respect to the operation defined on G if and only if H is a subgroup of G.
Let e G be the identity of G and eH the identity of H ; let x' be the element
in G which is the inverse of x E H. Prove that e G = eH and that x' E H.
2.6. 19.T. Let H be a subgroup of a group G and x E H. Prove that
xH = H x = H.
2.6.20. Let K be any nonempty subset of a group. Prove that
[[K]g]g= [K],.
and that the equality
[K]g = K
holds if and only if K is a subgroup of the group.
2.6.21.T.H. Let G = [x]g be a finite cyclic group of order n. For a natural
number d which is a divisor of n, denote by H d the set of elements
x d , X 2d , X 3d , . . . , x(n/d)d = x n . Prove that:
1) Hd is a subgroup of G;
2) if d 1 =1= d 2 then H d1 =1= H d2 ;
3) G has no other subgroups except H d , where d runs over the set of all
divisors of n.
2.6.22.T.H. Let G = [x]g be an infinite cyclic group. For each nonnegative
integer m denote by Hm the set of elements of the form x km (k = (), :t 1, + 2, . . .).
Prove that:
1) Hmis a subgroup ofG;
2) if ml =1= m2, then H m1 =1= H m2 ;
3) the H m are the only subgroups of G.
2.6.23.H. Prove that every infinite group has an infinite set of subgroups.
2.6.24.T.H. Find all groups which have exactly (1) one subgroup, (2) two
subgroups, and (3) three subgroups.
2.6.25. In the multiplicative group of all complex numbers different from
zero, find:
1) [ i]g
2) [ - J... + V3 l ]
22,.
3) [ V2 + Y2 l ]
2 2 g

48
Algebraic Operations of a General Type
4) [-  it
5) [2, - 5]g
6) the intersection of each of the subgroups stated above with the
subgroup of all real number differen t from zero.
2.6.26. Suppose that in a group G,
a => HI ::::> Hi
where H 1 is a subgroup of G and H 2 is a subgroup of HI' Prove that H 2 is a
subgroup of G.
2.6.27. Prove that the intersection of any class of subgroups of a group is a
subgroup.
2.6.28. Let K be a nonempty subset of a group G. Prove that [K]g is the
intersection of all subgroups of G which contain K.
2.6.29. Find those groups which have the property that for any subset K,
[KJ, = [K],
2.6.30. Suppose all elements (different from the identity) of a group G have
order 2. Prove that G is abelian.
2.6.31. Let H., H'. , H 2' H 2 be subgroups of a group, where H. c H'. and
H 2 c H 2 . Prove that
(HtH'J) n H; n H=(HI n H)(H; n H'J)
2.6.32. Prove that the center of a group is always nonempty.
2.6.33. For what groups does the center of the group coincide with the group
itself!
2.6.34. Prove that every subgroup of the center of a group G (including the
center itselO is a normal subgroup of G.
2.6.35.T. Prove that a subgroup H of a group G is normal if and only if
xH=Hx
for all x E G.
2.6.36. Let H be an arbitrary subgroup of a group G, and N a normal
subgroup of G. Prove that H N is a subgroup of G, and that H N = N H.

Elementary Concepts of the Theory of Groups
49
2.6.37. Prove that the product of a finite number of normal subgroups is
also a normal subgroup, and that the intersection of any class of normal
subgroups is a normal subgroup.
2.6.38. Let G be the set of all triples of the form (k l , k 2 , 1) or (k l , k 2 , -1),
where the k i , i = 1, 2 are integers. Define an operation on G by the rule
(k J , k i , 1) (lh Ii' e) = (k 1 + 1 2 , k 1 + Ii' e)
(kt, k.,., -l)(lb Ii' e)=(k 1 +l i , ki+lb -e)
where 8 = + 1.
Prove that G is a group. Prove that HI = [(1,0, 1), (0, 1, 1)]g is a normal
subgroup of G and H 2 = [(1,0, 1)]g is a normal subgroup of HI' Is H 2 a
normal subgroup of G?
Remark. Compare this result with 2.6.26.
2.6.39. In the set Q consisting of the elements 1, -1, i,j, k, - i, - j, - k (here
the minus sign plays no other role than to distinguish elements which are
distinct), an operation is given by the multiplication table
1 -1 -i l -j j -k k
1 1 -1 -i I -j j -k k
-1 -1 1 I -I J -j k -k
1 l -I 1 -1 -k k j -J
-i -i 1 -1 1 k -k -j j
j j -j k -k 1 -1 -I l
-j -j j -k k -1 1 i - l
k k -k -j j I -I 1 -1
-k -k k j -j -I i -1 1
Prove that Q is a group. Find all subgroups of Q, and show that each of these
is a normal subgroup of Q.
Remark. The group Q is called the quaternion group. It plays a very
important role in all of algebra.
2.6.40. T. Prove that if an infinite group has a finite or a countable generating
set, then it is countable.

Chapter 3
COMPOSITION OF TRANSFORMATIONS
1. General Properties of the Composition of Transformations
Let X be any set. A mapping of X into itself is called a transformation
of X. Since a transformation is a special case of a mapping of sets, we will
naturally retain the terminology and notation of Chapter 1.2, with one dif-
ference. By convention we will denote transformations by lower-case Greek
letters, and elements of the set by lower-case Roman letters. In particular, if oc
maps x onto y, then y will be called the image of x under oc, and we write
ocx = y or oc(x) = y.
The set of all transformations of a set X will be denoted by Tx.
We introduce an operation of multiplication on the set Tx in the follow-
ing way (this multiplication is also called composition). Let oc, 13, Y E Tx.
Then oc = fJy if ocx = fJ(yx) for all x E X.
If X is any set of real numbers, every function defined on X with values
contained in X is a transformation of the set X, and the product of two such
transformations is the usual composition of functions.
Let us point out one difference in notations. Functions are usually
written in the formf(x), where this notation denotes the image of the element
x under the transformation f When considering functions we will usually
adopt this notation.
Let X be an arbitrary set, oc E Tx. Write all the elements of X on one line
and the image of each directly beneath it:
( X, y, z, . .. )
ocx, ocy, ocz, . . .
The resulting expression is called the permutation corresponding to the trans-
formation oc. Conversely, if under each element of X we write an element
from the set, we obtain a table
( X,
x',
)', Z, . . . )
, ,
y, Z,...
51

52
Composition of Transformations
which obviously is a permutation for the transformation oc of X defined by
, , ,
ocx = x , lXy = y, lXZ = Z , . . .
If the elements in the upper row of a permutation are written in another
order, and the elements in the lower row are changed accordingly, we obtain
a permutation which corresponds to the same transformation oc. Any two
permutations which correspond to oc will bt: considered as equal. We will
identify all of them with oc, and write
( X, y, z, . .. )
oc = lXX, ocy, ocz, . . .
The commas in this notation will, as a rule, be omitted.
Each transformation can be viewed as a permutation, even for infinite
sets. However, it is clear that the actual permutational notation is possible
only in the case of a transformation of a finite set.
The cardinality of the set lXX, i.e., the cardinality of the set of all images
under the transformation oc, is called the rank of lX, denoted by rlX.
The transformation i defined by ix = x for all x E X is called the identity
transformation. It is obvious that i is the identity of Tx.
F or a given set X, every set of transformations of X which is a semigroup
with respect to the multiplication of transformations defined above is called a
semigroup of transformations.
The only operation of transformations which we shall consider in this
section is the multiplication (composition) introduced above.
One should keep in mind that in some books the transformation oc which
is obtained by first applying 13 and then y (which we have denoted by oc = yfJ)
is denoted by fJy. This notation is quite natural when mappings are written
on the right, as opposed to what was stated in the beginning of the introduc-
tion to Chapter 1.2. Indeed, it is natural to write x(fJy) = (xfJ)y in this case.
Throughout this book we shall write mappings and transformations on
the left, and multiplication of transformations will always be carried out as
was shown first.
3.t.t.T. Prove that multiplication of transformations is associative.
Remark. Thus Tx is a semigroup for every set X.
3.t.2. Let X = {I, 2, 3,4, 5, 6, 7, 8}, lX, 13, y E Tx,
( 1 2 3 4 5 6 7 8 ) ( 1 2 3 4 5 6 7 8 8)
!X= 38124567 p= 2123183
( 1 2 3 4 5 6 7 6 8 )
y = 2 8 1 3 845

Properties of the Composition of Transformations
S3
Find ay, ya, fJ 2 a, ay3, afJy.
Remark. Note that ay =F ya, i.e., multiplication of transformations is
not commutative.
3.1.3. Let X = {I, 2,3,4,5,6, 7}, a, 13 E Tx,
( 1 2 3 4 5 6 7 ) ( 1 2 3 4 5 6 7 )
oc= 2313212 p= 2323112
Find the types of a, 13, afJ, and fJa in the semigroup Tx.
3.1.4. Let X be the set of all real numbers. What basic properties does the
operation of composition satisfy for each of the following sets of real functions
defined on X :
1) all polynomials;
2) all polynomials of even degree;
3) all polynomials of odd degree;
4) all polynomials of degree one;
5) all polynomials of degree one whose leading coefficient is one;
6) all power functions (i.e., functions of the form f(x) = x ft for any
natural number n)?
3.1.5. Let X be the set of all real numbers. Find the zeros (left, right, two-
sided) with respect to the operation of composition for each of the following
sets of real functions defined on X :
1) all continuous functions;
2) all continuous functions which are equal to the identity on the
interval [0, 1] (i.e., f(x) = x for all x E [0, 1]);
3) all polynomials of even degree;
4) all polynomials of odd degree.
3.1.6. Let X by any set consisting of at least two elements. Find all left zeros
of the semigroup Tx, and show there are no right zeros.
3.1.7. Let X be the set of all real numbers. Which of the following sets of real
functions defined on X are semigroups? Which are groups?
1) The sets given in 3.1.4;
2) sets given in parts (1) and (2) in 3.1.5;
3) all even functions [f( - x) = f(x)];
4) all odd functions [f( -x) = - f(x)];
5) all bounded functions (for each function f there exists a number
M J > 0 such that If(x)1  M J for all x);
6) all functions which assume the value 0 at x = 1.
3.1.8. Let X' be the set of all points in the plane, a the projection onto a given
line I, and 13 the symmetry with respect to a point lying on I.
1) Prove that the transformations a and 13 commute.
2) Find the types of a and 13 in Tx.

54
Composition of Transformations
3.1.9. Let X be the set of points in the plane,  the symmetry with respect to
a line 11 and 13 the symmetry with respect to a line 1 2 which is parallel to 1 1 ,
Prove that fJ and fJ are parallel translations.
3.1.10. Prove that every parallel translation of the plane is the product of two
symmetries with respect to parallel lines.
3.1.11. Let X be the set of real polynomials, , 13, Ye E Tx (c is any real num-
ber ).
[f(x)] = f2(x)
fJ[f(x)] = f'(x)
Ye[f(x)] = cf(x)
[f(x) E X]
Which of these transformations commute?
3.1.12. Let X = {I, 2,..., n},  E Tx,
 = ( 1 2 n 3 4 ... n - 1 )
333 1 1 ... 1
Find all transformations 13 E Tx which commute with .
3.1.13. Let N be the set of all natural numbers, , 13 E TN'
(n) = n + 1 (n EN),
{ n - 1, n > 1
p(n) = 1 n = 1 (n E N)
,
1) Find all elements in the semigroup [, fJ]s.
2) Find all irreducible generating sets of [, fJ]s.
3.1.14. Let N be the set of all natural numbers,
( 1 2 3 ... n . . . )
 = 2 3 4 ... n + 1 ...
( 1 2 3 ... n . . . )
13=
1 1 2 ... n - 1 ...
Find all inverses of  and 13 in:
1) the semigroup TN;
2) the semigroup [, fJ]s (see 2.5.16).
3.1.15. Prove that Tx is a regular semigroup for any set X.
3.1.16. Let  be a transformation of a set X, and let P« be the binary relation
on X defined by (x, y) E P« if y = x. Prove that:
1) P«PfJ = P«fJ for all , 13 E Tx;
2) P« # PfJ if  # 13;
3) the set of all binary relations P« ( E Tx) is a semigroup with respect to
the operation of multiplication of binary relations, and that this
semigroup is isomorphic to Tx;
4) Pi (i is the identity transformation) is the diagonal.

Properties of the Composition of Transformations
55
Remark. Transformations can thus be regarded as a special case of
binary relations.
3.1.17. Let ex be a one-to-one transformation of a set X onto itself, and let P
be any transformation of X. Prove that
r(exp) = r(pex) = rp
3.1.18. Prove that the rank of the product of two transformations does not
exceed the minimum of the ranks of the factors.
3.1.19. Let X be the set of all infinite sequences of real numbers. Consider
the subset T' of the semigroup Tx consisting of all transformations of the
form
Pm,n(a l , a2, a3'...) = (mal + n, ma2 + n, ma 3 + n,...)
where n is a nonnegative integer and m is any natural number.
1) Prove that T' is a semigroup.
2) Prove that the collection of all transformations of the form PI,O'
nl,l' PPb O , PP2'O" · ., where the Pi are primes, is an irreducible
generating set for T'.
3) Find all idempotents in T'.
3.1.20. Let T' be the semigroup defined in the preceding problem. Which of
the following subsets of T' are subsemigroups, ideals (left, right, two-sided),
or normal subsets:
1) M I = {Pm,n: m fixed, m > I};
2) M 2 = {Pm,n: m = n};
3) M 3 = {Pm,n: n is a multiple of a fixed number I} ;
4) M 4 = {Pm,n: m is a multiple of a fixed number I} ;
5) M 5 = {Pm,n: m and n are multiples of a fixed prime p}?
3.1.21. Let X = {I, 2, 3,4, 5, 6, 7, 8}, P, ex l , ex 2 , ex 3 E Tx,
( 1 2 3 4 5 6 7 8 )
P = 3 138 141 8
( 1 2 3 4 5 6 7 8 )
ex l = 3 8 1 7 5 6 4 2
( 1 2 3 4 5 6 7 8 )
ex 2 = 4 1 3 2 4 5 4 1
( 1 2 3 4 5 6 7 8 )
ex 3 = 3 1 6 3 6 8 4 6
Which of the equations exty = P (i = 1, 2, 3) are solvable? How many solutions
does each solvable equation have?
3.1.22. Let ex, P E Tx. In order for the equation exy = P to be solvable it is
necessary and sufficient for PX c exX. Prove.
3.1.23. For which pairs of transformations ex, P E Tx does the equation
exy = P have a unique solution?

56
Composition of Transformations
3.1.24. Let N be the set of all natural numbers, cx l , CX 2 , cx 3 , 13 E TN'
( 1 2 3 4 ... 2n - 1 2n . . . )
13=
1 1 3 3 ... 2n - 1 2n - 1 ...
( 1 2 3 ... n . . . )
cx I =
1 1 2 ... n - 1 ...
( 1 2 ... n. . . )
CX2 =
2 4 ... 2n ...
_ ( 1 2 3 4 5 ... 2n 2n + 1 .. . )
cx 3 -
1 2 2 4 4 ... 2n 2n . . .
Which of the equations YCX i = f3 are solvable? Which of these have unique
solutions?
3.1.25. For what pairs of transformations cx, f3 E Tx is the equation ycx = 13
solvable? When does a unique solution exist?
Remark. It follows from 3.1.22 and 3.1.25 that composition oftransfor-
mations is neither left nor right invertible.
3.1.26.H. Find the conditions under which
1) a transformation cx is a right divisor of the identity transformation i
in the semigroup Tx;
2) a transformation cx is a left divisor of i in Tx;
3) a transformation cx is idempotent.
3.1.27. Let X be the unit interval [0, 1] and let T' c Tx be the set consisting
of all transformations of the form
{ 0, if x < c
cx x =
c 1, if x  c
{ O, if x  c
f3 x =
c 1, if x > c
where c is any real number in X.
1) Prove that T' is a semigroup.
2) Find all idempotents in T'.
3) Find all left and right ideals of T'.
3.1.28. Let X = {I, 2, 3,4, 5, 6, 7, 8}. Which of the following subsets of Tx
are normal subsets:
1) all transformations whose rank does not exceed three;
2) all transformations of rank 5 ;
3) the set of all transformations X M of rank 6 satisfying the properties:
for any cx, 13 E X M, cxX = fJX = M, and cxx = cxy if and only if fJx = fJy
(x, Y EX, M c X);

Properties of the Composition of Transformations
57
4) the set of permutations
( 1 2 3 4 5 6 1 8 )
38385513'
( 1 2 3 4 5 6 7 8 )
51513385'
5) the set of permutations
( 1 2 3 4 5 6 7 8 ) ( 1 2 3 4 5 6 7 8 )
38137562' 44258311
3.1.29. Let a be any real number, and let A be the set of real functions f(x)
which are differentiable at a and such that f(a) = a. Prove that:
1) A is a semigroup with respect to composition of functions;
2) the mapping 0 defined by O[f(x)] = f'(a) [f(x) E A] is a homomor-
phism of A into the multiplicative semigroup of real numbers.
3.1.30. Let X be any set. Let n be a fixed natural number. Show that the set
J n of all transformations in Tx whose rank is less than n is a two-sided ideal
of the semigroup Tx.
3.1.31.0. Prove that if J is a two-sided ideal of the semigroup Tx and rx E J,
rrx = n (n is a natural number), then J contains every transformation whose
rank does not exceed n.
3.1.32. Let X = {I, 2, 3, 4, 5, 6, 7, 8}. Which of the following subsets of Tx
are groups:
( 1 2 a 4 5 6 7 8 )
8 383 1 158
( 1 2 3 4.5 678 )
1 5 1 588 3 1
1) MJ={G
2 3 4 5 6 7 8 ) ( 1 2 3 4 5 6 7 8 )
6711676' 71677161'
( 1 2 3 4 5 6 7 8 )}
6 7 1 667 1 7
[( 1 2 3 4 5 6 7 8 )]
2) M 2 = 2 2 2 3 5 5 6 6 s
[( 1 2 3 4 5 6 7 8 )]
3) Ma = 2 3 4 1 6 7 8 5 s
{( I 2 3 4 5 6 7 8 ) ( 1 2 3 4 5 6 7 8 )
4) M. = 3 3 3 6 3 6 3 6' 6 6 6 1 6 1 6 1 '
( 1 2 3 4 5 6 7 8 ) ( 1 2 3 4 5 6 7 8 )}
1 1 1 6 1 6 1 6' 666 3 636 3

58
Composition of Transformations
3.1.3-3. Let G be a group of transformations ofa set X. Prove that the follow-
ing conditions hold for all ex, 13 E G :
1) X = f3X;
2) exx = exy implies fJx = fJy.
3.1.34.H. Prove that if a transformation  is contained in a group of trans-
formations of a set X, then
1) exx -# y for all x, y E X, x -# y;
2) for each y E X there exists x E X such that x = y.
3.1.35.H. Show that if a transformation  of a set X satisfies parts 1 and 2
of 3.1.34, then  is contained in some group of transformations of X.
3.1.36. Let X denote the set of all complex numbers, oc, f3, y,  E Tx,
z = z, fJz = Z2, yz = 1 - z, z = zz
(z EX, Z is the conjugate of z). Prove that every irreducible generating set of
the semigroup [, 13, y, ]s has exactly four elements.
3.1.37.H.Let X be a set with n elements. Prove that every generating set for
the semigroup Tx contains a transformation of rank n and a transformation
of rank n - 1.
3.1.38.H. Let X be a set with n elements. Prove that the set of all transforma-
tions of rank n - 1 is a generating set for the ideal J n (see 3.1.30).
3.1.39.H. Let X be a set with n elements. Every generating set for the semi-
group Tx containing two permutations of rank n - 1 is irreducible. Prove.
Remark. It follows from 3.1.37, 3.1.38, and 3.1.39 that if X is a set con-
sisting of n elements, then every irreducible generating set for the semigroup
Tx contains permutations of rank n and a single permutation of rank n - 1.
3.1.40.0. The set M consisting of a transformation of rank n - 1 and all
transformations of rank n is a generating set for the semigroup Tx, where X
is a set with n elements. Prove.
2. Invertible Transformations
Let  be a transformation ofa set X. The transformation -1 of the same
set X is called the inverse of  if  - 1 = i =  - 1, where i is the identity
transformation.
A transformation is called invertible if it has an inverse.
We will show (see 3.2.7 and 3.2.8) that a transformation is invertible
if and only if it is one-to-one and onto. Hence, if an invertible transformation
is written as a permutation, then all elements of X appear in the lower row
of the permutation and are distinct.
The set of all invertible transformations of a set X is a group with respect
to composition of transformations. The identity transformation is the
identity of this group (see 2.2.6). Any subgroup of this group, including the
group itself is called a group of transformations.

Invertible Transformations
S9
In what follows the phrase "group of transformations" will refer only to
groups of transformations considered with respect to composition.
3.2.1. Which of the transformations in 3.1.2 and 3.1. are left or right divisors
of the identity transformation? Which are invertible? Find the inverses of the
invertible transformations.
3.2.2. Let X be the set of all real numbers. Which of the following functions
defined on X are invertible transformations of X :
1) fl (x) = x n (n natural number);
2) f2(x) = 2 2x ;
3)f3(x) = mx + b(b,mEX,m ¥- 0);
4) f4(x) = sin x?
3.2.3. Let  be an invertible transformation of X, and let PN be the binary
relation on X defined in 3.1.16. Find
1) pr1P« and pr 2 P«;
2) the left and right cuts of the relation P« over each element.
3.2.4. Prove that every invertible transformation has a unique inverse.
3.2.5. Let  and p be invertible transformations of a set X. Prove that p is
invertible, and find its inverse.
3.2.6. T .H. Prove that the set of all invertible transformations of an arbitrary
set X is a group of transformations. What is the identity of the group? Find
the inverse element of each transformation  in the group.
3.2.7.T. Show that an invertible transformation  of a set X satisfies the
following two conditions:
1)  is one-to-one, i.e., x ¥- y for all x, y EX, x ¥- y;
2) X = X, i.e., for each y E X there exists x E X such that x = y.
3.2.8. T .H. Prove that every transformation satisfying conditions (1) and (2)
of the preceding problem is invertible.
3.2.9.H. Suppose an invertible transformation  of a set X is given by a
permutation,
 = ( a a c .. . )
a' b' c' ...
Write -l in permutational form.
3.2.10. Which of the two conditions of 3.2.7 are satisfied for the transforma-
tions given in 3.1.11, 3.1.19, and 3.1.36?
3.2.11. Using problems 3.2.7 and 3.2.8, determine which of the following
transformations are invertible, and find the inverse of each:
1) the transformation given in 1.2.5;
2) the transformation given in 1.2.6;
3) the transformation given in 1.2.8;
4) the transformations referred to in the preceding problem.

60
Composition of Transformations
3.2.12. Let k, I, m, n be rational numbers with kn - 1m =f:. O. Let X be the
field of real numbers of the formf(t)/g(t), wheref(x) and g(x) are polynomials
over the field of rationals, and t is an arbitrary but fixed transcendental real
number. Prove that the transformation of X defined by
kt + I
[ f(t) ] f mt + n
ex g( t) = g ( kt + I )
mt + n
is invertible, and find its inverse.
3.2.13. Let G be the set of all transformations in problem 3.2.12 (for all
rational numbers k, I, m, n satisfying kn - 1m =f:. 0).
1) Prove that G is a group of transformations.
2) Determine whether the set of all transformations  with m = 0 and
n = 1 is a group.
3) Describe all transformations in G of order two.
3.2.14. Let X be the set of all real numbers different from zero and one.
Let G be the set consisting of the following transformations of X :
a: 1 x = x,
a: 2 x = l/x,
a:3X = 1 - x,
a:4X = x/(x - 1),
a:sx = (x - 1)/x,
a:6X = 1/(1 - x)
1) Prove that G is a group of transformations, and construct the multi-
plication table for G.
2) What is the identity of G? Which elements in G are equal to their
inverses?
3) Prove that G is isomorphic to the group of all invertible transforma-
tions of a set with three elements.
3.2.15. Let X be the set of all real numbers, and let G be the set of all trans-
formations of X of the form a:a,b(X) = ax + b, where a and b are arbitrary real
numbers, a =f:. O. Prove that G is a group of transformations. Show that G
does not have a finite generating set.
3.2.16. Let G be the group in the preceding problem. Which of the following
subsets of G are subgroups, and which are normal:
1) the subset H of transformations a: 1 ,b ;
2) the subset H' of transformations a:a,a;
3) the subset H" of transformations a:a,O?
3.2.17. Let G be the group of invertible transformations of a set X and G
the collection of binary relations P<x from 3.1.16 for all a: E G. Prove that G
is a group with respect to multiplication of binary relations, and that G is
isomorphic to G.

Invertible Transformations
61
3.2.18. Let X be the set of all points in three-dimensional space. Determine
which of the following transformations of X are invertible:
1) rotation of the space about an axis through a given angle;
2) projection of the space onto a given plane;
3) symmetry of the space with respect to a given plane (reflection with
respect to a given plane);
4) the transformation  such that p(x, y) = 2p(x, y), where p(x, y) is
the distance between the points x and y.
3.2.19. Let X be the set of points in the plane. Which of the following sets of
transformations of X are groups:
1) all parallel translations of the plane;
2) all rotations of the plane about a given point;
3) all rotations of the plane about all points in the plane through a
fixed angle cp ;
4) all rotations of the plane about all points and all parallel translations
of the plane;
5) all axial symmetries of the plane?
3.2.20. Let
f (X., X2' Xa, X4) =
2 I 2 +2 oj 2 +2 .) '» +2 n 2 +2 29 +2 22 +
= X t X 2 XiXa xiX, X2 X a XiX, Xa X ,
+ XIXiXaX, - 5
How many invertible transformations of the set of variables {Xl' X 2 , X 3 , X 4 }
do not change f(Xl' X 2 , X 3 , X 4 )?
3.2.21. Let
f (Xh Xi' X3) = XXi - XIX + XXa - XIX: + X:Xa +
+ XiX: + 3XIX2X3 - 5
Find all invertible transformations of the set of variables {Xl' X2, X 3 }
which do not change f(xi , X2, X3)' Show that these transformations form a
group.
3.2.22. Let F(x I , X2, . . . , x n ) be any polynomial. Prove that the set of all
transformations of the set of variables {x I' X 2 , . . . , Xn} which leave F fixed
forms a group of transformations.
3.2.23. Let
V n (X., Xi' ..., x n ) =
= (Xi - XI) ... (XII - Xt) (xa - Xi) ... (x n - X n _l) =
= n (Xi -XJ)
J<:.i

62
Composition of Transformations
Prove that every invertible transformation of the set of variables {Xl'
X 2 , . . . , xn} either leaves V n fixed or changes its sign only.
3.2.24. Let f,,(x I , X2 , . . . , x n ) be the polynomial in the preceding problem.
Prove that the transformation ex of the set of variables {x I' X 2 , · . . , xn}
defined by (xXi = Xj' (xXj = Xi' (XXk = Xk (k # i,j) changes the sign of fn only.
3.2.25. Prove that the order of the group of all transformations of a set X
with n elements is n! .
3. Invertible Transformations of Finite Sets
In this section we will consider invertible transformations of finite sets.
Since we are concerned with the composition of transformations of a set,
the nature of the elements in the set is of no consequence. So for convenience
we will only consider in this section transformations of sets of natural
numbers.
he group of all invertible transformations of the set X = {I, 2, . . . , n}
is called the symmetric group of degree n, denoted by Sn. Each transformation
in Sn will be called a permutation of degree n or a transformation of degree n.
A permutation of the form
( Xl X2 ... Xk-l Xk YI ... Yn-k )
X2 X 3 ... Xk Xl YI ... Yn-k
is said to be a cycle of length k, and is denoted by (x I , X 2' . . . , Xk)' Of course
this notation can begin with any Xi. In addition, the commas are often
omitted.
Two cycles (Xl' X2, · · . , Xk) and (YI, Y2, · · . , Yk) are called disjoint (or
independent) if the sets {Xl' X2, · · . , Xk} and {YI, Y2, · · · , Yk} are disjoint.
We will prove that each permutation can be represented as a product
of disjoint cycles (see 3.3.5). When a permutation is represented in such a
way, we shall say that it is decomposed into disjoint cycles.
Cycles of length two are called transpositions.
A permutation of degree n is said to be even if it can be decomposed into
a product of an even number of transpositions, and odd if it can be decom-
posed into an odd number of transpositions.
We will prove (see 3.3.14) that every permutation of degree n can be
decomposed into a product of transpositions, and that every such decom-
position of a given permutation is even or everyone is odd (see 3.3.16).
Hence every permutation of degree n is either even or odd.
The set of all even permutations of degree n is a subgroup of Sn (see
3.3.17). This subgroup is called the alternating group of degree n.

Invertible Transformations of Finite Sets
63
3.3.1. Let, p, y, b, 't' E Ss,
 = (123)(4568), P = (34)(52618), y == (134)(2357)(1846)
b = (82143)(12)(15), 't' = (874312)(56)
Find 3, p2, yb't', y 4 b 2 , and tby.
3.3.2. Find the orders of the following elements of S 12 :
 = (1,3,2,5,4,6,7,8,12,10,9,11
p = (2, 1, 5, 8, 4)
3.3.3. Prove that any two disjoint cycles commute.
3.3.4. Decompose the following permutations into products of disjoint
cycles:
p = (
 = ( 1 2 3 4 5 6 7 8 )
231 568 7 4
2 3 4 5 6 7 8 9 10
12 8 11 6 7 5 3 2 4
( 1 2 3 4 5 6 7 8 9 )
y = 965 1 8 7 2 3 4
11 12 )
10 1
3.3.5.T. Prove that every permutation can be decomposed into a product of
disjoint cycles.
3.3.6. Let
 = (X 11 X12." X 1 kt)(X 2I X22'.. X2k2)." (X ll X,2'.. X,k,)
Prove that
(X-I = (X 1kl · · · X,2Xll)' · · (X2k 2 · · · X22X21)(Xlkl · · · X12 X ll)
3.3.7. Find all powers of the cycle  = (x 1 , X 2 , · · . , x n ).
3.3.8. Prove that if a permutation  is decomposed into disjoint cycles of
lengths m l , m 2 , . . . , m k , then the order of  is the least common multiple of
the numbers m l , m 2 , · · · , m k .
3.3.9. Let  be a permutation written as a product of cycles, and let p be any
permutation. Prove that pp-l is obtained by replacing every number in the
cycle decomposition of  by its image under p.
3.3.10. By using the rule stated in the preceding problem compute b-l,
t- 2yt 2 , and p- 5bP5, where , p, y, b, t are the permutations in 3.3.1.
3.3.11. Let  = (123)(456)(789), P = (147)(258)(369), y = (456)(789). Prove
that  commutes with both p and y, and that  can be decomposed into a
product of p and y.

64
Composition of Transformations
3.3.12. Find all elements of the group Sn which commute with the cycle
(Xl' X2,. · ., x n ), where Xl' X2,' . . , X n is a permutation of the numbers
1, 2, . . . , n.
3.3.13. Find all elements in SlO which commute with the permutation
 = (xlx2x3x4xs)(x6x7xsx9xlo)eSlo,whereallofthexi,i = 1,2,..., 10, are
distinct.
3.3.14.H. Prove that every permutation can be decomposed into a product of
transpositions.
3.3.15. Let
 = (123)(456789),
( 1 2 3 4 5 6 7 8 9 )
P = 1 594 7 6 8 2 3
Prove that:
1)  can be decomposed into a product of 9, 11, and 15 different trans-
positions, but cannot be decomposed into a product of 5 transposi-
tions;
2) /3 can be decomposed into a product of 4, 6, 8, and 10 different
transpositions;
3)  can be decomposed into a product of any odd number of trans-
position greater than five;
4) /3 can be decomposed into a product of any even number of trans-
positions greater than two.
3.3.16.H. Prove that any two distinct decompositions of a given permutation
into transpositions have the same parity.
3.3.17.T.H. Prove that the set of all even permutations of degree n is a
subgroup of Sn, and that it has order n !/2.
3.3.18. Prove that the subset
K = {e, (12)(34), (13)(24), (14)(23)}
of the group S4 is a commutative subgroup. Construct the multiplication
table for K.
Remark. The group K is called the Klein group or the four-group.
3.3.19.H. Prove that each of the following sets of permutations generates the
symmetric group Sn:
1) the set of all cycles;
2) the set of all transpositions;
3) the set of transpositions (1 2), (2 3),..., (n - 1 n);
4) the set of transpositions (1 2), (1 3),..., (1 n);
5) the set consisting of the two permutations (XIX2) and (X 1 X2 . . . x n ),
where Xl' X2, . . . , X n is any permutation of the numbers 1, 2, . . . , n.

Endomorphisms
65
3.3.20. Prove that each of the following sets generates the alternating group
of degree n:
1) the set of all 3-cycles;
2) the set of cycles (123), (124),. . ., (12n).
3.3.21.8. Prove that each of the following sets is an irreducible generating
set for S6:
1) Mt = {(12), (34), (56), (23) (45) }
2) M i = { (12), (34), (123) (456) }
3) M3 = {(12), (23), (24) (156) }
3.3.22. Let A be a generating set for the group S4 consisting of two elements.
Prove that neither of these elements can belong to the Klein group (see
3.3.18).
3.3.23.8. Prove that for every  E S s(C( =f:. e), there exists PES 5 such that
Ss = [, P].
3.3.24.8. A permutation is called regular if it can be decomposed into
disjoint cycles of the same length. Prove that every power of a cycle of
degree n is a regular permutation. Prove that the length of each of the disjoint
cycles in this decomposition divides n.
3.3.25. Prove that every regular permutation is a power of some cycle (see
3.3.24).
3.3.26.H. Let  and ' be regular permutations (see 3.3.24) which commute,
are of degree mn (where m and n are relatively prime), and do not have fixed
points. Suppose in addition that  has order m and ' has order n. Prove that
' is a cycle of length mn.
3.3.27. For what integers m do there exist elements in the symmetric group
S4 of order m?
4. Endomorphisms
Let there be given certain relations on a set X. As an example, one can
consider binary relations, in particular, orderings, algebraic operations, etc.
A .transformation of X is called an endomorphism of X with respect to the
given relations if every system of elements connected by one of the relations
is mapped into a system of elements which are also connected by this same
relation.
For example, if an algebraic operation 0 is defined on a set X, then every
transformation  of X which satisfies the condition
if a 0 b = c, then a 0 b = c
(a, b, c E X)

66
Composition of Transformations
is an endomorphism of X with respect to this operation. If an ordering is
defined on X, then every transformation  of X satisfying
if x  y, then x  y
(x, Y E X)
is an endomorphism of X with respect to this ordering. Linear transforma-
tions of a vector space are endomorphisms of the space with respect to the
two operations defined on this space. Let a relation p be defined on three-
dimensional space by requiring three points PI , P 2 , P 3 to be p-related if they
are collinear and P 2 lies between PI and P 3 . Then all affine transformations
(see 3.4.27) are endomorphisms of the space with respect to p.
We say that an endomorphism of a set X preserves the relations defined
on X.
It is easy to verify that the set of all endomorphisms of an arbitrary
set X with respect to relations defined on X forms a semigroup of trans-
formations (see 3.4.7).
An invertible transformation  of a set X which is an endomorphism
with respect to relations defined on X is called an invertible endomorphism
of X with respect to these relations. If a-I is also an endomorphism with
respect to these relations,  is called an automorphism of X.
We will prove (see 3.4.12) that the set of all automorphisms of X with
respect to relations defined on X is  group of transformations.
If it is clear from the context which relations are being considered, we
shall omit the phrase "with respect to relations defined on X."
Let us point out that in group theory the term "endomorphism" is
usually used in a more restricted sense, namely, an endomorphism is a
mapping which preserves the operation of the group. A subgroup H of a
group G is called a completely characteristic subgroup if H is mapped into
itself by every endomorphism of G which preserves the operation of G.
3.4.1. Consider the set R of all real numbers with respect to the usual order-
ing. Which functions define on all of Rare endomorphisms?
3.4.2. Define a binary relation p on the set N of natural numbers by (m, n) E p
if n divides m. Which of the following transformations of N are endo-
morphisms :
1) the transformations  and /3 in 3.1.14;
2) the transformation -r defined by -rn = n 2 (n E N);
3) the transformation y defined by
{ I, if n = 1
yn =
Pn, if n > 1
where Pn is the largest prime which divides n;
4) the transformation b m defined by c5 mn = mn, where m is a fixed natural
number, n EN?

Endomorphisms
67
3.4.3. Let N denote the set of all natural numbers. Which of the transforma-
tions defined in 3.4.2 are endomorphisms of N preserving the operation of
addition or multiplication?
3.4.4. Define an operation on the set Z of integers by m 0 n = m + n if
m and n are natural numbers, and undefined otherwise.
1) Prove that the transformation  defined by
2/, if I  0
(l) = -I, if I < 0 and is odd
1/2, if I < 0 and is even
is an invertible endomorphism of Z.
2) Is  an automorphism of Z?
3.4.5. Let A be a set on which a closed operation is defined. Prove that every
invertible endomorphism of A is an automorphism.
Remark. Compare this result with the preceding problem.
3.4.6. Prove that the transformation  in 3.2.12 is an automorphism of X
with respect to the usual operations addition, subtraction, multiplication,
and division. Further, show that  leaves each rational number fixed.
3.4.7.T. Let a family of relations be defined on a set X. Prove that the set of
all endomorphisms of X which preserve these operations forms a semigroup
under multiplication (composition) of transformations.
3.4.8. Let A be a multiplicative set, a E A, and a the transformation of A
which maps each element of A onto a. Prove that a is an endomorphism of A
if and only if a is an idempotent.
3.4.9. Let Q denote the additive group of rationals, roE Q. Prove that the
transformation A, defined by A,(X) = rx(x E Q) is an endomorphism of Q.
3.4.10. Let Q be the additive group of rationals. Prove that the semigroup of
endomorphisms of Q consists of all A" r E Q (see 3.4.9).
3.4.11.T. Using the results of the last two problems, prove that every endo-
morphism of the additive group of rational numbers, different from the null
endomorphism, is an automorphism (the null endomorphism maps each
number onto zero).
3.4. 12.T. Let X be a set on which a family of relations is defined. Prove that
the set of all automorphisms of X which preserve these relations forms a
group of transformations.
3.4.13.0. Let X be a finite set on which a family of relations is defined.
Prove that every invertible endomorphism of X is an automorphism. Does
this statement hold for any set X?
3.4.14. Let G denote the group of all nonsingular 2 x 2 matrices with
rational entries, d the determinant of the matrix A E G, written in the form

68
Composition of Transformations
d = (m/n)2 'A , where m and n are odd, and lA is an integer. (It is obvious that
every rational number d can be written in this form.)
1) Prove that the transformation  E TG defined by
( 1 lA )
oc( A) = 0 1
is an endomorphism of G with respect to matrix multiplication.
2) Find the image of the matrix
( )
under this endomorphism.
3) Prove that the center of G is not a completely characteristic subgroup.
3.4.15. Find all completely characteristic subgroups of the quaternion
group (see 2.6.39).
3.4.16. Define a binary relation p on the set X = {I, 2,3,4,5, 6} by (m, n) E p
if m and n are relatively prime. How many endomorphisms of X have rank
less than three?
3.4.17. Let X = {a, b, c, d, f} and define an ordering on X by a < b < t
for t = c, d,f.
1) Find all automorphisms of X.
2) Prove that the group of automorphisms of X is isomorphic to the
symmetric group S3 (cf. Chapter 3.3)
3.4.18. Suppose an ordering on a finite set X is given by a  t, where a is
fixed and t is any element of X. Describe all endomorphisms of X.
3.4.19. Let X be the ordered set in problem 3.4.18. Let X' be the set obtained
by adjoining an element v to X which is not comparable to any element of
X (v  t, t  v, t E X). Prove that the groups of automorphisms of X and X'
are isomorphic.
3.4.20. Let p be a nonempty binary relation on a set X, and J the set of all
transformations of X of rank one. Prove that the semigroup of all endo-
morphisms of X contains J if and only if p is a reflexive relation.
3.4.21. Let p be a reflexive binary relation on a set X. Prove that the semi-
group of endomorphisms of X with respect to p coincides with the semigroup
of all transformations of X if and only if p is the diagonal or the universal
binary relation (see Chapter 1.3).
3.4.22. Let X be an ordered set. We say that X is connected iffor any x, Y E X
there exists a chain t. = x, t 2 ,. .., t n = Y (t.,..., t n E X) in which any two
adjacent elements are comparable (t i  t i + 1 or t i + 1  t i ). Prove that if X
is connected then, for any endomorphism , the set a.X is also connected.

Endomorphisms
69
3.4.23.8. Prove that every invertible endomorphism of a linearly ordered
set is an automorphism. Does this statement hold for any ordered set?
3.4.24. Let N be the set of natural numbers with the usual ordering. Prove
that the group of automorphisms of N is the unit group. Does this statement
hold for the set of all integers with the usual ordering?
3.4.25. Let p be an equivalence relation on a set X, a, b, c E X. Prove that the
transformation  defined by
{ a, if (x, c) E p
x =
b, if (x, c)  p
is an endomorphism of X.
3.4.26. Let X be an ordered set, with a  b, C  d (a, b, c, d E X, a =F b,
c =F d). Show that there exists an endomorphism of X mapping a onto c and
b onto d.
3.4.27.8. Define a relation p on three-dimensional space by requiring three
points P 1 , P 2 , and P 3 to be p-related if they lie on the same line and P 2 is
between P 1 and P 3 . Prove that every affine transformation of the space is an
endomorphism with respect to p (an affine transformation is an invertible
transformation which maps lines onto lines). Are affine transformations
necessarily automorphisms of the space with respect to p?
3.4.28. Let X be the set of all real numbers of the form a + bJ2 + cJ3 +
dJ 6, where a, b, c, d are rational numbers, and let G x be the group of all auto-
morphisms of X with respect to the usual arithmetic operations for which
each rational number is a fixed point (see 3.4.12).
1) Find ,.J2 and J3 for any  E G x .
2) Prove that G x is finite.
3.4.29. Let G x be the group in the preceding problem. Prove that the subset
H of G x consisting of all automorphisms which leave J2 fixed is a normal
subgroup of G x .
3.4.30.8. Let X be the field in problem 3.2.12. Prove that the group of all
automorphisms of X with respect to the usual arithmetic operations is
infinite.
3.4.31. Let P be the set of endomorphisms of an ordered set X with the
property that X is linearly ordered for every  E P. Prove that P is a two-
sided ideal of the semigroup of all endomorphisms of X.
3.4.32.8. Let X = {I, 2, . . . , n} have the usual ordering.
1) Find all idempotents of rank n - 1 in the semigroup of endomor-
phisms of X.
2) Prove that the set of all idem po tents of rank n - 1 forms a generating
set for this semigroup.

70
Composition of Transformations
3.4.33. Let X be the ordered set in problem 3.4.18. Prove that the semigroup
of endomorphisms of X is regular.
3.4.34.8. Prove that the semigroup of endomorphisms of the ordered set X
in problem 3.4.17 is not regular.
5. Groups of Isometries
Let R be the set of all points on the real line. A transformation of R
which preserves the distance between any two points on R is called an
isometry of R.
For each nonnegative real number c we define the binary relation Pc
on R by (P1' P 2 ) E Pc if the distance between the points P 1 and P 2 is c. Then the
isometries of R are the only endomorphisms of R which preserve all of the
relations Pc'
In addition to isometries of the line we also consider isometries of the
plane and of space, i.e., transformations which preserve the distance between
any two points. As in the case of lines, isometries of the plane and of space
are those endomorphisms which preserve all of the Pc.
Every transformation which is the result of a rigid displacement is
obviously an isometry. Such isometries are usually called isometries of the
first kind. There are isometries which are not of the first kind. For example, a
reflection of the plane about a line (otherwise called a symmetry with respect
to the line) is an isometry which is not of the first kind. Isometries not of the
first kind are said to be isometries of the second kind. However, we shall not
have any occasion in this book to distinguish between these two kinds of
isometries.
Each of the sets of isometries of (1) the real line, (2) the plane, and (3)
space is a group of transformations (see 3.5.1). Subgroups of these groups are
called groups of isometries.
Let X be the set of all points on the real line, plane, or space, and let F
be a figure in X (i.e., F is a subset of X). By a self-coincidence of F in X we
mean an isometry of X which maps F onto itself.
For each figure F in X we define the binary relation t F on X by
(P 1 , P 2 ) E tF if P 1 , P 2 E F. We will prove that the set of self-coincidences of F
consists of all automorphisms of X with respect to t F and all relations Pc
defined above (see 3.5.11). Hence it follows that the set of all self-coincidences
of a figure is a group of transformations.
We can say that the group of self-coincidences of a figure characterizes
the "degree of symmetry of the figure." In other words, determining the
group of self-coincidences of a figure F is equivalent to knowing its symmetry.
The determination of the symmetry of F is often very important in the study
of its structure. In particular, an important facet in crystallography is the
study of the group of self-coincidences of figures having special forms.

Groups of Isometries
71
3.5.1. Let X be the set of points on the real line, plane, or space. Prove
that:
1) every isometry of X is an invertible transformation;
2) the set of all isometries of X is a group of transformations.
3.5.2. Prove that the group of all isometries of the real line has a subgroup
which is isomorphic to the additive group of real numbers.
3.5.3. Find all elements of finite order and of order two in the group of all
isometries of the plane.
3.5.4. Prove that the set of all axial symmetries (reflections about an axis)
of the plane is a generating set for the group of all isometries of the plane.
3.5.5. Prove that the set of all parallel translations of the plane is a normal
subgroup of the group of all isometries of the plane.
3.5.6. Let F be a figure in space, and let G be the class of all isometries of
space for which any point in F is a fixed point. Prove that G is a group of
isometries.
3.5.7. Find the order of the group G in the preceding problem where
1) F is a line;
2) F consists of two intersecting lines;
3) F consists of two skew lines.
3.5.8. Let F be a figure in space and A the set of all isometries  of space with
(F) c F. Prove that A is a semigroup with an identity. For what elements
 E A is  - 1 E A?
3.5.9. For which of the following figures will the semigroup A in the pre-
ceding problem be a group:
1) a sphere;
2) a half-plane;
3) a regular polyhedron?
3.5.10. Let F be the set of all points in the first quadrant of the plane and A
the semigroup defined in problem 3.5.8. Prove that the only subgroup of
[A]g is the set consisting of all elements of infinite order in [A]g and the identity
of this group. Describe this subgroup.
3.5.11. Let F be a figure on the line, plane, or space. Prove that the self-
coincidences of F are either automorphisms with respect to 't F or relations
Pc (see the text, above).
3.5.12. What is the order of the subgroup of all self-coincidences of each of
the following figures in the group of all isometries of the line:
1) a ray;
2) the unit interval;
3) the figure consisting of an infinite set of disjoint closed intervals of
equal length which are so situated that for each closed interval there
exist closed intervals lying to the left and right of it and the distance

72
Composition of Transformations
between the ends of any two adjacent closed intervals is some fixed
constant?
3.5.13. Find the subgroup G of all self-coincidences of a line in the group of
all isometries of the plane. Prove that G is an infinite, nonabelian group.
3.5.14. Find the subgroup of all self-coincidences of each of the following
figures in the group of all isometries of the plane:
1) a rhombus;
2) a square;
3) an isosceles triangle.
3.5.15. Let the vertices of a regular n-gon (i.e., a regular polygon with n sides)
with center 0 be AI' A 2 ,. . ., A,., and let B I , B 2 ,. . ., B,. be the midpoints
of the sides (where B i is the midpoint of AiA i + I for i = 1,2,..., n - 1, and
B,. is the midpoint of A,.A I ). Let F be a figure which is composed of all the-
triangles OAiB i , i = 1,2, 0 . 0 , n. Prove that the set of all self-coincidences of
F is a cyclic subgroup of the group of all isometries of the plane. What is the
order of this subgroup?
3.5.16. Find the subgroup of self-coincidences of a regular n-gon in the
group of all isometries of the plane.
3.5.17. Suppose that a figure F consists of two equal squares having a
common vertex and so situated that the diagonals of the squares lie on the
line passing through the common vertex. Find the subgroup of all self-
coincidences of F in the group of all isometries of the plane.
3.5.18. Prove that every group of self-coincidences of a polyhedron with n
vertices F is isomorphic to a subgroup of the symmetric group S,.o
3.5.19. Prove that the group of self-coincidences of a regular n-angular
pyramid (n  4) is isomorphic to the group of self-coincidences of a regular
n-gon.
3.5.20. Prove that the group G of all self-coincidences of a regular n-angular
pyramid (n > 4) possesses a unique commutative subgroup of order n.
Describe this subgroup (see the preceding problem).
3.5.21. Let G be the group in problem 3.5.20. For what values of n do any
two distinct reflections about a plane of symmetry of the pyramid form a
generating set for G?
3.5.22. An n-angular dihedron is a body consisting of an n-angular pyramid
and its mirror image in the plane of the base. What is the order of the group of
self-coincidences of an n-angular dihedron for n =1= 4?
3.5.23. Prove that the set
0= {e, (12), (34), (35), (45), (345), (354), (12) (34), (12) (35),
(12) (45), (12) (345), (12) (354)}
is a group which is isomorphic to the group of self-coincidences of a triangular
dihedron.

Groups of Isometries
73
3.5.24. Let P be a vertex of a regular tetrahedron T. Prove that the set of all
self-coincidences of T which leave P fixed is a group which is isomorphic to
S3'
3.5.25.8. Describe the group G of self-coincidences of a regular tetrahedron
and prove that G is isomorphic to 8 4 .
3.5.26. Let lp be any isomorphism of the group of self-coincidences of a
regular tetrahedron onto 8 4 (cf. the preceding problem). Which of the self-
coincidences correspond to elements in A4 under lp?
Remark. These self-coincidences of a tetrahedron form the so-called
rotation group of a regular tetrahedron.
3.5.27. Prove that the set of all self-coincidences of a cube which leave a given
vertex A fixed is a group. Describe this group.
3.5.28.8. What is the order of the group of all self-coincidences of a cube?
3.5.29.8. Prove that the set G of rotations of a cube about all axes of
symmetry is a subgroup of the group of self-coincidences of a cube, and that
G is isomorphic to 8 4 (this group is called the rotation group of a cube).
3.5.30. Prove that the set of rotations of a cube about its diagonals and about
the lines connecting the midpoints of opposite edges is a subgroup of the
rotation group of a cube (see 3.5.28). What subgroup of S4 is this group iso-
morphic to?
3.5.31. An icosahedron is a regular polyhedron which is bounded by 20
triangles. Describe the group of all self-coincidences of an icosahedron which
leave a given vertex P fixed.
3.5.32.8. Prove that the order of the group of self-coincidences of an icosa-
hedron is 120.
3.5.33. Let F be a figure consisting of all points P with coordinates
( .!.k+.!.l Jf3 k- J f 3 l )
2 2' 2 2
in some rectangular coordinate system, where k and I are any integers.
1) Which of the self-coincidences of F belong to the group ofisometries
of finite order that do not leave any points of F fixed?
2) Prove that there exist groups of self-coincidences of F of orders 2, 3,
6, and 12, but not of order 5.
3.5.34. Let F be a figure consisting of all points in the plane with integral
coordinates in some rectangular coordinate system.
1) Find the group G of self-coincidences of F.
2) Describe all subgroups of order 4 in G.
Remark. Finite groups of self-coincidences of figures like those in
3.5.33 and 3.5.34 are called crystal groups or Fedorov groups. They play an
important role in crystallography.

74
Composition of Transformations
6. Partial Transformations
Let X be a set; a partial transformation of X j$ a mapping C( of a subset
M of X into X.
The set N = ex(M) will be denoted by rex, and M, the domain of ex, will be
denoted by dex. By using this notation we can say that ex is a mapping of dex
onto rex.
For convenience we consider the empty transformation 0 to be a partial
transformation, where dO and rO are the empty set.
For a given partial transformation ex, it is sufficient to give dex and a
rule which associates an element exx E X with each x E dex. The set rex will then
be completely determined.
The set of all partial transformations of a set X will be denoted by P x .
If dex = X, then the partial transformation ex becomes a usual transforma-
tion of X (cf. Chapter 3.1). Consequently, Tx (the semigroup of all trans-
formations of X) is contained in Px.
We define an operation of multiplication (composition) on the set Px in
the following way: if ex, f3 E Px, then y = exf3 is the transformation such that
dy consists of all x E df3 with f3x E dex, and yx = ex(f3x) for all x E dYe
If ex, f3 E Tx, then it is obvious that the multiplication defined here coin-
cides with the multiplication of transformations defined in Chapter 3.1.
Let ex E Px. Define the binary relation Pa on X by (x, y) E Pa if y E dex
and exy = X. Obviously pr 1 Pa = rex and pr 2 Pa = dex. We will say that the
binary relation Pa corresponds to the transformation ex.
We will prove (see 3.6.15) that distinct binary relations correspond to
distinct partial transformations, and that PaPfJ = PafJ (ex, f3 E Px). Consequently,
multiplication of partial transformations can be considered as a particular
case of multiplication of binary relations.
One-to-one partial transformations, i.e., one-to-one mappings of one
subset of X onto another, playa very significant role.
If ex is a one-to-one partial transformation of X, then its inverse ex - 1
exists, and is also a one-to-one partial transformation (Chapter 1.2). Obviously
Pa - 1 = (Pa)* (cf. Chapter 1.3).
If ex is a partial transformation such that dex = rex = M and exx = x for
every x E M, then ex is called the partial identity on the set M, denoted by eM'
As with ordinary transformations, partial transformations can be written
as permutations by placing the elements of dex in the upper row and writing
the image of each directly below it.
Suppose X is a set of real numbers. Each real function f(x) determines
a partial transformation ex on X, where dex is the domain off(x) and exx = f(x)
for all x E dex. The converse also holds: each partial transformation of X is a
real function.

Partial Transformations
75
It should be noted that usually in the theory of functions one only
considers the composition of two functions f(x) and g(x) when the range of
g(x) is contained in the domain of f(x). However, we will not impose this
condition.
In this section, except for problems 3.6.25 and 3.6.26, we will not
consider any other operations on functions than composition. By virtue of
this, f(f(x» will be denoted by f2(x).
3.6.1.T. Prove that multiplication of partial transformations is associative.
Remark. Thus Px is a semigroup under multiplication of partial trans-
formations.
3.6.2. Let X = {I, 2, 3,4, 5,6, 7,8,9, 10}, a. l , a. 2 , PI' P2 E P x ;
a. = ( 2 3 8 10 ) ( 1 2 7 9 10 ) ( 1 3 5 7 9 10 )
1 1 4 1 5 (X2 = 3 4 3 1 10 Pl = 2 1 4 2 5 3
( 1 2 4 5 6 8 9 10 )
P2 = 5 3 9 1 7 5 2 7
Find:
1) a., a. 1 a.2, a.2a. 1 , P ;
2) the types of a. l , P1' and P2 in Px.
3.6.3. Let X be any set, and find the types of the following elements in
Px:
1) (Xl = (:) (x,yeX)
2) (X2 = (: ;) (x, ye X)
3) a.3 = ( Xl X2 ... x n - 1 )
X 2 X 3 ... X n
(X 1 , X2, · . · , X n E X, X n rt {X 1 , . . . , X n - 1})
3.6.4. Let
X = {I, 2, 3,4, 5, 6, 7, 8}, a. E Px
a. ( 1 3 4 5 6 7 8 )
8472513
Find a.- 1 , a.a.- 1 , and a.- 1 a..
3.6.5. Let a. be a one-to-one partial transformation of a set X. Find a.a. - 1
and a. - I a.. When does a.a. - 1 = a. - I a.?

76
Composition of Transformations
3.6.6. Given the functions
fl(x)=ln l X , fi(x)=+ V xi-l, f3(x)=sinx
-x
h(x)= arcsin x, f5(X)=+ y x'1 - 
where their domains consist of all points for which the corresponding formula
makes sense. Find:
1) r[fl(x)], d[f2f3(x)], r[f2f3(x)];
2) ff{x), fsf4(x), ff4{x);
3) all powers of f2(x).
3.6.7. Which of the following real functions h(x) are one-to-one partial
transformations? For those that are, find their inverses, and the products
hfi-1(x) and fi-1h(x):
1) the functions given in the preceding problem;
2) the function
x - ax + b
16 ( ) - ex + d
where a, b, c, d are fixed rational numbers such that ad - bc =F 0,
and d[f6(x)] is taken to be the set of all real numbers where the
denominator is different from zero.
3.-6.8.H. Let f(z) be a complex polynomial which is not a constant, with
d[f{z)] the set of all complex numbers. Prove that r[f{z)] is also the set of all
complex numbers.
3.6.9. Let X be any set,  E Px. Determine when 2 = 0 and 3 = O.
3.6.10. Let X be any set. Determine when d(l 2) = d(2)' where h 2 E Px.
3.6.11. Prove that for any 1' 2 E Px, the equality r(l 2) = rl holds if
and only if for each Y E rl' ther exists x E r2 such that lX = y.
3.6.12. Let X be any set. Determine when:
1) a partial transformation is an idempotent;
2) a one-to-one partial transformation is an idempotent.
3.6.13. Let X be any set. Given  E Px, determine when p E Px exists such
that (1) p = ex, and (2) p = ex.
3.6.14. Let X be any set,  E Px,  =F O. Determine under what conditions
thereexistspEPx,P =F 0, such that(I)p = 0,and(2)p = O.
3.6.15. Let , p E Px, and let Pa, PfJ be the binary relations corresponding to
, p. Prove that if  =F P then Pa =F PfJ' and that PafJ = PaPfJ.
3.6.16. Using the notation of the preceding problem, prove that the set
of all binary relations Pa taken over all  E Px is a semigroup under multipli-
cation of binary relations and that this semigroup is isomorphic to Px.

Partial Transformations
77
3.6.17. Given a binary relation p on a set X, determine when there exists a
partiaJ transformation  E Px such that P = Pa'
3.6.18. Let Pa be the binary relation on a set X corresponding to  E Px.
Determine for which transformations  the relation Pa is (1) reflexive, (2)
symmetric, and (3) transitive.
3.6.19. Let M eX. Prove that:
1) the set R of all  E Px for which r c M is a right ideal of P x ;
2) the set L of all f3 E Px for which df3 c M is a left ideal of Px.
3.6.20. Prove that the set L c Px consisting of all  such that d =F X is a
left, but not a right, ideal of Px.
3.6.21.T. Let X be any set. Prove that the set of all one-to-one partial trans-
formations of X is a semigroup.
3.6.22.H. LetX = {1,2,...,n}, = (12),f3 = (1,2,...,n),y = e{2,...,n}.Prove
that the semigroup of all one-to-one partial transformations of X is generated
by the permutations , f3, and y.
3.6.23. Let
2x-3
f(x)= x+ 1
be a real function Nhose domain is all real numbers except -1. Findf2(x)
and f3(x).
3.6.24.8. Let R be the set of all functions of the form
f( ) - ax+b
x - ex + d
as defined in 3.6.7. Does R form a group under multiplication of partial
transformations?
3.6.25. Let G be the set of all functions of the form
f(x) = ax+b
eX+d
where a, b, c, d are rational numbers such that ad - bc =F O. An operation is
introduced on G in the following manner:
if
( ) alx+bl
gx -
- c t x+d 1
then
f( ) ( x ) _ (aal+bcl)x+(abl+bdl)
x og -(ea 1 +c 1 d)x+(cb 1 +dd 1 )
Prove that G is a group.
Remark. Compare this result with the preceding problem.

78
Composition of Transformations
3.6.26. Let G be the group in the preceding problem. Show that G is iso-
morphic to the group G in 3.2.13.
3.6.27. Let C x be the subset of P x consisting of the empty transformation and
all elements a. e Px such that ra. = x, where x is a fixed. element in X. Prove
that the sets Cx, x E X, are the only nonempty minimal right ideals of Px (i.e.,
ideals which do not properly contain any right ideals).
3.6.28.T.H. Let A be a semigroup of one-to-one partial transformations
which satisfy the conditions
1) the inverse of every transformation in A is also contained in A;
2) there exists a set M such that for each C( E A, da. = ra. = M.
Prove that A is a group.
3.6.29.T.H. Prove that every group of one-to-one partial transformations
satisfies conditions (1) and (2) of the preceding problem.
3.6.30. Let B be a semigroup of one-to-one partial transformations of a set
X with the property that the inverse of every element in B also lies in B.
Prove that B is a regular semigroup in which every element has a unique
inverse (see 2.5.16).
Remark. The converse also holds: if each element in a semigroup S
has a unique inverse, then S is isomorphic to a semigroup T of one-to-one
partial transformations in which the inverse of each transformation in T is
also contained in T.

Chapter 4
GROUPS AND THEIR SUBGROUPS
1. Decomposition of a Group by a Subgroup
Let H be a subgroup of a group G, x E G. The set xH is called a right
coset of H in G, and Hx is called a left coset of H in G. If G is written as
the union of its mutually disjoint right cosets of H :
O=x(J.H U xH U ... U x£HU ...
then such a partition is called the right decomposition of G by H. The set
{x«, xfJ' . . . , x, . . .} is called the set of representatives of the right decomposi-
tion of G by H.
Left decomposition and its set of representatives is defined analogously.
One should keep in mind that sometimes what we have called right de-
composition is called left decomposition, and conversely.
As we will show below (see 4.1.16), the sets of co sets in the right de-
composition and in the left decomposition have the same cardinality. If
the number of such classes is finite, then this number is called the index
of H in G. When there is an infinite number of cosets we say that H has in-
finite index.
In the literatut:e, the index is often denoted by [G: H]. When there is an
infinite number of cosets, the index is sometimes defined as the cardinality of
the set of these cosets.
Let F and H be subgroups of G (in particular, F can be equal to G) and
x E G. The set FxH is called the double coset of the pair (F, H) in G [or some-
times the double modulo (F, H)].
If G is written as the pairwise disjoint union of double co sets of the pair
(F, H):
. o = FXa,H U FxH U ... U FXEH U ...
then such a partition is called the decomposition of G by the pair (F, H)
[or the decomposition modulo (F, H)]. The set {x«, xfJ' . . . , x,. . .} in this
79

80
Groups and Their Subgroups
case is called the set of representatives of the decomposition of G by the
pair (F, H).
If G is commutative, the right and left decompositions obviously
coincide. In such a case the decomposition of G by the pair (F, H) coincides
with the decomposition of G by the subgroup F H (see 2.6.36).
4.1.1.H. Prove that if an element t is contained in a right coset xH of a sub-
group H in a group G, then tH = xH.. An analogous result holds for left
cosets.
4.1.2.H. Let H be a subgroup of a group G (x, y E G). Prove that two right co-
sets xH and yH are either equal or disjoint. The same is true for left cosets.
4.1.3. Let G be a group, H a subgroup of G, and x e G. Show that x is con-
tained in the right coset xH and the left coset Hx.
4.1.4.T.H. Prove that for every subgroup H of a group G, there always
exists both a right and a left decomposition of G by H.
4.1.5. Let there be given two right decompositions of a group G by a sub-
group H. Prove that they represent the same partition of the set of all elements
of G. The same result holds for left cosets.
4.1.6. Find the right decomposition of the symmetric group S3 by the sub-
group H = {e, (12)}.
4.1.7. Find the left decomposition of the alternating group A4 by the sub-
group H = {e, (123), (132)}.
4.1.8. Find the right and left decompositions of the quaternion group K
(see 2.6.39) by the subgroup H = {I, -I}. Compare these decompositions,
and explain the result of the comparison.
4.1.9.H. Find the decompositions of a cyclic group of order 10 by each of its
subgroups.
4.1.10. Find the decomposition of the infinite cyclic group generated by x
by the subgroup generated by x 3 .
4.1.11. For any group G, what are the decompositions of G by the identity
suBgroup and by G itself?
4.1.12. Let S be a set of representatives in the right decomposition of a
group G by a subgroup H. Define a mappingf of G into itself by f(z) = XES,
where zH = xH, for each z e G. Prove that this rule does indeed define a
single-valued function. Prove that f is a mapping of G onto S satisfying the
following properties: for every z e G and he H,
I) f{f(z»=/{z)
2) z-t j{z) E H
3) f{zh)=/(z)

Decomposition of a Group by a Subgroup
81
4.1.13. Let H be a subgroup of a group G and f a mapping of G into itself
possessing the following three properties: for every Z E G, h E H,
I) f(f(z»=f(z)
2) z-lf(z) E H
3) f(zh)=f(z)
Prove thatf(G) is a set of representatives in the right decomposition of
G by H.
Remark. Compare this result with 4.1.12.
4.1.14. Formulate conditions on a mapping of G onto a set of representatives
in the left decomposition of a group G by a subgroup H similar to 4.1.12,
and obtain results analogous to 4.1.12 and 4.1.13.
4.1.15. Let {x(X' xfJ' . . . , x, . . .} be a set of representatives in the right
decomposition of a group G by a subgroup Hand {h(X, hfJ' . . . , h, . . .} a set of
elements in H, each of which is associated with an element in the set ofrepre-
sentatives. Prove that {x(Xh(X, xfJhfJ' . . . , xh, . . .} is also a set of representatives
of the right decomposition of G by H, and that every set of representatives of
the right decomposition of G by H can be obtained from the original set
{x(X' x fJ ,..., x,...} in such a way.
Formulate and prove a similar property for left decompositions.
4.1.16.H. Let {x(X' xfJ, . . . , x, . . .} be a set of representatives in the right
decomposition of a group G by a subgroup H. Prove that {x; 1, xi 1, . . . ,
x 1, . . .} is a set of representatives in the left decomposition of G by H.
Remark. This implies that the set of right cosets and the set of left cosets
in the right and left decompositions of G by H, respectively, have the same
cardinality. This is the basis for introducing the notion of index of H in G.
4.1.17. Let H and H' be subgroups of a group G, where H' c H c G, let S
be a set of representatives of the right decomposition of H by H'. Prove that
SS' is a set of representatives of the right decomposition of G by H'.
4.1.18. How many distinct sets of representatives are there in the right
decomposition of a group of order 12 by a subgroup of order 31
4.1.19. Let K be a nonempty subset of the group G, x E G. Prove that the
three sets K, xK, and Kx have the same cardinality.
Remark. In particular, all right cosets and left cosets have the same car-
dinality as K itself (if K is finite all cosets have the same number of elements as
K).
4.1.20.T.H. Let G be a finite group of order n. Let H be a subgroup of order
h in which the index of H in G is k. Prove that n = hk.

82
Groups and Their Subgroups
Remark. Hence we arrive at the important conclusion that in a finite
group the order of every subgroup, as well as its index, is a divisor of the order
of the group. *
4.1.21.T.H. Prove that the order of every element of a finite group is a factor
of the order of the group.
Remark. Compare this result with 3.3.27.
4.1.22. Prove that if a subset K of a group G is either a right or a left coset of
some subgroup H in G, then for all x, y, z e K,
X y-l Z E K
4.1.23.H. Let K be a nonempty subset of a group G such that
X y-l Z E K
for all x, y, z e K. Prove that there exists a unique subgroup H of G such that
K is a right coset of H in G, and a unique subgJ.'oup H' of G such that K is a
left coset of H' in G.
Remark. Compare this result with the preceding problem.
4.1.24. Determine which of the sets below are cosets of the symmetric
group Ss by one of its subgroups:
1) K 1 ={(2 3 4), (12 3 4)}
2) Ki={(l 2), (1 2 3), (1 2 3 4)}
3) Ka= {e, (1 2 3 4), (1 3) (2 4), (1 4 3 2)}
4) K, = {( 1 2), (1 3), (1 4), (1 5)}
5) Ka= {(I 2), (1 5 2) (3 4)}
4.1.25. Is the set of all matrices whose determinant is equal to a given number
c =f: 0 a coset (and if so, by what subgroup) of the group of all nonsingular
complex n x n matrices?
4.1.26.T.H. Let F and H be subgroups of a group G. Prove that there always
exists a decomposition of G by the pair (F, H).
4.1.27. Find the decomposition of the symmetric group S4 by the pair
(F, H)
F = {e, (1 2 3), (1 3 2)}, H= fe, (1 2) (3 4)}
4.1.28. Find the decomposition of the symmetric group S3 by the pair
(F,H), whereF = H = {e,(1 2)}.
*This result is known as Lagrange's Theorem. [Trans.]

Conjugate Classes
83
4.1.29. Let N be a normal subgroup and H any subgroup of a group G.
Prove that the decomposition of G by the pair (N, F) coincides with the right
decomposition of G by F N (see 2.6.36).
4.1.30.T.H. Let HI and H 2 be two subgroups of a finite group G having
orders m 1 and m2, respectively. Prove that the set H 1 H 2 consists of m 1 m2/ d
elements, where d is the order of the subgroup HI n H 2 .
4.1.31. Prove that the alternating group A4 has no subgroup of order 6.
Remark. Compare with 4.1.20.
2. Conjugate Classes
Let G be a group, a, bEG. If there exists an element x E G such that
x-lax = b, then we say that b is a conjugate of a [obviously in this case a is
also a conjugate of b, since a = (X- l )-lb(x- l )]. A set K of elements in G
such that any two elements in K are conjugates and no element in K is a
conjugate of any element outside of K, is called a conjugate class of elements of
G. Two subsets M 1 and M 2 of G are said to be conjugates if there
exists an element XE G such that x- 1 M IX = M 2 [and therefore also
(X- l )-lM 2 {x- l ) = M l ].
For a subset M of G, the set of all x E G which commute with M
xM =Mx
is called the normalizer of M. If M consists of a single element g, then the
normalizer of M is called the normalizer of g.
Since the relation of conjugacy is an equivalence relation on a group
(see 4.2.1), the set of elements which are conjugates of a given element turns
out to be a conjugate class. Therefore the set of all elements of a group can be
divided into disjoint conjugate classes.
Suppose a finite group of order n has m conjugate classes, and the number
of elements in each of these classes is k l , k 2 , . . . , k m . Then we have
n=k 1 +kJ+...+km
I t is particularly useful to consider those k i for which k i = 1. As we will see.
below (see 4.2.18), the elements in the center are the only ones whose conju-
gate classes consist of the element itself. Thus the above equality can be
written
n=c+k 1 +kJ+...+ks
where c is the order of the center and k i > 1 (i = 1,2,..., s). It will be useful
to take into account 4.2.10 when applying this equality.

84
Groups and Their Subgroups
4.2.1.T. Prove that the relation of conjugacy is an equivalence relation on a
group.
4.2.2. Let G be a group. Prove that if two elements x and yare conjugates in
G, then their orders are equal.
4.2.3. Prove that any two subsets M 1 and M 2 of a finite group which are
conjugates have the same number of elements.
4.2.4. Partition the elements in the symmetric group S3 into conjugate
classes.
4.2.5. Partition the elements of the quaternion group (see 2.6.39) into con-
jugate classes.
4.2.6. Prove that the permutations
( 1 2 3 4 5 6 )
x= 2 5 3 6 1 4 '
( 1 2 3 4 5 6 )
Y= 5 3 4 2 1 6
are conjugates in the symmetric group S6, and find the number of Z E 8 6
such that
Z-IXZ=y
4.2.7.T. Let N be the normalizer of the subset M of a group G. Prove that N
is a subgroup of G.
4.2.8. In the group of all real nonsingular 2 x 2 matrices, find the normalizers
of the following elements:
x=( ).
( -1 0 )
y 0 -1 '
z=(  )
4.2.9.T. Let N be the normalizer of the subset M of a group G. Let there be
given a left decomposition of G by N:
Q= Nxfl. U Nx U ... U NX e U ...
Prove that all of the sets
-1 M - 1 M -1 M
XfI. Xu, X p X, ..., Xe X e ' ...
are distinct, and that every set which is a conjugate of M is equal to one of the
x  1 M x  ( = li, P, . . .).
4.2.10.T.H. Let G be a finite group of order n, and let M be a subset of G.
Prove that there are n/m subsets of G which are conjugates of M, where m is
the order of the normalizer of M.

Conjugate Classes
85
Remark. It follows immediately from 4.2.10 that in a finite group G
the number of elements in each conjugate class is a factor of the order of G.
4.2.11. Assume that a finite group G has order n, x E G has order m, and the
number of conjugates of x is k. Prove that k is a factor of the integer n/m.
4.2.12. Let k be the number of elements in some conjugate class of a finite
group G of order n. Let c be the order of the center of G. Prove that k is a
factor of the integer n/ c.
4.2.13. Find all finite groups which have exactly two conjugate classes.
4.2.14. Define an operation on the set of all pairs of integers (n, m) by the
fornlula
(nit ml) (n'), m'J) = (nl + n'), (-1 )n 2 ml + mi)
Prove that this set forms a group with respect to this operation. Find the
normalizer of each element x, and the number of conjugates of each x.
4.2.15. Let G be a group, and suppose that the normalizer of an element a
in G is a subgroup N. Prove that for any x E G, the normalizer'of x-lax is
X-I N x.
4.2.16. Let G be a finite group, x E G, k the number of conjugates of x in G,
and k' the number of conjugates of x n in G. Prove that k' divides k.
4.2.17.T. Prove that a subgroup A of a group G is a normal subgroup if and
only if the only conjugate of A is A itself.
4.2. 18.T. Prove that Z EGis contained in the center of G if and only if Z
itself is its only conjugate.
4.2.19. In the symmetrIc group Sn, two permutations x and yare written as
disjoint cycles:
x = (all · · . a lk J(a21 . . . a2k2) · . . (a pl . . . apk p ) (k l  k 2  . . .  k p )
y = (bll...bllt)(b21...b212)...(bqlv..bqlq) (11  1 2  ...  lq)
Prove that x and yare conjugates in Sn if and only if they have the same
types,i.e.,p = qandk l = 1 1 ,k 2 = 12,...,k p = Ip.
4.2.20. Partition the elements of the symmetric group 54 into conjugate
classes.
4.2.21. Partition the elements of the alternating group A4 into conjugate
classes in A4'
Remark. Compare this result with 4.2.20.

86
Groups and Their Subgroups
4.2.22. Determine which of the following matrices are conjugates in the
group of all nonsingular 2 x 2 matrices over the reals :
m 1 =(_: :). mi=( :). m3=(_)
4.2.23. Let K 1 , K 2' K 3 be three conjugate classes of a group G. Prove that if
K 1 n K2 K 3 -=F 0, then K 1 c K2 K 3'
4.2.24.H. Let K 1 , K 2 , K3 be three conjugate classes of a finite group G.
Let K 1 = K 2 K 3 , and k 1 , k 2 , and k3 be the number of elements in K 1 , K 2 ,
and K 3' respectively. Prove that k 1 divides k2 k 3 .
4.2.25.T. Let H be a subgroup of a group G, and xe G. Prove that x- 1 Hx
is also a subgroup of G.
4.2.26.H. Let HI and H 2 be two subgroups of a finite group G, having orders
m 1 and m2, respectively. Prove that the set HI X H 2 has m 1 m 2 /d elements,
where d is the order of the subgroup X-I H 1 X n H 2 (see 4.2.25).
4.2.27.T.H. Suppose there is given a decomposition of a finite group G
by a pair of subgroups (F, H) (cf. Chapter 4.1)
O=FXtHU FX2HU... U FxmH
Let G, F, H, and X i - 1 FX i n H have orders n, a, b, and d i , i = 1, 2, . . . , m,
respectively. Prove that
ab + ab + ab
n=T d ...+r
1 I m
3. Normal Subgroups and Factor Groups
According to the definition given in Chapter 2.6, a subgroup N of a
group G is called a normal subgroup if for each x e N and a e G,
a-1xae N
We will show (see 2.6.35) that a subgroup N of G is normal if and only if
xN = N x for all x e G.
Let M be a subset of a group G. The intersection of all normal subgroups
of G which contain M is a normal subgroup of G (see 2.6.37). This normal
subgroup N M contains M, and is contained in every normal subgroup of G
containing M. Therefore N M is called the minimal normal subgroup containing
M, or the normal subgroup generated by M.
One should keep in mind that the normal subgroup generated by M
contains the subgroup [M], but in general is distinct from it.

Normal Subgroups and Factor Groups
87
Let N be a normal subgroup of a group G. Then the set of right cosets of
N in G is a group with respect to multiplication of cosets (see 4.3.12). We call
this group the factor group of G by N, and denote it by GIN.
By considering the set of left co sets of N in G, we obtain the same group
G/ N (see 4.3.13).
Factor groups play a particularly important role in the theory of
groups because of their connection with homomorphisms of a group.
Namely, every factor group of G by a normal subgroup N is a homomorphic
image of G, and conversely if G' is a homomorphic image of G, then G' is
isomorphic to some factor group of G (see 4.3.26 and 4.3.28).
Let H be a subgroup of G. We denote by PH the binary relation on G
defined by (x, y) e PH if xH = yH.
4.3.1. Find all normal subgroups of the symmetric group S3'
4.3.2. Determine the normal subgroups generated by each of the following
subsets of S 4 :
M. = [(12), (1324)],.,
M i = {e, (123), (132)},
M3 = {e}
4.3.3. Is the subset of all rotations of an n-angled dihedron (n =1= 4) about its
axis (see 3.5.12) a normal subgroup of the group of all self-coincidences of the
dihedron.
4.3.4. Prove that the Klein group is a normal subgroup of the symmetric
group 8 4 (see 3.3.18).
4.3.5.8. Prove that the set N of all matrices whose determinant is equal to
one is a normal subgroup of the group G of all nonsingular n x n matrices
over the reals.
4.3.6. Prove that if G is a group and H is a subgroup of index 2 in G, then H
is a normal subgroup of G.
4.3.7. Prove that the alternating group An is a normal subgroup of the
symmetric group 8n.
4.3.8. Let G be the multiplicative set of all triples of integers with an operation
defined by
(k b k i , ka) </b l'b Is) = (kJ + < - 1 )kd Ib k'J + Ii' ka + 1 8 )
Prove that G is a group, and that the subgroup H = [(1,0, 0)] is a normal
subgroup of G.
4.3.9. Prove that every subgroup of a commutative group is normal.
Remark. There exist noncommutative groups with such a property. For
example, the quaternion group (see 2.6.39).

88
Groups and Their Subgroups
4.3.10.T. Let H be a subgroup of a group G, Hx = XHX-l (x e G), and
N = n Hx. Prove that:
xeG
1) N is a normal subgroup of G;
2) N is maximal in the set of normal subgroups of G which are contained
in H (if N 1 is normal in G and N 1 c: H, then N 1 c: N).
4.3.11. Let N be a normal subgroup of G. Prove that for any pair of elements
x,yeG
(xN) (yN) =xyN
4.3. 12.T.H. Let N be a normal subgroup of G. Prove that the set GIN of
right cosets of N in G forms a group relative to multiplication of cosets. Find
the identity of GIN. Find the inverse of each element xN.
Remark. As was stated in the introduction to this section, G/N is called
the factor group of G by N.
4.3.13. Let N be a normal subgroup of a group G. Prove that the set of left
cosets of N in G is a group with respect to multiplication of cosets, and that
this group coincides with the factor group GIN.
4.3.14. Let G be a group and E the subgroup consisting of the identity element.
Determine G/G and GIE.
4.3.15. Prove that the factor group of the symmetric group S4 by the Klein
group is isomorphic to S3 (see 3.3.18).
4.3.16. Letting G and N be the groups in 4.3.5, prove that GIN is isomorphic
to the multiplicative group of real numbers different from zero.
4.3.17.H. Let G' be the group in 3.2.19, (4). Prove that the group N of parallel
translations of the plane is a normal subgroup of G', and that G'IN is iso-
morphic to the group of all rotations of the plane about a fixed point.
4.3.18.T. Let N be a normal subgroup of a group G, let G have order m,
and N have order n. Prove that the order of GIN is min.
4.3.19. Let H be a subgroup of a group G. Prove that the relation PH is left
compatible (cf. Chapter 2.4).
4.3.20. Let H be a subgroup ofa group G. Prove that PH is a congruence on G
if and only if H is a normal subgroup of G (see the preceding problem).
Remark. By using 2.4.14, we see that GlpH is a multiplicative set if and
only if H is a normal subgroup of G.
4.3.21. Let H be a normal subgroup of G. Prove that the factor set G/PH
is a group, and that this group is equal to the factor group G/H.
4.3.22. Let G and H be the groups in 4.3.8. Prove that G/H is isomorphic to
the additive group of Gaussian numbers, i.e., numbers of the form a + bi,
where a and b are rational numbers.

Normal Subgroups and Factor Groups
89
4.3.23. Let G and H be the groups in 3.2.16. Prove that GIH is isomorphic
to the factor group given in 4.3.16.
4.3.24. Let G x and H be the groups in 3.4.29. Prove that GxlH is isomorphic
to H.
4.3.25.H. Prove that the symmetric group S3 is a homomorphic image of the
symmetric group S4.
4.3.26.T.H. Let N be a normal subgroup of a group G. Prove that GIN is a
homomorphic image of G.
4.3.27.T.H. Let f be a homomorphism of a group G onto a group G'. Prove
that:
1) the set N of all elements in G which are mapped onto the identity of
G' under f is a normal subgroup of G;
2) for a fixed element g' E G', the set of all elements in G which are
mapped onto g' by f is a right coset of N in G.
Remark. The normal subgroup N is called the kernel of f.
4.3.28.T.H. Prove that if a group G' is a homomorphic image of a group G,
then G' is isomorphic to some factor group of G.
Remark. Compare the results of 4.3.26 and 4.3.28 with those of 2.4.18-
2.4.20.
4.3.29. Let G be a noncommutative group, Z the center of G. Prove that GIZ
cannot be a cyclic group.
4.3.30.T. Let N be a normal subgroup of a group G, and H any subgroup of
G. Let H be the subset of GIN consisting of those cosets which contain at least
one element of H. Prove that:
1) H is a subgroup of GIN;
2) if N c H, then N is a normal subgroup of Hand H = H/N.
4.3.31.T. Assuming the notation and the two conditions from the preceding
problem, prove that if H is a normal subgroup of G, then H is a normal
subgroup of GIN.
4.3.32. Let G and N be the groups in 4.3.5. Let HI be the subgroup of G
consisting of all triangular matrices having zeros below the main diagonal,
and H 2 the subgroup of H 1 consisting of all matrices with determinant + 1.
In the notation of 4.3.30, find H 1 and H 2 .
4.3.33.T. Let H be a subgroup of the factor group GIN. Prove that:
1) the subset H of G, where H is the union of all cosets contained in H ,
is a subgroup of G containing N;
2) if H is a normal subgroup of GIN, then H is a normal subgroup of
G.
4.3.34. Find all subgroups of the symmetric group S4 containing the Klein
group (see 3.3.18) by making use of 4.3.33 and 4.3.15. Which of these sub-
groups are normal in S4?

90
Groups and Their Subgroups
4.3.35. Let f be a homomorphism of a group G I onto another group G 2 .
Prove that:
1) if G 2 is not commutative, then neither is G I ;
2) if G 2 is infinite, then G I is too;
3) if f(a) has infinite order for some a E G I , then a also has infinite
order.
4.3.36. Let G be a group. Let A be a set of isomorphisms of G onto itself
which forms a group under multiplication (composition) of transformations.
Denote by H the set of all ordered pairs (x,f), where x E G and f E A. Define
an operatIon on H by
(Xl ,fl)(X2,f2) = (xlfl(x2),flf2)
Prove that H is a group. Show that the set G' of all pairs (x, i) (x E G), where
i is the identity mapping of G, is a normal subgroup of H, G' is isomorphic to
G, and H/G' is isomorphic to A.
4. Subgroups of Finite Groups
We say that a group G is a p-group if the order of every element in G is a
power of the prime number p.
If the order of a finite group G is divisible by pk, for some prime p and
some k > 0, but is not divisible by pk+ I, then every subgroup of G having
order pk (that such subgroups exist, see 4.4.6) is called a Sylow subgroup of
G relative to p, or simply a p-Sylow subgroup of G.
4.4.1.T.H. Prove that every group of order pn, where p is a prime number and
n > 0, has a nontrivial center (i.e., the order of the center is greater than 1).
4.4.2.H. Prove that for every prime number p there exists a unique (up to
isomorphism) noncyclic group of order p2. In addition, show that this group
is commutative.
4.4.3.H. Prove that if the order of a group G is divisible by a prime number p,
then there exists an element g E G whose order is p.
4.4.4.T.H. Prove that every finite p-group has order pn for some n.
4.4.5.H. Prove that every abelian group of order pq is cyclic, where p and q
are distinct primes.
4.4.6.T.H. Let G be a finite group whose order is divisible by pk, where p is a
prime. Prove that G contains a subgroup of order pk.
Remark. In particular, it follows that a finite group G contains a p-Sylow
subgroup for every prime p which divides the order of G.
4.4.7.H. Let P be a p-Sylow subgroup of a finite group G, and let N be the
normalizer of P. Prove that every element of N whose order is a power of p
is contained in P.

Subgroups of Finite Groups
91
4.4.8. Find all Sylow subgroups of the symmetric group S4'
4.4.9. Let G 1 = [x] be a cyclic group of order ten and G 2 = [y] a cyclic
group of order four. Let G be the set consisting of the 40 ordered pairs (gl' g2)
(gl E G 1 , g2 E G 2 ) with multiplication
(xal, yb 1 ) (xas, ybs) = (xat +3 b 1 a2, yb 1 + b2)
Prove that G is a group. Find all Sylow subgroups of G.
4.4.10.T.H. Prove that in a finite group, any two p-Sylow subgroups are
conjugates.
4.4.1J.T.H. Let p be a prime number which divides the order of a finite
group G, and let s be the number of distinct p-Sylow subgroups.
Prove that s divides the order of G, and is of the form s = 1 + kp, for
some nonnegative integer k.
4.4. 12.H. Prove that every noncyclic group of order six is isomorphic to the
symmetric group S3.
4.4.13. Suppose a group G has order pq, where p and q are distinct primes.
Prove that G has a normal subgroup N whose order is a prime number.
4.4. 14.H. Let G be a group of order pq, where p and q are primes, p < q.
Prove that if p does not divide q - 1, then G is cyclic.
4.4.15. Let P be a p-Sylow subgroup of a finite group G, N p the normalizer
of P, and H a subgroup of G such that N p c H. Prove that the normalizer of
H is equal to H.
4.4.16. Let G be a finite group, N a normal subgroup of G, and p a prime
number which divides the order of G but does not divide the index of N in
G (cf. Chapter 4.1). Prove that every p-Sylow subgroup of G is contained in N.
4.4.17. Prove that every group of order 50 has a proper normal subgroup.
4.4. 18.T.H. Let H be a subgroup of a finite group G. Let H have order pk,
where p is prime. Prove that H is contained in some p-Sylow subgroup of G.
4.4.19.H. Suppose that there exists a unique p-Sylow subgroup for every
prime p which divides the order of a finite group G. Prove that G has a non-
tri vial center.
4.4.20.H. 'Let G be a finite p-group. Prove that for every integer m which
divides the order of G there exists a normal subgroup of G of order m.
Remark. Note, however, that there exist finite groups such that even
though a number m divides the order of the group, there is no subgroup of
order m (see 4.1.31).
5. Commutators and the Commutator Subgroup
To each pair of elements x, y in a group G we associate the element

92
Groups and Their Subgroups
x- 1 y-l xy , which is called the commutator of x and y, denoted either by
k(x, y) or (x, y). We will use the latter notation. Thus
(x, y) = x-.y- 1 xy
The notation [x, y] = x - 1 Y - 1 xy is also used in the literature.
The subgroup of G generated by all of its commutators is called the
commutator subgroup, or derived group.
4.5.1. Prove that
XY=YX(X, y)
for any elements x and y in a group G.
Remark. This equality explains the role of commutators. The commu-
tator acts as a ','correction" for the degree of noncommutativity of x and y,
i.e., x and y commute "up to their corresponding commutator." The equality
(x, y) = e denotes that x and y commute.
4.5.2. What are the commutators in an abelian group?
4.5.3. Prove that if all commutators in a group G are equal to e, then G is
a belian.
Remark. Compare the results of 4.5.2 and 4.5.3.
4.5.4. Let G be a group, and suppose that an element Z EGis a commutator
in G. Prove that for every x E G, x-1zx is also a commutator.
4.5.5. Prove that (x, y) - 1 = (y, x).
4.5.6. In the symmetric group Sn (n  4), find the commutators
(Xh Xi), (X(J X3), (X., y)
(X2' XI)' (X 4 , XI)' (y, XI)
of the elements
x.=(12),
xi=(l 2 3),
x3=(1 234), y=(l 3)(2 4)
4.5.7. Which elements in the quaternion group (see 2.6.39) are commutators?
4.5.8. In the group of 2 x 2 matrices over te integers with determinant
equal to + 1, find the commutators (x, y), (y, z), and (z, x) of the elements
x = ( )
( t 2 )
y= ,
-1 -1
z=(_  )

Commutators and the Commutator Subgroup
93
4.5.9. Find the commutators (u, v), (v, w), and (w, u) in the group of nonsingu-
lar 3 x 3 matrices, where
( 1 0 0 )
u= 0 2 0 ,
o 0 3
( 1 1 1 )
'V= 0 1 1 ,
o 0 1
( 0 0 1 )
w= 0 2 0
300
4.5.10. In the symmetric group Sn, n  5, find the commutator (x, y) for the
3-cycles x = (u, c, b) and y = (a, b, v), where a, b, c, u, v are distinct numbers.
4.5.11.T. Prove that the commutator subgroup is a normal subgroup of a
group.
4.5.12. T. Prove that the factor group of a group by its commutator subgroup
is a belian.
4.5.13.T. Let N be a normal subgroup of a group G. Prove that GIN is
abelian if and only if N contains the commutator subgroup of G.
Remark. This shows that the commutator subgroup of G is the univer-
sally minimal element (under inclusion) in the set of all normal subgroups N
such that GIN is abelian.
4.5.14.8. Find the commutator subgroup of the symmetric group Sn.
4.5.15. Let {xa, xp,..., x,...} be a generating set of a group G, and let N
be a normal subgroup of G which contains all (x, x,,). Prove that N contains
the commutator subgroup of G.
4.5.16. Suppose that the commutator subgroup of a group G is contained in
the center of G. Prove that for any x, y, Z E G
(x y, z) = (x, z) (V, z)
(x, yz)=(x, y)(x, z)
(x n , y) = (x, yn) = (x, y)n
I
-n(n-l)
(xy)n = xnyn (y, x) 2
(x, (y; z» (y, (z, x» (z, (x, y» = e
4.5.17.8. Prove that in every finite p-group G, the commutator subgroup of
G is different from G.
4.5.18.8. Suppose the commutator subgroup of a finite group has order
two. Prove that the index of the commutator subgroup is even.
4.5.19.8. Let there be given a sequence of groups such that each one is a
subgroup of the following one:
0 1 C O 2 C.. · C an C On+l C.. ·

94
Groups and Their Subgroups
Prove that H = U G n is a group, and that the commutator subgroup of H
n
is equal to the union of the commutator subgroups of the G n .
4.5.20. Let N be a normal subgroup of a group G. Prove that all commutators
of GIN are cosets of the form kN, where k is a commutator of G.
4.5.21. Find the commutator subgroup of the group of all rotations of the
plane about all possible points in the plane and all parallel translations of the
plane [see 3.2.19, (4)].
4.5.22.". Find the commutator subgroup of the group of all nonsingular
2 x 2 matrices.
6. Solvable Groups
Let G ,be a group. A sequence of subgroups of G,
a = Ko ::::J K 1 ::::J ... ::::J Kn-l ::J Kn :::> ...
in which each K i is the commutator subgroup of K i - 1 , is called a derived
series, or commutator series.
If some term in the derived series is the identity subgroup, i.e., if Km = e
for some m, then the group G is called solvable. From Km = e, it obviously
follows that all successive terms are equal to the identity: e = Km = Km+ 1 =
Km+2 = . . . . In such a case we say that the derived series "breaks off" at the
identity subgroup and is stabilized there.
It is obvious that for a finite, nonsolvable group, Kn = Kn + 1 =F e must
occur for some n. By the same token, all successive terms must be equal, i.e.,
Kn = Kn+l = Kn+2 = ..., and the derived series is stabilized at some
nonidentity subgroup.
For an infinite nonsolvable group, the derived series is either stabilized
at some nonidentity subgroup or is not stabilized at any subgroup.
There are several equivalent definitions of a solvable group. We will
obtain some of these in the problems below as necessary and sufficient
conditions for solvability in the sense that we have defined it above.
The concept of a solvable group is one of the most important notions
in group theory. Besides the role it plays in group theory, it is also important
in the theory of the solvability of algebraic equations by radicals, which is
based on the theory of finite groups. J:
.1. Prove that every abelian group is solvable.
.6.2. Prove that the quaternion group (see 2.6.39) is solvable.
4.6.3.8. Prove that every group of order pq, where p and q are primes
(possibly equal), is solvable.

Solvable Groups
9S
4.6.4.T. Prove that in order for a group G to be solvable, it is necessary and
sufficient that it have a finite sequence of subgroups,
Q= HI:) Hi::J...  H m _ l ::) Hm =e
where each term H k contains all commutators of the preceding subgroup
H k - 1 .
Remark. In many cases this condition is more convenient to apply when
testing whether a given group is solvable.
4.6.5.T.H. Prove that the symmetric groups Sn are solvable for n = 1,2, 3,4.
4.6.6.T.H. Prove that the symmetric groups Sn are not solvable for
n = 5, 6, 7, . . . .
 . . Prove that every subgroup of a solvable group is solvable.
. .H. Prove that every factor group of a solvable group by a normal
s group is solvable.
4.6.9.T.H. Suppose a group G possesses a normal subgroup N such that
both N and GIN are solvable. Prove that G is solvable.
4.6.10.T. A normal series of a group G is a finite sequence of subgroups
Q=Ft  F i :)...  F n - 1 ::J Fn=e
where each term is a proper normal subgroup of the preceding one.
Show that if a group has a normal series in which every factor group
Fk-11F k is abelian, then it is solvable.
.4.6.11.T. Prove that every solvable group has a normal series (see 4.6.10)
in which each factor group Fk-11F k is abelian.
Remark. Compare this result with 4.6.10.
4.6.12.T. A composition series of a group G is a normal series (see 4.6.10)
in which each term is a maximal (under inclusion) proper normal subgroup
of the preceding term.
In other words, this means that there does not exist a group F' in G
which is a normal subgroup of some F i and such that
F i :) F' +1' F' =1= Fi' p' =1= F;+1
Suppose G has order n = PIP2 . . . Pm' where the Pi are primes (which
can be either distinct or equal). Prove that if G has a normal series consisting of
m + 1 terms, then this series is a composition series and G is solvable.
What type of groups are the factor groups FilF i + I?

96
Groups and Their Subgroups
4.6. 13.T.H. Prove that a finite solvable group of order g = PIP2 . . · Pm'
(where the Pi are distinct or equal prime numbers) has at least one composition
series, and that each such series consists of m + 1 terms.
4.6. 14.T.H. Prove that every finite p-group (i.e., a group of order pn, where
p is a prime) is solvable.
4.6. 15.H. For what values of n is the alternating group An solvable?
4.6.16. Define an operation on the set G = {..., Z-2, Z-l, Zo, Zl, Z2"'.} by
{ z n+m' if n is even
ZnZm=
Z n-m' if n is odd
Prove that G is a group with respect to this operation. Is G solvable?
4.6.17. Define an operation on the set of infinite sequences of integers by
(ab ai' aa, ..., an, ...) (b l , b i , b a , ..., b n , ...) =
= (at + bit (- l)b l ai + b i , (- 1)b 1 +bi aa + b a , ...
( _ 1) b1+b2+ba+...+bn-l a +b )
· · · , n n' ...
Prove that G is a group with respect to this operation. Is G solvable?
4.6. 18.H. Prove that every group of order 275 is solvable.
4.6. 19.H. Prove that every group of order 100 is solvable.
4.6.20.H. Prove that the group G in problem 3.2.19, part 4 is solvable.
7. Nilpotent Groups
The following two systems of subgroups of a group olay a role similar to
that played by the commutator series considered in the preceding section.
The upper central series of a group G is the sequence of subgroups
Zo =e c:: ZI c:: ZI <= ... c: Zn C Zn+l c...
in which Zn+ 1 (n = 0, 1,2,. . .) is the set of elements x E G such that xZn
commutes with all elements in the factor group G/Z n (Zn is a normal subgroup
ofGforalln;see4.7.1).Ifforsomen,Zn = Zn+l,thenevidently Z n = Zn+l =
Zn + 2 = · · · , and we say that the series is stabilized at Zn' If the upper central
series is stabilized at G, we say that it reaches G. Of course, it is possible for
this series to be stabilized at some Zn different from G. Finally, it is possible
that for some infinite groups Zn =F Zn + 1 for every n, i.e., the series is not
stabilized at any Zn'
A group G is nilpotent if its upper central series reaches G (finite nil-
potent groups were also called special groups in some of the earlier literature).

Nilpotent Groups
97
If Zn is the first term in the upper central series which is equal to G, then n
is called the nilpotency class for G.
The lower central series of a group G is the sequence of subgroups
Ho = a =::J Ht => Hi =::J ... :::J Hn :::J Hn+l :::> ...
in which H n + 1 (n = 0, 1,2,. . .) is the subgroup of G generated by all the
commutators x- 1 y- 1 xy, where x E Hn, y E G.
The impQrtance and some of the properties of these series, and various
relations between them, will be elucidated in the exercises below.
4.7.1.T. Prove that each term in the upper central series of a group G is a
normal subgroup of G.
4.7.2. Prove that every abelian group is nilpotent. Determine the nil-
potency class for each.
4.7.3. Determine which of the following groups are nilpotent, and find their
nilpotency class:
1) the symmetric groups Sn (n = 1, 2, 3, . . .);
2) the quaternion group (see 2.6.39) ;
3) the group in 2.6.38;
4) the group G = [(1 2 3 4); (1 2 3); (2 5); (5 6)];
5) the group G = [(1 2 3 4)(5 6 7 8); (1 5 3 7)(2 8 4 2); (9 10 11)].
4.7.4.H. Let G be a nilpotent group whose order is the product of k prime
numbers (either equal or distinct). Prove that the nilpotency class for G
does not exceed k - 1.
4.7.5.T. Prove that every nilpotent group is solvable.
4.7.6. Let G be the quaternion group (see 2.6.39), and let cp be the isomor-
phism of G onto itself defined by
cp= ( l -li-i} -j k -k )
1 - 1 } - j - Ii k - k
The set A = {8, cp, q>2, q>3} (where cpo = e is the identity mapping) forms a
group.
Construct the group H from G and A as shown in 4.3.36, and find the
lower and upper central series of H.
4.7.7. Prove that the term Zn in the upper central series of a group G contains
all commutators X-I y-l xy, where x E Zn+ l' Y E G.
4.7.8.T.H. Prove that if the upper central series of a group G reaches Gat the
nth step, i.e., Zn = G, then the lower central series reaches the identity at the
nth step, i.e., H n = e.

98
Groups and Their Subgroups
4.7.9.T.H. Prove that if the lower central series of a group G reaches the
identity at the nth step, Hn = e, then the upper central series reaches G at the
nth step, Zn = G.
4.7.10.T.H. Prove that the upper and lower central series of a nilpotent
group have the same length.
Remark. Exercises 4.7.8, 4.7.9, and 4.7.10 together imply that it is
possible to define a nilpotent group and its nilpotency class by starting with
the lower central series.
4.7.11. Let the commutator subgroup of a noncommutative group G lie
in the center of G. Prove that G is nilpotent. Determine its nilpotency class.
4.7. 12.T. Prove that every subgroup of a nilpotent group is nilpotent.
4.7.13.T. Prove that a factor group of a nilpotent group by any of its normal
subgroups is nilpotent.
4.7.14.T.H. Prove that every finite p-group is nilpotent.
4.7.15.H. Determine which of the following groups are nilpotent, and find
their nilpotency class:
1) 0 1 = [( 1 2 3), (4 5), ( 1 2) ]
2) Og=[(l 8 3 6)(2 7 4 5),
(9 15 11 13) (10 14 12 16),
(1 6 3 8) (2 8 4 7) (9 12 11 10) (13 16 15 14)]
3) 0 3 = [(1 2 3 4) (5 6 7 9), (1 5 3 7) (2 9 4 6),
(8 1 0 11), ( 1 2 3 4), ( 1 3 2) ]
4) a, = [( 1 2 3 4 5 6 7 8) (9 10 12 13 14 16 15 11 ),
(1 11)(2 15)(3 16)(4 14)(5 13)(6 12)(7 10)(8 9)]
4.7. 16.H. Let G i be a Prgroup (i = 1,2,..., n), where all of the primes
PI' P2, · . . , Pn are distinct. Let H be the set of all sequences (Xl' X2, · · · , x n ),
where Xi E G i (i = 1, 2, . . . , n). Define an operation on H by
(Xh x, ..., x n ) (y., Y, ..., Yn) = (XJYb x2Y, ..., xnYn)
Prove that H is a nilpotent group, and that its nilpotency class is equal to the
largest of the nil potency classes of the G i .
Remark. We have proved that every finite nilpotent group is isomorphic
to a group of this form.
4.7.17. Let G i be nilpotent groups with nilpotency classes k i (i = 1,2, 3,. . .).
Let H be the set of all sequences (Xl' X2, X 3 , . . .), where Xi E G i , and define
an operation on H by
(X., X, X3, ...) (y., Y, Ys, ...) = (X1Yb xt,Vg, xsYs, ...)
Determine when H is a nilpotent group, and find its nilpotency class in this
case.

Automorphisms of Groups
99
4.7.18. Prove that every finite nilpotent group G has a sequence of normal
subgroups
Q= No:) Nt:) Nt:)...:) N n - l :) Nn=e
such that every factor group N k-l/ N k (k = 1,..., n) is cyclic.
4.7.19. Prove that in a finite group G the set of all normal subgroups of G
which are nilpotent groups has a universally maximal (under inclusion)
element, i.e., a nilpotent normal subgroup which contains all other nilpotent
normal subgroups.
4.7.20. For the group G in 4.7.17, determine when the set of normal sub-
groups of G which are nilpotent has a universally maximal (under inclusion)
element.
Remark. Compare this result with 4.7.19.
4.7.21.H. Let PI' P2, and P3 be distinct prime numbers, and let G be a nil-
potent group of order PIP2P3 . Prove that G is abelian.
8. Automorphisms of Groups
An automorphism of a group G is an isomorphism of G onto itself.
It is obvious that the concept of an automorphism of a group G coincides
with the concept of an automorphism of the set G with respect to the operation
on the group (cf. Chapter 3.4). Thus automorphisms of G are transformations
on the set of elements of G. As we did with transformations, we will sometimes
define automorphisms by means of permutations.
Let G be a group, x e G. We will denote by t x the transformation on G
defined by
tx(a) = xax- l
(a e G)
We will prove that t x is an automorphism of G for every x e G (see 4.8.12);
such an automorphism is called an inner automorphism.
A subgroup of G is said to be characteristic if it is mapped onto itself by
every automorphism of G.
4.8.1. Let G be the additive group of integers. Which of the following trans
formations of G are automorphisms :
1) (Xlm = m + 1
2) (X2m = 2m
3) (X3m = - m (m e G)

100
Groups and Their Subgroups
4.8.2. Let G be the multiplicative group of complex numbers different from
zero. Which of the following transformations are automorphisms of G :
1) atz = Z (z is the conjugate of z);
2) a2[f(cos cp + i sin cp)] = ,.2(COS cp + i sin cp);
3) a3[f(cos cp + i sin cp)] = r[cos(cp + n12) + i sin(cp + nI2)]?
4.8.3. Let cp be an automorphism of a group G. Prove that:
1) cp(a) has the same order as a for every a E G;
2) cp(e) = e (e is the identity of G);
3) cp(a- 1 ) = [cp(a)] - 1 (a E G);
4) if M = {x(%, xp, . . .} is a generating set of G, then M' = {cp(x(%),
cp(x p ), . . .} is also a generating set of G;
5) every conjugacy class is mapped onto a conjugacy class by cp.
4.8.4. Let K be the quaternion group (see 2.6.39). Which of the following
permutations are automorphisms of K:
1) a 1 = ( - 1, i) ;
2) a2 = (i, -1)(j, - j)(k, - k);
3) a3 = (i, j, k)( - i, - j, - k);
4) a 4 = (i, - j, k, - i, j, - k) ;
5) as = (i, - i)(j, - j)?
4.8.5. Find the group of automorphisms of the group [(12)(34)(56), (34)].
Remark. Note that the group of automorphisms of a commutative
group need not be commutative.
4.8.6. Find the group of automorphisms of the infinite cyclic group.
4.8.7. Let G be a cyclic group of order n. Prove that the transformation (X
defined by a(x) = Xk (x E G) is an automophism of G if and only if k and n
are relatively prime.
4.8.8. Prove that the transformations described in the preceding problem
exhaust all automorphisms of a finite cyclic group.
4.8.9. Let G be a group. Prove that the transformation of G which maps each
element onto its inverse is an automorphism of G if and only if G is commu-
tative.
4.8.10.H. Prove that the group of automorphisms of a finite cyclic group of
order greater than two is a commutative group of even order.
4.8.11.H. Describe the groups of automorphisms of the cyclic groups of
orders 12 and 14.
4.8.12.T. Let G be a group, x E G. Prove that the transformation t x defined
by tx(a) = xax- 1 (a E G) is an automorphism of G.
4.8. 13.T. Prove that the set of inner automorphisms of a group G is a normal
subgroup of the group of all automorphisms of G.

Automorphisms of Groups
101
4.8.14. Describe the innerautomorphisms t Ul ' t U2 whereu 1 = (12),u2 = (132),
of the symmetric group S 3 .
4.8.15. Describe the inner automorphisms t- 1 , t- h t j of the quaternion
group (see 2.6.39).
4.8.16. Determine for which groups G the group of inner automorphisms of
G consists of the identity only.
4.8.17. What is the order of the group of inner automorphisms of the group
G = [(1234)(56), (13)]?
4.8. 18.T. Let Z be the center of a group G. Prove that the group of inner
automorphisms of G is isomorphic to G/Z.
4.8.19. Prove that the group of inner automorphisms of the symmetric
group Sn (n  3) is isomorphic to Sn.
4.8.20.H. Prove that every automorphism of the ymmetric group Sn
(n  3, n "# 6) maps a transposition onto a transposition.
4.8.21.H. Prove that every automorphism of Sn (n  3, n "# 6) maps the
set of distinct transpositions of the form (ij 1), (ij2)' . . . , (ijs) into a set of trans-
positions of the form (k/ 1 ), (kI 2 ), . . . , (kl s ).
4.8.22.0. Prove that every automorphism of S4 is inner. What is the oder
of the group of automorphisms of S4?
Remark. The first assertion of this exercise is valid for every symmetric
group Sn (n "# 6), However, S6 has an outer automorphism.
4.8.23. Let G be the additive group of Gaussian integers (see 4.3.22). Describe
all automorphisms of G.
4.8.24. Prove that the group of automorphisms of the .group G in 4.8.23
is isomorphic to the group of 2 x 2 matrices over the integers with deter-
minants + 1.
4.8.25. Let G' be a characteristic subgroup of a group G, and let a be an auto-
morphism of G. Denote by a. the following transformation of G': if g' E G',
then a.(g') = ag'. Prove that a. is an automorphism of G'.
Remark. We say that a. is induced by a.
4.8.26. Prove that the alternating group A4 and the Klein group (see
3.3.18) are characteristic subgroups of the symmetric group S4'
4.8.27. Find the automorphisms of the Klein group (see 3.3.18) which
are induced by the following automorphisms of S4 :
t(12)' t(34)' t(1423)' t(132)
Remark. Note that distinct automorphisms of a group can induce the
same automorphism of a characteristic subgroup.

102
Groups and Their Subgroups
4.8.28.H. Prove that distinct automorphisms of the symmetric group 8 4
induce distinct automorphisms of the alternating subgroup A4' What is the
order of the group of automorphisms of A4?
4.8.29. Prove that every characteristic subgroup of a group G is normal.
4.8.30. Let G be any group. Prove that the following subgroups of G are
characteristic:
1) G itself;
2) e (the identity subgroup);
3) the center of G ;
4) the commutator subgroup of G.
Remark. Compare this exercise with 3.4.14.
4.8.31.H. Let G be a noncommutative group. Prove that the group of auto-
morphisms of G is not cyclic.
9. Transitive Groups of Transformations
A group G of transformations of a set X is called transitive if for any
x, y EX, there exists a E G such that ax = y. If G is not transitive, it is said to
be intransitive.
We will use the following notation in this section: for x EX, G x is the
subset of all transformations in G which leave x fixed.
A group G of transformations of X is said to be imprimitive if (1) it is
transitive, and (2) there exists a proper partition of X into classes M i , M j , . . . ,
of which at least one M k has more than one element, such that for every
a E G and every class M k the set a(M k) is also one of these classes.
The sets M i , M j ,. . . are called imprimitive systems. We will say that these
sets form an imprimitive series.
If G is a transitive group for which no such partition of X exists, then we
call G a primitive group.
We shall also consider in this section doubly transitive groups, which
form a particular case of the so-called multiply transitive groups. A group G
of transformations of a set X is said to be doubly transitive if for any two
ordered pairs (a, b), (c, d), a, b, c, d E X, a =1= b, c =1= d, there exists a E G such
that aa = c, ab = d. Obviously every doubly transitive group is transitive.
4.9.1. Let G be a group of transformations of a set X, and assume that there
exists a E X such that for every be X, there exists a E G mapping a onto b.
Prove that G is a transitive group of transformations.
Remark. The converse is obvious. If G is a transitive group of trans-
formations, then each element a has the desired property.
4.9.2. Prove that the symmetric group of any degree and alternating groups
of degree greater than 2 are transitive.

Transitive Groups of Transformations
103
4.9.3. Let G be an intransitive group of transformations of a set X. Prove
that there exists a proper partition of X into classes M j , M j ,. . . such that if
xeM j and aeG, then axeM j and if X1,X2eMj, then there exists fJeG
such that fJx 1 = x 2 .
Remark. The sets M j , M j ,. . . are called intransitive systems.
4.9.4. Which of the following groups are transitive:
a. = [(123) (456), (1346)], Q9 = [(1234) (56), (123)]
0 3 = [(1234) (56), (123) (567)]
where G 1 and G 2 are groups of permutations on six elements, and G 3 is a
group of permutations on seven elements? Find the intransitive systems of the
intransitive groups.
Remark. It is easy to see that 6 2 is isomorphic to symmetric group S4'
Consequently the property of transitivity of a group is not always preserved
under isomorphisms.
4.9.5.T. Let G be a transitive group of transformations of a set X, x e X.
Prove y
x is a subgroup of G;
2) if ax = y (a e G), then aG x consists of all transformations in G which
map x onto y.
4.9.6.H. Let G be a transitive group of transformations of a finite set X of
cardinality n. Prove that the index of G x (x e X) in G is n.
Remark. Hence the order of a transitive group of transformations of a
finite set consisting of n elements is divisible by n.
4.9.7. Let G be a transitive group of transformations of a set X. Prove that:
1) G x and G y (x, y e X) are conjugate subgroups in G;
2) if G' is a subgroup of G which is a conjugate of G x , then there exists
ye X such that G' = G y .
4.9.8.H. Let G be a transitive group of transformations of degree n. Prove
that if a =1= e is contained in the center of G, then a is a regular permutation
(see 3.3.24) which does not have any fixed points.
4.9.9. Prove that a transitive group of permutations of degree n contains at
least n - 1 permutations, each of which does not have any fixed points.
4.9.10. Let G be a transitive group of order I of permutations of degree
n, n =1= I. Assume that each permutation in G different from the identity either
does not have any fixed points or has exactly two fixed points. Prove that if
m is the number of permutations in G which do not have any fixed points,
then m satisfies the inequality tl < m < ll.

104
Groups and Their Subgroups
4.9.11. Let G be an imprimitive group, and M i , M j , . . . an imprimitive series.
Prove that all the sets M i , M j ,. . . have the same cardinality.
4.9.12. Prove that the symmetric group of any degree and the alternating
groups of degree greater than 2 are primitive.
4.9.13. Prove that a transitive group of permutations of prime degree is
primitive.
4.9.14. Prove that the group [(abed), (ae)], where {a, b, e, d} is a permutation
of the numbers 1,2,3,4, is imprimitive, and find all of its imprimitive systems.
4.9.15. Find all primitive and imprimitive subgroups of the symmetric
group S4'
4.9.16. Prove that the group G 1 in 4.9.4 is imprimitive, and that it has a
unique imprimitive series.
4.9.17. When is the group G = [(12... n)] primitive?
4.9.18. Determine which of the following groups are transitive, and which
are primi ti ve :
1) the group of all isometries of the plane (ef. Chapter 3.5);
2) the group of all parallel translations of the plane;
3) the group of all rotations of the plane about one point.
4.9.19. Let p be an equivalence relation on a set X with the property that
for each x E X, there exist y, Z E X, y =1= x such that (x, y) E p, (x, z)  p.
Let G be a transitive group of transformations of X. Prove that if each ele-
ment in G is an endomorphism of X with respect to p, then G is imprimitive
(cf. Chapter 3.4).
4.9.20.T.H. Prove that if some G x is not a maximal subgroup of a transitive
group G (i.e., there exists a subgroup G' of G such that G x c G' c G, G' =1= G x ,
G' =1= G) then G is imprimitive.
4.9.21.H. Prove that if a proper transitive subgroup G of the symmetric
group Sn contains a transposition, then it is imprimitive.
4.9.22.T.H. Prove that if G is a transitive group of permutations of prime
degree p and if G contains a transposition, then G is equal to the symmetric
group Sp'
4.9.23. Let G be an imprimitive group and M an imprimitive system.
Prove that the set of all transformations a E G having the property a(M) = M
is a subgroup of G.
4.9.24. Let G be an imprimitive group and H the set of all transformations in
G which map each imprimitive system of an imprimitive series into itself.
Prove that:
1) H is a normal subgroup of G;
2) H is an intransitive group.

Transitive Groups of Transformations
105
4.9.25. Let G be an imprimitive group which has a finite imprimitive series
M 1 , M 2' . . . , M n' Let cp be the mapping of G into the symmetric group Sn
defined in the following way: if a E G and a(M 1) = M h , a(M 2) = M i2 , . . . ,
!X(M n ) = M in , then
cp(a) = ( ..  ...  )
II l2 ... In
Prove that cp is a homomorphism of G into Sn. Find the set of elements of G
which are mapped onto the identity by cp (see the preceding exercise).
4.9.26.H. Prove that every normal subgroup N "# e of a primitive group of
transformations G is transitive.
4.9.27.H. Prove that the groups described in 4.9.24 exhaust the set of all
intransitive normal subgroups of an imprimitive group.
4.9.28. Let G be an intransitive group of transformations of a set X and
let M i be an intransitive system (see 4.9.3). For every a E G denote by ai
the transformation of M i defined by aix = ax. Prove that the set G i of all
transformations ai is a transitive group of transformations of Mi'
4.9.29. Assume the conditions and notation of the preceding problem. Let
r be the Cartesian product of the groups G i , G j ,... which correspond to all
intransitive systems of G. Prove that:
1) r is a group with respect to the following operation:
(a h a j , · . ,)(Ph Pj,. · .) = (aiPi' ajpj,. . .)
2) G is isomorphic to some subgroup of r.
Remark. Thus every intransitive group is, up to isomorphism, a subgroup
of the Cartesian product of transitive groups. The group G is called a sub-
direct product of the groups G i , G j , . . . .
4.9.30. Let G be a group of transformations of a set X, x, Y E X, X "# y.
Prove that if for every pair u, v E X, u "# v, there exists a E G such that ax = u,
!XY = v, then G is doubly transitive.
4.9.31. Which of the following groups are doubly transitive:
1) the symmetric group Sn;
2) the group G 1 in 4.9.4;
3) the group of all isometries of the plane (see 4.8.18)?
4.9.32. Determine for what values of n the alternating group An is doubly
transitive.
4.9.33. Determine the number of doubly transitive groups of transformations
vf degree four.
4.9.34.H. Prove that the order of a doubly transitive group of transforma-
tions of degree n is divisible by n(n - 1).

Chapter 5
DEFINING SETS OF RELATIONS
1. Defining Sets of Relations on Semigroups
Let S be a semigroup and K a subset of S. We will consider words in S
over K (cf Chapter 2.5).
If u = X I X 2 . . . X n and v = YIY2 . . . Ym are words in S over K, then by
uv we mean the word Xl · . . XnYl . · · Ym'
For K, a subset of a semigroup S, and X t X2 ... X n , a word over K, the
element in S which is the product of the elements Xl' X2,' . . , X n is called the
value of the given word in S (cf Chapter 2.5).
If S = [K]s, then each element in S is the value of some word over K.
Moreover, it is possible for an element in S to be the value of several words
over K. Ifx1x2 . . . X n and YIY2 . . . Ym are two words which have the same value
in S, then the equality
X 1 X 2 ... X n = YIY2 ... Ym
holds in S.
Every such equality is a relation on the semigroup S with respect to the
set K. In other words, a relation is a pair of words having the same value in S.
We will say that the words XtX2 . . . X n and YIY2 . . . Ym form a relation,
and write this either as Xt X 2 . . . X n = YtY2 . .. Ym or YIY2 .. . Ym = Xt X 2' . . Xn'
We will not write "with respect to K" ifit is clear from the context over
what set the word is being considered.
Let S = [K]s, and assume that the generating set K has the property
that each word over K has a unique value in S; then K is called a free generat-
ing set of the semigroup S. A semigroup is called a jree semigroup over K, or
simply a free semigroup, if it has a proper generating set K.
Let S = [K]s, and let F be a set of relations on S with respect to K. If
W t = W 2 is a relation in F, and VI' V2 are any words over K, then the following
relations obviously hold in S:
° 1 - 1 = fl 1 W j , w 1 f1 :a = wJv J , fl 1 - 1 V :a = fJ 1 'WSfJ 1
107

108
Defining Sets of Relations
Every relation in such a form, and also every relation of the form w = w,
where w is a word over K, is called a direct consequence (or immediate corol-
lary) of F. A relation Ul = U2 is said to be a consequence of F if there exists a
finitesequenceofrelationswithrespecttoK:u 1 = Vt'V I = V 2 ""'V n = U2,
in which each relation is a direct consequence of F.
It is obvious that if the words Ut and U 2 form a relation which is a con-
sequence of some set of relations in S, then U t and U2 will have the same value
in S. The converse does not always hold, i.e., two words VI and V 2 can have
the same value in S although the relation V t = V 2 need not be a consequence
of the given system of relations.
A set of relations F on a semigroup S with respect to a generating set K
is called a defining set oj' relations oj' the semigroup S with respect to K, or a
defining set of relations, if every relation on S with respect to K is a con-
sequence of F.
A defining relation is any relation in a defining set of relations. The
significance of the concept of a defining set of relations lies in the fact that the
defining set of relations determines the semigroup up to isomorphism (see
5.1.19).
In the problems we will encounter symbols of the form uo, which will be
omitted when they appear in a word. For example xOy5 will be written y5.
5.1.1. Let S = [u, v, wJs be a semigroup of transformations, where
u=(1 2),
'V = (n, n - 1, ..., 1 ),
W= ( 1 2 3 ... n )
1 1 3 ... n
Determine which of the following pairs of words over {u, V, w} form relations
on S:
tt = WU, t'J = vn-'Juv'Jwvn-iuv'J, t s = wvn-1uvw
t i = 'VlIV n - 1 WVUv n - 1 w, t s = 'V n - 1 uvwV n - 1 11VW, t 6 = (Vll)n-l
5.1.2. Let a, b, and c be distinct elements in a semigroup S, and assume that
the relations
a 6 = a 3 , a'Jb = atj, a 7 cb 6 a 3 bll = aBc, bca 3 = cba'J
aic = caS., aSea = a 3 c., be = eb, ba 2 = b
hold in the semigroup [a, b, c Js' Determine which of these relations is a con-
sequence of the others.
5.1.3. Let S be the semigroup in the preceding problem. Prove that the
following relations hold in S:
a'},e = at2bUeal0, (baS)'" = b, (abca)S = ac'Jbaca l

Defming Sets of Relations on Semigroups
109
5.1.4. Let S = [KJs and let F be a set of relations in S with respect to K.
Prove that:
1) the relation u = u is a consequence of F;
2) if u = v is a consequence of F then v = u is also;
3) if u = v and v = ware consequences of F, then so is u = w;
4) if u = v is a consequence of F, and W l , W 2 are any words in S over
K, then the relations W l u = W l v, UW 2 = VW 2 , and W l UW 2 = W l VW2
are consequences of F;
5) if U l = Vl and U 2 = V 2 are consequences of F, then so is U l U 2 = V l V2'
5.1.5.". Let S = [a, b, c, dJs be the semigroup in 3.1.36. Show that the follow-
Ing relations hold in A :
ab=ba, ac=ca, ad=da, bd=db, a 8 =a, ad=d, c 3 =c
a'J = c 2 , bd = d'J, b 4 c 3 a s d' b 3 a'Jd 8 cd = b"'cd l8 Cd
ba 3 cb'Jabda'J = ba 3 cb'J a d'J a 'J, c8b6a3c6a'Jb4 = ab 9
5.1.6.". Does the following set of relations
ab=ba, ad = da, bd=db, ad=d
bd=d i , ac=ca, a 3 =a, c 3 =c
form a defining set of relations for the semigroup S in the preceding problem?
5.1.7. Let S' = [a, b, d]s be a subsemigroup of the semigroup S in 5.1.5.
Prove that the set
ab=ba, ad=da, bd=db
ad=d, bd=d'J, a'Jb=b, a 3 =a
is a defining set of relations for S'.
5.1.8. Suppose the following relations hold in the semigroup S = [Xl' X 2 ]s:
X ._ X I X 4- I X I X -  X 2_ X
1 - I' i - -"'i' I i - -"'g I - 'J
X;XI = XIX; = XI' (XIX'J)3 XI = XI
Prove that S is finite.
5.1.9. Assume that a semigroup S has a finite generating set K, and that for
some natural number n, every sequence of 2n + 1 elements Xl' X 2 , . . . ,
X 2n + 1 E K satisfies the relation
XIX, ... x n =- Xn+IXn+i ... X'n+l
Prove that S is a finite semigroup and that S has a zero.

110
Defining Sets of Relations
5.1.10.T. Let X be any set, and C x the set of all finite sequences of elements
in X. Define a multiplication on C x by: if a = (Xl' X 2 , · · ., x n ), b = (YI'
Y2, . · . , Ym)' then ab = (x I , · . · , X n , Y I , · . . , Ym)' Prove that:
1) C x is a free semigroup;
2) C x has a unique free generating set;
3) if 8 is an arbitrary free semigroup, then there exists a set X such that S
is isomorphic to C x .
Remark. Thus every free semigroup has a unique free generating set.
5.I.ll.T. Prove that every semigroup is the homomorphic image of a free
semlgroup.
5.1.12. Let 8 = [x]s be a finite cyclic semigroup. Prove that there exists a
defining set of relations F on 8 with respect to the generating set {x} such that
F consists of exactly one relation.
5.1.13. Let N be the multiplicative semigroup of natural numbers. Let P
be the irreducible generating set consisting of all primes and the number one
(see 2.5.3). Prove that the set of all relations in the form pq = qp, pi = p
(p, q E P) forms a defining set of relations on N with respect to P.
5.1.14.T. Prove that in each semigroup there exists a definIng set ofrelatins
with respect to any generating set.
5.1.15. Assume that the set
X+l=Xk' XkXl=X1X k
for k = 1, 2, . . . , 10, and I = 1, 2, . . . , 10, is a defining set of relations for the
semigroup 8 = [Xl' X2,. . . , XIO]s. Prove that 8 is a finite commutative semi-
group having neither an identity nor a zero. How many elements are in 8?
5.1.16. Suppose that the semigroup 8 = [u, v]s has the following set of
defining relations with respect to {u, v} :
1l'J = ll, v'J = V, IIVII = U, vuv = v
1) Construct the Cayley table for 8.
2) Is 8 regular (cf Chapter 2.5)?
5.1.17. Let 8 = [K]s and let F be a set of relations on 8. Let cp be a mapping
of the set K into a semigroup T. Replace each element in every relation on S
by its image under cp. Prove that if this substitution yields a relation on T for
every relation in F, then it also yields a relation on T for every consequence
of F.
5.1.18.T.H. Prove that if the set F in the preceding problem is a defining set
of relations for 8, then the mapping cp can be extended to a homomorphism of
8 into T (cf. Chapter 2.2).

Defining Sets of Relations on Semigroups
111
5.1.19.T.H. Let F l' F 2 be defining sets of relations for the semigroups
S 1 , S 2 with respect to the generating sets K 1 , :( 2' respectively. Assume that
there exists a one-to-one mapping cp of K 1 onto K 2 such that if in every
relation in F 1 each element in K 1 is replaced by its image under cp, then a
relation in F 2 is obtained. Prove that if, in addition, every relation in F 2 can
be obtained in such a manner, then S1 an S2 are isomorphic.
5.1.20.H. Let S be a semigroup with a generating set K and a defining set of
relations of the form
ltVU = 'lJU'V
(a, 'V E K)
Is S commutative?
5.1.21. Given the semigroup S = [a, b, c]s, and a defining set of relations F
for S,
a'l = a,
ab = ba,
ac = ca,
bc=c
prove that:
1) every element in S can be written uniquely in the form akcmb n , where
k = 0, 1 ; m and n are nonnegative integers, and k, m, and n do not all
equal zero;
2) S has no identity.
5.1.22. Let S be the semigroup defined in the preceding problem. Find:
1) all idempotents in S (cf. Chapter 2.2):
2) all elements in S which commute with acmb n (where m and n are
fixed natural integers);
3) the types of all cyclic subsen1igroups of S (see 2.5.10).
5.1.23. Let there be given a semigroup S' = [a, b, c], and let F' be a defining
set of relations for S' consisting of all relations in the set F in 5.1.21 and the
relation c 5 = a.
1) Prove that S' is a homomorphic image of the semigroup S defined in
5.1.21.
2) Find the type of each element in S' (see 2.5.10).
3) Determine which elements generate cyclic subgroups of the semi-
group S'.
5.1.24.8. Let S be a free semigroup. Prove that:
1) S does not have an identity;
2) S is both left and right cancellative (cf. Chapter 2.2);
3) each element in S has a finite number of distinct left or right divisors.
5.1.25. Prove that a free semigroup has a unique irreducible generating
set.

112
Derming Sets of Relations
5.1.26. Let S = [a, b]s, and let the set
ba 9 = a,
aba = a,
bab = b,
b'1.a = b,
(ba)'1.=ba
be a defining set of relations for S. Prove that:
1) ba is the identity of S;
2) the only elements which ommute with a are powers of a and the
identity ba;
3) the only elements which commute with b are powers of b and the
iden ti ty ba;
4) for every XES, x =1= a, x =1= b, there exists an element y =1= ba which
commutes with x and is not a power of x.
5.1.27.H. Let S be the semigroup in the preceding problem. Find:
1) all regular elements in S (cf Chapter 2.5);
2) the inverse of each regular element;
3) all automorphisms of S.
5.1.28. Prove that the semigroup [ex, P]s in 3.1.13 is isomorphic to the
semigroup S defined in 5.1.26.
2. Defining Sets of Relations on Groups
Let G be a group and let K be a generating set of G in the sense of group
theory, i.e., G = [K]g. If K' is the set of all elements in G which are inverses of
elements in K, then G = [K U K']s (cf Chapter 2.6).
By considering the group G as a semigroup with K U K' U e as its
set of generators, where e is the identity 01G, we can apply all the terms
introduced in the preceding section to G (the concepts of word and relation
over the generating set K U K' U e, a consequence of a set of relations, a
defining set of relations, etc.).
However, we employ a different terminology in group theory. A word in
a group G over the set K U K' U e is called a word over K, and a relation on G
with respect to K U K' U e is called a relation on G with respect to K. In
what follows we will adhere to the "group" terminology, adding the phrase
"in the sense of group theory" where possible ambiguity might arise; we
will do the same for generating sets (cf Chapter 2.6).
Let G = [K]g, and let F be a set of relations on G with respect to K in the
sense of group .theory. A relation u = v is called a consequence of F in the
sense of group theory if it is a consequence (cf Chapter 5.1) of the set con-
sisting of all relations in F and all relations of the form i
ke= ek= k, kk- 1 =k- 1 k =e (kEKU KUe)

Defining Sets of Relations on Groups
113
Hence we consider as consequences of a set of relations F on a group G the
natural group relations given above in addition to the relations appearing
in F.
One should note the following simple, yet important, property. If
X 1 X2 · . · X n = YIY2 . . . Ym is a relation in a group, then the relation X 1 X 2 . . .
x"y'; 1 . . . Yl 1 = e is a consequence of it. Conversely, the first relation is a
consequence of the second (see 5.2.4). As a result, relations on a group are
usually given in the form u = e, where u is a word. Also note that each of the
I . d -1 -1 -1' f h h
re atlons Z l Z2 . . . Zn = e an Zn . . . Z2 Zl = e IS a consequence 0 t e ot er
(see 5.2.4).
We will consider the empty set of relations as a set of relations on a
group. If a relation u = v is a consequence of the empty set of relations in the
sense of group theory, then it is a consequence (see Chapter 5.1) of the natural
group relations given above.
If every relation on a group G with respect to a generating set K is a
consequence of a set of relations F in the sense of group theory, then F is called
a defining set of relations of G with respect to K.
The description of a group by a generating set and a defining set of
relations determines the group up to isomorphism (see 5.2.7).
Clearly the theory of relations in the sense of group theory is a particular
case of the theory of relations in the sense of semigroups. As has been already
noted, a different terminology is employed in group theory, partly because of
simplicity of notation, and partly due to tradition.
5.2.1. Let a = (12), b = (12345). Prove that the following relations hold in
the group [a, b] = 8 5 :
a'J = e, b a = e, (ab)" = e, (ab-'Jab'J)'J = a'J, (ba)" = e
5.2.2. Assume that the following relations hold in the group G = [u, v]:
fig = e, va = e, (uv)" = e, (uv-iuvg)g = IZg
Show that the following equalities are valid in G :
(Vll)4 = e, llv-tzlV g = v-gllVgll,
v- 1 uv = V 3 11V- I UV- l ll, (1IVIIV- I )3 = e
Remark. In particular, the following equalities hold in the group [a, b]
in S.2.1,
(ba)i = e, ab- 2 ab 2 = b- 2 ab 2 a,
b-tab = b 3 ab- t ab- 1 a, (abab- 1 )3 = e

114
Defining Sets of Relations
The validity of these last equalities can be cal<;:ulated directly in [a, b], but
after verifying 5.2.2, such a check is not necessary.
5.2.3. Let x and y be elements in a finite group G connected by the relation
yx = xl, where k is an integer different from O. Prove that every element in
the group [x, y] can be written in the form xmyn (where m and n are integers).
5.2.4.H. Let G be any group. Prove that:
1) if the relation X 1 X2'.' X n = YlY2 . . . Ym is valid in G, then so is
XI X 2 . . . XnY 1 . . . Y 11 = e, where each of these is a consequence of
the other in the sense of group theory;
2) if the relation Z1Z2 ... Zn = eholdsin G,thensodoesz; 1 ... Z21Z11 =
e, and each relation is a consequence of the other in the sense of
group theory.
5.2.5. Let G = [a, b], and
at. = b 3 = (ab)'J = e
Prove that:
1) the elements a 2 , b- 1 a 2 b, ba 2 b- 1 are pairwise commutative, and forma
conjugate class;
2) the subgroup [a 2 , b] is a normal subgroup of G.
5.2.6. Let G = [aI' a2, . . . , an-I]' and suppose the relations
akaj=ajak (k, j=l, ..., n-l; Ik-jl>l)
ai a i+l Q i = Qi+laiai+l (l = 1, 2, ..., n - 2)
hold. Prove that:
1) the following equalities hold in G :
ai 1ai+lai = ai+laia"iI' ai b = bai_b al b9 = b 9 a n _t
where b = a 1 a2 . . . an-I' and ai is arbitrary;
2) b n is in the center of G.
5.2.7.T.H. Let there be given the groups
0 1 = [/(d g = [K. UK; U eds
a= [K2]g= [K U K U e2]S
and defining sets of relations F l' F 2 of G 1 , G 2 with respect to the generating
sets K 1 , K 2 , respectively, in the sense of group theory. Prove that if there
exists a one-to-one mapping cp of the set Kl U K U e 1 onto K 2 U K; U e2
satisfying
cp (el) = e2, cP (ki 1 ) = [1> (k 1 )]-1 (ki E K 1 )

Defining Sets of Relations on Groups
liS
and the set F 1 is in a one-to-one correspondence with F 2' then G I is isomor-
phic to G 2 .
5.2.8.T.H. Let there be given a group G = [Kg] = [K U K' U e]s, and a
defining set of relations F of G witp respect to K. Let cp be a mapping of the
set K U K' U e into a group G', where each relation in F and each relation
of the form ke = ek = k, kk- 1 = k-1k = e (keK U K' U e) is mapped
onto a relation on G'. Prove that if G' = [cp(K)]g, then G' is isomorphic to a
factor group of G.
5.2.9. Let a = (1234), b = (132) be elements in the symmetric group 8 4 ,
Prove that:
1) 8 4 = [a, b] ;
2) the relations a 4 = b 3 = (ab)2 = e hold in 8 4 ,
5.2.10. Let G be the group in 5.2.5, and let c = a 2 ba- 1 . Show that the follow-
ing equalities hold in G:
c 9 = e, a- 1 b- 1 a- 1 = b, b- 1 e = cb, be = eb- 1 , ae = eb- 1 a
a-Ie = b- 1 a t , a- 1 b = beat, atb = ea, ab- 1 = ea'J
5.2.11. Assume the conditions and notation from the preceding problem.
Show that every elenlent in G can be written in the form ct% 1 bt% 2 a t%3 , where
o  eli  i, i = 1, 2, 3.
5.2.12. Assume the conditions and notation of 5.2.10. Suppose that the
equalities
cf1.1bf1. 2 af1.3 = elb2a8 (0  r:l;  I, 0  ,  i, 1= 1, 2, 3)
hold in G only if ell = PI' el2 = P2' and el3 = P3' Prove that the set of relations
a" = b 3 = (ab)'J = e
is a defining set of relations on G.
5.2.13.H. Prove that the set of relations
a"=b 3 =(ab)t=e
is a defining set of relations for the symmetric group 8 4 with respect to the
generating set {a,b}, where a = (1234), b = (132).
Remark. Problems 5.2.11 and 5.2.12 should be borne in mind for those
problems which verify that a given set of relations forms a defining set of
relations.

116
Defining Sets of Relations
5.2.14. Let K be the quaternion group. Prove that:
1) K = [i, j] ;
2) all of the following relations in the set F below are valid in K:
F = {i'= 1, j'= 1, i'J=ji, ijl=j}
5.2.15. Prove that the set of relations F in the preceding problem is a defining
set of relations for the quaternion group K.
5.2.16. Let G 1 = [a, b, e] c 8 6 , where a = (12), b = (34), e = (56). Prove
that the set of relation
ai=b'J=c'J =e, ab=ba, ac=ca, be =cb
is a defining set of relations for G 1 with respect to the generating set {a, b, c}.
5.2.17. Let G 2 = [a,b] c 8 6 , where a = (1234), b = (56). Prove that the
set of relations
a'=b'J=e,
ab = ba
is a defining set of relations for G with respect to the generating set {a, b}.
A
5.2.18. Suppose the following relations hold in the group G = [a, b]:
at = b i = (ab)' = e
Prove that:
1) the order of G is less than or equal to eight;
2) (ab)2 lies in the center of G.
5.2.19. Let G = [(1234), (13)] and a = (12)(34), b = (13). Prove that G =
(a, b], and the set of relations a 2 = b 2 = (ab)4 = 'e is a defining set of relations
for G.
5.2.20. Let G be the group in 4.6.16. Prove that G = [Zl' Z2], and that the
set of relations z = Zo, Z l Z2 = Z21Z1 (where Zo is the identity of G) is a
defining set for G.
5.2.21.". Let [a] and [b] be infinite cyclic groups,
G = {(a CZ , b P ): aCZ E [a], b P E [b]}
Define an operation on G by:
(ar.tJ, b1) (ar.t 2 , b2) = (ar.t1 +aj, bpl (_1)r.t2 +2)
Prove that:
1) G is a group;
2) if e = (a, bO), d = (aO, b), then G = [e, d], and the set consisting of
the relation de = ed- 1 is a defining set for G.

Free Groups
117
5.2.22.H. Let [a] and [b] be cyclic groups of orders m and n, respectively,
and let r be any natural number. Let
Gn = {(a, b P ): a E [a], P E [b]}
and define an operation on G by
(ar.t t , bl) (a Ct2 , b2) = (ar.t 3 , b3)
where
as = al +  (mod m), s = lrt1 + 2 (mod n)
Prove that
1) G" is a group if and only if",. = 1 (mod n);
2) if Gn is a group, then Gn = [c, d], where c = (a, bO) and d = (aO, b);
3) the set of relations
cm=e, d"=e, d=cd,.c- 1
v/here e is the identity of Gr: n , is a defining set of relations for Gn.
5.2.23.H. Let p and q be primes, p < q. Prove that the groups Gq and G:q
in the preceding problem are isomorphic for any rand s which satisfy the
conditions
r P = 1 (mod q), sP = 1 (mod q)
3. Free Groups
A generating set K of a group G is called a free generating set if all of its
elements are different from the identity and the empty set of relations with
respect to K is a defining set of relations for G in the sense of group theory.
As was noted in Chapter 5.2, this means that every relation in G with respect
to K is a consequence of the natural group relations: ke = ek = k, kk- 1 =
k-1k = e, where k E K, k = e, or k is the inverse of an element in K.
A group is said to be free if it has a free generating set. It will be proved
in 5.6.31 that if a free group G has a free generating set with n elements, then
every other generating set of G also has n elements.
The number of elements in a free generating set of a free group G is
called the rank of G. If a free generating set is infinite we say that the free
grou p is of infinite rank.
In problems 5.3.1-5.3.6 we will illustrate a method for constructing
free groups. This will enable us to exhibit the existence of free groups of
any given rank.

118
Defining Sets of Relations
The importance of free groups stems from the fact that every group is
isomorphic to a factor group of a free group (see 5.3.13). In addition, the
concept of a free group is sometimes used in the definition of a defining set
of relations on a group (see 5.3.15 and 5.3.16).
One should keep the following remarks in mind when considering "free
semigroups" (introduced in Chapter 5.1). A free semigroup can never be a
group. Of course a free group is a semigroup, but it is not a free semigroup
(see 5.3.21).
5.3.1. Let F K be a free semigroup with free generating set K. Let cp be a one-
to-one mapping of K onto itself, with cp(k) =1= k, cp2(k) = k for all k E K.
Define the binary relation P on F K to consist of the pairs
(Wb kr.p (k) WI)' (w 'W'jkr.p (k», (Wtw'j, 'Wlkr.p (k) 'Wg), (Wb w.)
(kr.p (k) Wb WI), (Wikr.p (k), Wi), (Wtkr.p (k) WI, 'WIWI)
where k E K, WI' W2 E F K' Prove that:
1) P is reflexive and symmetric;
2) the transitive closure p' of p is a congruence on F K'
Remark. From 2.4.18, 2.4.14, and 2.4.8 it follows that F K = FK/P' is a
semIgroup.
5.3.2. Assuming the conditions and notation of the preceding problem,
show that for k 1 , k 2 E K, k 1 =1= k 2 , we have (k 1 , k 2 )  p'.
5.3.3. Assuming the conditions and notation of 5.3.1, prove that the semi-
group F K = F K/ P' is a group. Which class is the identity? Which class IS the
inverse element of the class containing k E K?
5.3.4.". Prove that the group F K in 5.3.3 is a free group.
5.3.5. Prove that if two sets K and L have the same cardinality then F K is
isomorphic to F L (see 5.3.3).
5.3.6. Prove that for every free group G, there exists a set K such that G is
isomorphic to the group F K defined in 5.3.3.
Remark. Hence groups of the form F K exhaust all free groups up to
isomorphism.
5.3.7. Let F be the free group with free generating set {a, b}. Prove that the
normal subgroup of F generated by {a 2 , b 2 , (ab)2} (cf Chapter 43) consists
of all words afZlbfJlafZ2bfJ2... afZkb fJk for which III + ll2 + . . . + llk and PI +
P 2 + . · . + P k are even.
5.3.8.". Let F be the free group with {a, b} its free set of generators. Find
the index of the following subgroups of F:
1) HI = rag, bl, ab);

Free Groups
119
2) Hi=[a];
3) H3 = [ai, bi, a-tbia, b-taib, (ab)']
Which of these subgroups are normal?
5.3.9. What kind of free groups are commutative?
5.3.10. Let F be a free group of rank greater than one. Prove that:
1) every element in F different from the identity has infinite order;
2) the center of F is the identity subgroup.
5.3.11. Let K be a free generating set for a group F. Prove that every mapping
of K into F can be extended to an endomorphism of F (cf. Chapters 1.2
and 3.4).
5.3.12.H. Prove that every free group of rank greater than one has an
infinite number of free generating sets.
5.3.13.T.H. Let G = [K] be a group. Prove that G is a homomorphic image
of a free group which has a free generating set of the same cardinality as K.
5.3.14. Let F be a free group with K a free generating set of F, cp a homomor-
phism of F onto a group G, and K the set of generators of G consisting of all
qJ(x) (x E K). Let H be the normal subgroup of F consisting of all elements
which are mapped onto the identity by cpo Prove that:
1) if
XIX;J ... x:nE H
then the relation
q> (Xl)tl 1 q> {Xi)tl s ... q> (xn)tl n = e
holds in G with respect to K ;
2) if
q> (Xl)'l t ... q> (xn)'An = q> (Yt)P 1 ... q> (Ym)P m
where
Xb Xi' ..., X n , Yh ..., Ym E K
then
X flt X tln Y -m Y -l E H
t ... n m ... 1
Remark. If
- tl 1 flS tin E h
U-X 1 X i ... X n

120
Defming Sets of Relations
then we will call the relation
q:> (XtYl l q:> (Xi)(J.j ... q:> (xn)(J.n = e
the relation corresponding to u in F.
5.3.15. Assume the conditions and notation of 5.3.14. In addition, let M
be a set of generators of H (cf. Chapter 4.3). Prove that the set of all relations
on G corresponding to elements in M is a defining set of relations of G with
respect to K .
5.3.16. Assume the conditions and notation of 5.3.14. Let D be the defining
set of relations of G with respect to K consisting of relations of the form
u = e. Prove that the set of all elements in F which correspond to a relation
in D generates H (cf. Chapter 4.3).
Remark. The set of relations on G corresponding to elements in M (see
5.3.15) is sometimes called the defining set of relations of G.
5.3. 17.H. Let F be a free group of rank n, and let H be the subgroup of F
generated by the squares of all elements in F. Prove that H is a normal
subgroup of F, and find the order of F/H.
5.3.18. Let H be the normal subgroup in the preceding problem. Prove that if
{Xl'" ., xn} is a free generating set of F, then H, as a normal subgroup, is
generated by
{xl, ..., x:. (XIXg)i, ..., (Xn_IXn)i}
(cf. Chapter 4.3)
(cf. Chapter 4.3).
5.3.19.H. Let {x, y} be a free generating set ofa group F, and H a completely
characteristic subgroup of F containing x2yxy-l. Prove that if H =1= F,
then the index of H in F is 9.
5.3.20. Let {x I , . . . , xn} be a free generating set of a group F. Prove that
each of the following mappings of the set {x I , . . . , xn} onto itself can be
extended to an automorphism of F:
1) q:>l} (Xi) = Xi' Cf(l} (x}) = x,, q:>,} (Xk) = Xk,
2) l (Xl) = xi l , l (x}) = x},
3) 6 1j (x}) = XiX} (I '# j), 61} (Xk) = Xk'
k -:j:. i, j
l-:j:.j
k:j:.}
where i,j = 1, 2, . . . , n (cf. Chapter 1.2).
Remark. It is well-known that these automorphisms generate the group
of automorphisms of the free group F.
5.3.21. Prove that a free group is not a free semigroup.

Groups Dermed by Sets of Relations
121
5.3.22. Prove that a necessary and sufficient condition for a group F to be
free is for F to have a generating set K with the property that every mapping
of K into any group G can be extended to a homomorphism of F into G.
4. Groups Defined by Sets of Relations
Let K be an arbitrary set. A word over K is a finite sequence of elements
in K written on a line with no symbols separating them.
The concept of a word over a set was considered earlier (cf Chapter 2.5
and in the earlier sections of Chapter 5), but only for the case where the given
set was a subset of a semigroup.
For a given set K select an arbitrary, but fixed, set K' which has the same
cardinality as K, and such that K n K' = 0. Let t/J be a one-to-one mapping
of K onto K', and let e be any element such that e rt K UK'. Consider a
binary relation p on the set of all words over K U K' U e. The question arises
whether there exists a group G with the following properties: K is a generating
set of G, e is the identity, t/J(k) is the inverse of k for all k E K, and the set of all
relations U 1 = U2, where (u 1 , u 2 ) E p, is a defining set for G in the sense of
group theory.
We shall show that such a group G always exists (see 5.4.1). Moreover,
from 5.2.7, G is unique up to isomorphism.
One should keep in mind that it is possible for various elements in K
to be equal in this group. In particular, the group can consist of the identity
alone.
On the other hand, if G is a group and K is a generating set of G, then
there always exists a defining set of relations on G with respect to K in the
sense of group theory; for example, the set of all relations on G with respect
to K.
If K is a generating set and F is a defining set of relations on a group G
with respect to K, then we say that G is defined by the generating set K and the
set of relations F.
It is natural to ask certain questions concerning the group G which is
defined by a generating set K and a defining set of relations F. The most
important of these is : how can one determine if two given words in G have
the same value with respect to K? If G is concrete, i.e., if its elements (and
how they are multiplied) are known, then no such question arises. In fact,
we can always determine whether the values of two words in G are equal by
conlputing them directly.
If a method (algorithm) can be devised for testing whether the values of
any two words are equal in G then we say that the word problem is solvable
in G. The great amount of difficulty which often occurs when determining the
solvability of the word problem is well-known. In fact, there are some

122
Defining Sets of Relations
classes of groups for which the word problem is not, in general, solvable, i.e.,
no such algorithm as stated above exists for all pairs of words.
The second most important question concerns when two groups defined
by generating sets and sets of relations are isomorphic.
Finally, we raise the questions: if G is defined by a generating set and a
set of relations, is it finite? Is it commutative? Does it reduce to the trivial
group?
5.4.1.T.8. Let K be any set, and K' a set with the same cardinality as K,
with K n K' = 0. Let e be any element such that eft K UK', and let t/J
be a one-to-one mapping of K onto K'. Prove that for every binary relation
p on the set of all words over K U K' U e, there exists a group G such that
G = [K]g, e is the identity of G, t/J(k) = k- l for every k E K, and the set of all
relations U I = U 2 , where (u I , u 2 ) E p, is a defining set of relations on G with
respect to K. Y
5.4.2.T. Assume that the group G I is defined by a generating set K and a
set of relations F I ; and that the group G 2 is defined by the same generating
set K and a set of relations F 2' Prove that if Fie F 2' then:
1) G 2 is a homomorphic image of G 1 ;
2) if every relation in F 2 is a consequence of the set of relations F I'
then G 2 is isomorphic to G I .
5.4.3.8. Which of the following groups are commutative:
1) G I , defined by the generating set {Xl' X2, X 3 ,. . .} and the set of
relations
xi = Xb xi = Xi' ..., X = Xn_b ...
2) G 2 , defined by the generating set {x, y} and the relation xy2 = y2x;
3) G 3 , defined by the generating set {Xl' X 2 , X3, . . .} and the set of
relations
X 9 _ X 9 X 9_ X 9 X 9_ X 9
1 - 2' 1 - 3"'. , 1 - n,."
5.4.4.8. Prove that all three groups in the preceding problem are infinite.
5.4.5.8. Determine the order of the group defined by the generating set
{ X, y} and the set of relations
xyx-iyx = e, y3 = e
5.4.6.8. Let a group G be defined by the generating set {x, y} and the rela-
tion xy = yx S , where s is any integer different from zero. What is the order'
of the subgroup [y]?

Groups Defined by Sets of Relations
123
5.4.7.H. Assume that a group G is defined by the generating set {x, y} and
the relation x 2 = y2. Which of the following elements are equal in G:
III = (xy)3, U2=YXYXY', l'3 = x 3 y3, u,=xy 9 x 3
"6 = x9y-t x 3 yx, U6 = x- 1 yx 6
5.4.8.H. Let a group G be defined by the generating set {x, y} and the set
of relations
X'J. = y2, xy6 = y3 Xl
Which of the following elements are equal in G :
-I -1 ....I I -1 -I -1
tJl=X Y xy, Vi=-' V3=Y' Vi=XY xY XY
VG=x9yt, V6=(xy)3, V7=X 3 y3, V8=X'-l yx 6
Remark. Compare this result with 5.4.7.
5.4.9. Let G = [K], where K is a finite set, and let the set of relations
x 9 =e, (xy)t=e
(x, y E K)
be a defining set for G. Prove that G is a finite group, and find its order.
5.4.10.H. Let a group G be defined by the generating set {x, y} and the
relationsxy = y-lx,x2 = y2.
1) Prove that every element in G can be written in the form xiyi, where
i = 0, 1 and j is any integer.
2) Determine when two elements x il y/1 and xi2yj2 (i 1 ,i 2 = 0,I;jl,j2
arbitrary) are equal, and when one is the inverse of the other.
5.4.11. Prove that the group G = [a, b] with the defining set of relations
a 3 = e, b'1 = e, a-tba = b l
is a cyclic group of order 3.
5.4.12. Let a group G be defined by a generating set K and a set of relations
F, and let y = x IX2... x:' m , where y E G, Xl' X2'.'" X m E K. Prove that K' =
K U {y} is a generating set of G, and that the set consisting of all relations
in F and the relation y = XI X2 . . . x:, m is a defining set of relations of G with
respect to K'.
5.4. 13.T. Let a group G be defined by a generating set K and a set of rela-
. F h . h . h I t . al a2 a ( K
tlons , w IC contaIns t e re a Ion y = Xl X2 ... X m m y, Xl' . . . , X m E ;
al, . . . , am integers). Let F' be the set of relations obtained by substituting
the word XIX2 . . . x:, m in place of y in all relations in F. Prove that the set

124
Defming Sets of Relations
K' = K"{y} generates G. Further, show that F' is a defining set of relations
of G with respect to K'.
5.4.14.8. How many elements of finite order and how many of infinite order
lie in the group G defined by the generating set {x, y} and the relation
(xy)k = e (k natural number)?
5.4.15.8. Find all elements of finite order in the group defined by the
generating set {x I , x 2 , . . . , x n } and the relation
(XtX2 ... xn)2 = e
5.4.16.8. Let G I = [x, Y I , . V 3] have as a defining set of relations
-1 -1 -1
X YIXY2 = X Y2 X Y3 = X YaXYI = e,
x3=y:=y=y:=e, YJY2=Y2Yh YJY2Y3=e
.
Prove that G I is isomorphic to the group G 2 defined by the generating set
{a, b 1 , b 2 } and the set of relations
ll :l - b 2 - b 2 - e (b b )  - e
- 1- i-' 12 - ,
b 1 a = ab-1" bia = ab.b'J.
5.4.17. Suppose that a group G is defined by a generating set K and the
relation
XIX · .. X n = x n + 1 x n + 2 · .. X m
where Xl' X2, . . . , X m are distinct elements in K. Prove that G is a free group.
5.4.18.8. Let G = [a, b], and assume that the set of relations a 2 = b 2 =
(ab)4 = e is a defining set of relations on G. Find:
1) the number of generating sets of G consisting of two elements;
2) the center of G ;
3) the group of automorphisms of G.
5.4.19. Let G be a noncyclic commutative group of order eight. Show that
either G has a generating set with two elements a, b which satisfy the relations
a 4 = b 2 = e, or G has a generating set with three elements a, b, c which
satisfy a 2 = b 2 = c 2 = e.
5.4.20.8. Determine the number of nonisomorphic commutative groups of
order eight.
5.4.21. Let G be a noncommutative group of order eight. Prove that:
1) G contains no elements of order eight;
2) G contains an element of order four;
3) if b rt [a], where a is an element of order four, then b -lab = a 3 ;
4) if be [a], where a is an element of order four and b 2 =F e, then b 2 = a 2 .

Free Products of Groups
125
5.4.22.H. Let G be a noncommutative group of order eight. Prove that G is
isomorphic to either the quaternion group or the group [(1234), (13)].
5.4.23.H. How many nonisomorphic groups of order eight are there?
5.4.24.H. Let p and q be primes, p < q, and let G be a noncyclic group of
order pq. Prove that:
1) G has a unique subgroup N of order q, and that N is a normal sub-
group of G;
2) G has q subgroups of order p.
5.4.25.H. Let G be the group defined in the preceding problem. Prove that
there exist elements a, bEG satisfying
aP=e, bQ=e, a- 1 ba=b T
where r  1 (mod q), r P = 1 (mod q), such that G = [a, b].
5.4.26.H. Prove that every noncyclic group of order pq, where p and q are
distinct primes with p < q, is isomorphic to some group G:q as defined ip
problem 5.2.22.
5.4.27.H. Let p and q be distinct primes. Show that there exist at most two
nonisomorphic groups of order pq. Describe their structure.
5.4.28.H. How many nonisomorphic groups of order 14 are there? How
many of order 15?
5.4.29.H. Determine the number a(n) of nonisomorphic groups of order
n for n  10.
5. Free Products of Groups
A group G is called the free product of its subgroups G a , a E A, (where
the cardinality of A is greater than one) if:
1) G a =F e for all a E A ;
2) G a n G p = e for all a, pEA, a =F P;
3) G' = U G a is a generating set of G;
aeA
4) the set consisting of all relations on G a , for all a E A, is a defining set
of relations on G with respect to G'.
The representation of G by a free product of subgroups G a , a E A, is
called a decomposition of G into a free product, and is denoted by
a = II * a, or a = a tI. * a:I: ... :If Q; :51 ...
eEr
Another way of defining the free product of subgroups is given in 5.5.10 and
5.5.11. This equivalent statement is often taken as the definition of free
product.

126
Defining Sets of Relations
5.5.1. Let a group G be defined by the generating set {ai, a 2 , a3, a 4 } and
the set of relations a: = e, a = e; aj = e, al 2 = e. Can G be decomposed
into a free product?
5.5.2. What kind of free groups are decomposable into a free product?
5.5.3. Let G be the free product of its subgroups A and B, each of which is a
free group. Prove that G is a free group.
5.5.4. Let a group G be defined by the generating set {at, a 2 , a 3 } and the
set of relations a: = a2, aj = e. Let G i (i = 1,...,4) be subgroups of G such
that G 1 = [a 1 ,a2], G 2 = [a2]' G 3 = [a 2 ,a3], G 4 = [a3]. Can G be decom-
posed into a free product of any of the G i ?
5.5.5. Let a group G be defined by the generating set {ai' a2, a3} and the
relation a: = e. Prove that G is the free product of a cyclic group of order 3
and two infinite cyclic groups.
5.5.6. Can an abelian group be decomposed into a free product of its
subgroups?
5.5.7. Let a group G be the free product of the subgroup A = [ai' a2, a3]
having the defining set of relations a l ai I = a, af = a and the subgroup
B = [b l , b 2 ] subject to the relation b: = b l b. Prove that M = {ai, a 2 , a 3 ,
b l , b 2 } is a generating set of G, and the set at ail = a, af = a, b: = b t b is
a defining set of relations of G with respect to M.
5.5.8. Is the free product of two finite subgroups necessarily finite?
5.5.9. Let G = n* G a , where every a E U G a , a =F e, has infinite order.
aeA aeA
Does G have any elements of finite order?
5.5.10.T. Suppose that there are subgroups G a =F e (a E A) of a group G
satisfying the following condition: every g E G (g =F e) can be uniquely
written in the form
g=al a ') ... an; ai*e; aiE OrJ.i (l=l, 2, ..., n)
where ai =F ai+ 1 . Prove that G = n* Ga.
aeA
5.5.11.T. Prove that if G = n* G a , then every g E G (g i= e) can be uniquely
aeA
written in the form
g= al Q 2 ... an; Qi :j:. e;
Qi E Oa.
t
(i=l, 2, ..., n)
where ai =F ai+ I'
5.5.12. Let G = n* G a , and in every subgroup G a select a subgroup G =F e.
aeA
Prove that the subgroup G' c: G which is generated by all of the subgroups
G is the free product of its subgroups G (a E A).

The Direct Product of Groups
127
5.5. 13.T. Let G = G 1 * G 2 , and let N be the normal subgroup generated by
G 2 (cf. Chapter 4.3). Prove that G 1 is isomorphic to GIN.
5.5.14. Let A, B, C be subgroups of a group G, and let G = A * B, G =
A * C. Does it follow that B is isomorphic to C?
5.5.15. Let G be the free product of two cyclic subgroups of orders 5 and 7.
Is G a periodic group?
5.5.16. Let a group G be the free product of a finite number of subgroups,
each of \yhich has a finite generating set. Prove that G also has a finite
generating set.
5.5.17. A group G is called complete if for every a E G and every natural
number n, the equation x n = a can be solved in G. Under what conditions is
a group which is decomposed into a free product of complete subgroups
complete?
5.5.18. What is the center of a free product of groups?
5.5.19. Let G = A * B, a E A, a =1= e, be B, b =1= e. Can ab lie in A?
5.5.20.H. Let G be the set of all partial transformations on the set of complex
numbers K defined by
A( ) - az+b
z - cz + d
(z E l()
(a, b, c, d are rational numbers with ad - bc = 1). Prove that G is a group
under the operation defined in 3.6.25, and that it can be decomposed into
the free product of a cyclic group of order 2 and a cyclic group of order 3.
5.5.21. .Suppose that G = G 1 * G 2 , and that G'l and G are proper sub-
groups of G', where G't is isomorphic to G 1 , G is isomorphic to G 2 , and
G' = [G'l, G]. Is G' a homomorphic image of G?
5.5.22. Let G = G 1 * G 2 * G 3 . Prove that
Q= 0 1 * (Ot 11= 0 3 )=(0 1 * Ot) * 0 3
.5.23. Let G be a free noncyclic group. Show that there exist an infinite
number of decompositions of G into free products of its subgroups.
5.5.24.T.H. Let there be given a family of groups G a (a E A), G a =1= e, where the
index set A has more than one element. Prove that there exists a group G
having subgroups G a (a E A) which are isomorphic to the G a (a E A), such
that G is the free product of the G a (a E A).
6. The Direct Product of Groups
We say that a group G is a direct product of its subgroups G a (a E A)
if:

128
Defming Sets of Relations
1) G a n G p for all a, pEA, a =1= P;
2) G ' = U G a is a generating set of G;
aeA
3) for all a E G a , be G p , (a, pEA, a =F P)
ab = ba
and the set consisting of these relations and all relations on the G a
is a defining set of relations on G with respect to G ' .
Two other equivalent definitions of the direct product which are
frequently used are given below (see 5.6.13, 5.6.14, and 5.6.30).
A representation of G as a direct product of subgroups G a is called a
decomposition of G into the direct product. We will denote this by
0= IT 0e, or O=Ov. X O X... X Oe X ...
eer
A subgroup A of G is called a direct factor of G if there exists a subgroup
B of G such that G = A x B. A group is called decomposable if it admits a
proper subgroup which is a direct factor of it.
Two decompositions of a given group G into direct products are said
to be isomorphic if there exists a one-to-one correspondence between the
factors of these decompositions such that corresponding factors are iso-
morphic groups.
Let a group G be a direct product of its subgroups G a (a E A), and
suppose that some of these Ga(a E A', A' c A) are decomposed into direct
products of their subgroups G a = n G a ". Then G is the direct product of
"eA oc
all the subgroups G(l" (a E A', 11 E Aa) and the G a (a E A "A') (see 5.6.3). Such a
decomposition is called an extension of the decomposition G = n Ga.
aeA
5.6.1. Let a group G be defined by the generating set {aI' a2, a 3 , a 4 } and the
set of relations a 7 1 = a 2 l3 = a 3 3 = a 4 5 = e a.a. = a .a. ( i J . = 1 2 3 4 ) D e -
, I) )1' ",.
compose G into a direct product of cyclic subgroups.
5.6.2. Let G = A x B, where A has the generating set {a l , a 2 } and defining
relation a = a2, and B = [b l ,b 2 ,b 3 J has the defining relation bi = b 2 b 3 .
Prove that M = {aI' a 2 , b l , b 2 , b 3 } is a generating set of G and that {at = a 2 
bi = b 2 b 3 , a 2 b 3 = b 3 a 2, a l b 2 = b 2 a l , b 2 a 2 = a 2 b 2 , a l b l = b l a l , a l b 3 =
b 3 a l , a2 b l = b l a 2 } is a defining set of relations on G with respect to M.
5.6.3.T. Let G be a direct product of its subgroups G a (a E A), and let some
of its subgroups G a (a E A', A' c A) be decomposed into direct products
G a = n G a ". Prove that G is a direct product of all the G:x" (a E A', 11 E Aa)
"eA oc
and G a (a E A "A').
5.6.4. Prove that it is impossible for a group G to be decomposed both into a
direct product of cyclic subgroups and into a free product of cyclic subgroups.

The Direct Product of Groups
129
Remark. We have thus shown that a group cannot be decomposed into
both a direct product and a free product of any of its proper subgroups.
5.6.5.H. Determine which of the following groups are decomposable into
a direct product of subgroups:
1) a finite group of prime order;
2) an infinite cyclic group;
3) the quaternion group;
4) a group of order 51 ;
5) the additive group of rational numbers.
5.6.6. Prove that the multiplicative group of all real numbers different from
zero is decomposable into a direct product of the multiplicative group of
positive numbers and a cyclic group of order 2.
5.6.7. Let G 1 and G 2 be finite subgroups of a group G, where the orders of
G 1 and G 2 are relatively prime and each element in G 1 commutes with every
element in G 2 . Prove that [G 1 , G 2 ] = G 1 x G 2 .
5.6.8. Let A be the subgroup of the symmetric group Sg generated by the
permutations a = (1234)(5678) and b = (1537)(2846). Is A decomposable
into a direct product?
5.6.9. Let H be the subgroup of S4 generated by a = (12)(34), b = (13)(24),
and c = (14)(23). Show that H can be decomposed into a direct product.
How many such nonisomorphic decompositions of H exist?
5.6.10. Prove that
1) the direct product of abelian groups is abelian;
2) 'the direct product of complete groups is complete;
3) the direct product of periodic groups is periodic.
5.6.11. Prove that the factor group of a free group by its commutator
subgroup is either an infinite cyclic group or a direct product of infinite
cyclic groups.
5.6.12. Prove that a group G of order pq, where p and q are distinct primes,
is decomposable into a direct product if and only if G is commutative.
5.6.13.T. Let G = n G(l' Prove that every g E G (g =1= e) can be written
(leA
uniquely, up to the order of the factors, in the form
g= alai ... an; al =1= e; al E Oa.l
(i = 1, 2, ... J n)
(*)
where rxi "# rx i + 1 ·
5.6.14.T. Let G(l (rx E A) be a collection of subgroups of a group G, where the
elements in any}two subgroups G(l and G p (rx "# fJ) commute, and suppose

130
Defining Sets of Relations
that every element g E G (g "# e) can be written uniquely, up to the order of
the factors, in the form
g = a 1 a'j ... alP ai =t= e, a i E Oa.. (l = 1, 2, ... J n)
,
where (Xi "# (Xi + 1 for all i = 1, 2, . . . , n - 2, n - 1. Prove that G can be
decomposed into a direct product of its subgroups G(% ((X e A).
Remark. The element g(% e G(% in the decomposition (*) of g (see 5.6.13)
is called the component of g in the direct factor G(% relative to the given de-
composition. If in the stated decomposition there are no factors belonging
to G(%, then the component of g in G(% is considered to be the identity. If
G = G 1 X G 2 X ... x G n , then every ge G can be written uniquely in the
form g = g 1 · . . gi · . · gn, where gi is the component of gin G i . This decomposi-
tion is a special case of the one given above (*), where (*) contains additional
factors equal to the identity.
5.6.15. Let D be the additive group of all complex numbers. Is it possible for
D to have two direct decompositions
D=AXB, D=AXC
such that for some element de D, the component of d in A relative to the
first decomposition is different from the component of d in A relative to the
second decomposition?
5.6.16. Suppose that a group G is decomposed into a direct product of finite
subgroups G l' G 2 , . . . , G n , of orders m 1 , m2, . . . , m n . Prove that G is a finite
group. What is the order of G?
5.6.17. Let g e G = n G(%; fix f3 e A, and let gp be the component of g E G
(%eA
in G p. Prove that the mapping lp : g -.. gp is a homomorphism of G onto G fJ
Remark. This shows that the component of a product is equal to the
product of the components.
5.6.18. Let G be a group having exactly one normal subgroup different
from G and the identity. Show that G is indecomposable.
5.6.19. Prove that the center of the direct product is equal to the direct
product of the centers of the factors.
5.6.20. Prove that the commutator subgroup of the direct product is equal
to the direct product of the commutator subgroups of the factors.
5.6.21.T. Prove that if A is a direct factor of a group G, then every normal
subgroup A' of A is a normal subgroup of G.
5.6.22. Let the group G be decomposed into the direct products
0= A X B,
Prove that Band C are isomorphic.
O=AXC

The Direct Product of Groups
131
5.6.23. Let G = A x B. Let F be a subgroup of G containing the direct factor
A. Show that F = A x (F n B).
5.6.24. Let G be the multiplicative group of all n x n matrices over the
real numbers with positive determinants. Denote by N 1 the set of all matrices
in G with determinant 1, and by N 2 the set of matrices of the form AEn,
where A is a positive number and En is the n x n identity matrix. Show that
G=N 1 xN 2 .
5.6.25.T. Prove that if every descending chain of normal subgroups of a
group G is finite, then G cannot be decomposed into the direct product of an
infinite number of subgroups.
5.6.26.T.H. Suppose that every decreasing chain of normal subgroups of a
group G is finite. Show that every decomposition of G can be extended to a
decomposition in which each factor cannot be decomposed into the direct
product of subgroups.
5.6.27. Let G be a group which is decomposed into the direct product of
subgroups G = G 1 X G 2 X ..., such that the order of each G i is a prime.
Under what condition will G have a unique decomposition into indecom-
posable factors?
5.6.28.T.H. Suppose that every increasing chain of direct factors of a
group G is finite. Show that every decreasing chain of direct factors must also
be finite.
5.6.29.T.H. Let the set G be the Cartesian product (cf. Chapter 1.1) of the
groups G 1 ,. .., G n . Define an operation on G in the following manner. If
gt = (a., ..., ai' ..., an)'
ai E ai'
g9.=(bb ..., bi' ..., b n ),
b i E Oi
then gt'g2 = (albl,...,aibi,...,anbn)' where aibi is the product in
G i (i = 1,..., n). Prove that G is a group relative to this operation, and that G
can be decomposed into a direct product of subgroups G i which are iso-
morphic to G i .
5.6.30. Let G be the direct product of n infinite cyclic groups. Let H be the
subgroup of all elements x 2 (x E G). Prove hat G/H is a group of order 2 n ,
and can be written as the direct product of n cyclic groups of order 2.
5.6.31.T.H. Let G be a free group with a free generating set consisting of n
elements. Prove that every free generating set of G also contains n elements.
Remark. This proves the uniqueness of the rank of a free group for the
finite case. The uniqueness of the rank in the general case follows immediately.

132
Defining Sets of Relations
5.6.32.H. Let G be the Cartesian product of the finite groups G 1 , G 2 ,. . .,
G i , . . . . Define an operation on G by:
if
gl = (aj, a2, ..., ai' .. .), g9. = (b h b9., ..., bb ...)
then
gl · g9. = (al b h a2 b 2' ..., aibl, ...)
where aibi is the product in G i . Prove that G is a group under this operation.
Can G be decomposed into the direct product of subgroups G i (i = 1,2,...)
which are isomorphic to G i ?
Remark. Compare this result with 5.6.29.
5.6.33.T. Prove that a group G can be decomposed into t direct product
of its subgroups G(%, ex E A, if and only if the following three conditions are
satisfied:
1) G(% is a normal subgroup of G for every ex E A;
2) G = [U G (%] ;
(%eA
3) the intersection of each GfJ (f1 E A) with the subgroup generated by
the remaining G(% (ex "# (1) is the identity.

Chapter 6
ABELIAN GROUPS
1. Elementary Properties of Abelian Groups
The present chapter is devoted to commutative (abelian) groups. For
the remainder of this chapter we will only consider abelian groups, where this
property will sometimes not be stated explicitly.
When an abelian group G is defined by a generating set M and a set of
relations F, we will tacitly assume that the relation xy = yx (x, Y E M) is
included in F.
Additive notation is often used when dealing with abelian groups.
Thus one writes x + y instead of x · y, nx instead of x n , and denotes the
identity of the group by O. However, we shall retain our usual multiplicative
notation throughout this chapter.
An abelian group G is called periodic (or torsion) if the order of each of
its elements is finite (cf. Chapter 2.6), and torsion free (or aperiodic) if each of
its elements (except the identity) has infinite order. If G contains elements,
different from the identity, of finite order and elements of infinite order, it is
said to be mixed.
If, for some prime p, every element of a periodic group G has order a
power of p, Le., G is a p-group (cf. Chapter 4), then G is called a primary
group with respect to p, or a p-primary group.
The set F of all elements of finite order in an abelian group G forms a
subgroup, which is called the maximal periodic subgroup of G, or the periodic
part of G (see 6.1.1). The set of all elements of a periodic group G whose
orders are powers of some prime p forms a subgroup of G and is called the
primary component with respect to p, or the p-component of G (see 6.1.7).
A finite set of elements g l' g2, . . . , gk in a group G will be called linearly
dependent if there exist integers n 1 , n2,' . ., n k , not all equal to zero, for which
n 1 n2 n k
gl g'J. ... gk = e
A set of elements that does not have this property is called linearly inde-
pendent. In the additive notation, the property of linear dependence takes
133

134
Abelian Groups
the form n1g l + n2g2 + . .. + nkg k = 0, and consequently coincides with
the usual definition of linear dependence.
An infinite set of elements of a group G is called linearly independent
if all of its finite subsets are linearly independent. Otherwise the set is called
linearly dependent. We say that an element g EGis linearly dependent on the
set E c: G if there exist a number m and elements gl, g2,' . ., gk E E such that
gm E [g I , · · · , gkJ.
If a group G has a finite maximal linearly independent set, then every
maximal linearly independent set contains the same number of elements
(see 6.1.21). In this case the number of elements in a maximal linearly
independent set is called the rank of G. If G does not have a maximal linearly
independent set consisting of a finite number of elements, then we say that
G has infinite rank. A periodic group is considered to have rank zero.
6.1.1. T. Prove that the periodic part of a grou p G is a completely characteris-
tic subgroup of G.
6.1.2. Find the periodic part of the multiplicative group of all complex
numbers different from zero.
6.1.3. Let T be the periodic part of a group G, and H a subgroup of G.
Prove that H n Tis the periodic part of H.
6.1.4. Prove that the additive group of complex numbers is the direct
product of two additive groups which are isomorphic to the additive group
of real numbers.
6.1.5. Prove that every set of elements of a group G which contains an element
of finite order is linearly dependent.
6.1.6. Prove that a set of elements VI' V2, . . . , V k of a group G is linearly
dependent if and only if one of the Vi is linearly dependent on the others.
6.1.7. Prove that every primary component of a group G is a completely
characteristic subgroup of G.
6.1.8. Let T be the p-primary component of a group G and let H be a sub-
group of G. Prove that H n Tis the p-primary component of the group H.
6.1.9.T.H. Prove that every periodic group can be decomposed into the
direct product of primary subgroups relative to distinct primes.
6.1.10.T. Prove that every periodic group can be decomposed in only one
way into the direct product of primary subgroups relative to distinct primes.
6.1.11. Prove that the additive group of all rational numbers is a torsion-free
group and, further, that it can be represented as a union of an ascending chain
of cyclic subgroups.
6.1.12. Is the group of rational numbers under addition isomorphic to the
group of positive rational numbers under multiplication?

Elementary Properties of Abelian Groups
135
6.1.13. Let Hand K be subgroups of G and H c: K, where the index of H
in K is finite. Prove that the index of the subgroup [H, a] in the group [K, a]
is finite and divides the index of H in K (a E G).
6.1.14. LetGbeap-primarygroupandkanaturalnumbersuchthat(k,p) = 1.
Is the equation Xk = a always solvable in G (a E G)? How many solutions are
there?
6.1.15. Prove that the set of all nth powers of elements in a group G is a
completely characteristic subgroup of G.
6.1.16. Find all abelian groups which do not have any proper homomorphisms,
i.e., homomorphisms which are not isomorphisms and do not map onto the
identity element.
6.1.17. Let G be a group which is decomposed into the direct product of
cyclic groups. Does there exist an element a E G, a #: e, for which the equation
x n = a is solvable for any natural number n?
6.1.18. Prove that the direct product of a family of p-primary groups is a
p-prlmary group.
6.1.19. Let G be a group which is the direct product of cyclic groups of
orders 5 and 3 and four infinite cyclic groups. Determine:
1) the periodic part of G;
2) the number of periodic subgroups of G;
3) the factor group of G by the periodic part;
4) the homomorphic image of G under a homomorphism ex whose
kernel is a cyclic subgroup of order 5.
6.1.20. Prove that every finitely generated abelian group G which is not-
periodic has a finite maximal linearly independent set.
6.1.21. Let A and B be two finite maximal linearly independent subsets of a
group G. Show that A and B contain the same number of elements.
6.1.22. Let G be an abelian group and T its periodic part. Prove that the
rank of G is equal to the rank of G/T.
6.1.23. Prove that the rank of an abelian group G is invariant under iso-
morphism (Le., the rank of any group isomorphic to G is equal to the rank of
G).
6.1.24.T. Let G be decomposed into a direct product of a finite number of
cyclic groups. Prove that the number of infinite cyclic factors does not
depend on the choice of the decomposition of G.
6.1.25. Let G be a p-primary abelian group. Prove that the set of all elements
of G whose orders are less than or equal to p is a characteristic subgroup.
Remark. The indicated subgroup is called the lowest layer of G.
6.1.26. Suppose that a p-primary abelian group G is decomposed into a
direct product of cyclic groups. For any natural number n, denote by A(n)

136
Abelian Groups
the product of all direct factors whose orders are pn (if such do not exist, then
set A(n) = e). Let B(n) denote the direct product of the lowest layers A\n),
A\n+ 1), . . . of the groups A(n), A(n+ 1), . . . . Prove that:
1) An) is isomorphic to B(n)/B(n+l);
2) B(n) contains only those elements a in the lowest layer for which the
equation xpn- 1 = a is solvable;
3) the subgroups A\n) and B(n) do not depend on the particular de-
composition of G into a direct product.
6.1.27.T.H. Let G be a primary group which is decomposed into a direct
product of cyclic groups. Prove that any two decompositions of G into a
direct product of cyclic groups are isomorphic.
6.1.28.T.H. Let G be a finite or countable group which is decomposed into a
direct product of cyclic groups. Prove that any two decompositions of G
whose factors are cyclic groups which are either infinite or are finite and
primary are isomorphic.
Remark. The assumption of countability can, in fact, be eliminated.
2. Finite Abelian Groups
Let G be a finite abelian group. We shall show that G can be decomposed
into the direct product of cyclic primary subgroups, and that any two such
decompositions are isomorphic (see 6.2.7 and 6.2.9). Let G be decomposed
into the direct product of primary cyclic groups relative to the primes
PI , P2, . . · , Pk' Assume further that the number of primary factors cor-
responding to Pi(i = 1,..., k) is li' and that the factors have orders
ait a is ail
Pi , Pi , ". , Pi
where CXi!  CXi2  . . .  CXili' The numbers piini (i = 1, 2, . . . , k; n i = 1,
2, . . . , li) are called the invariants of G. These invariants determine the group
G up to isomorphism. Finite groups having distinct invariants are not iso-
morphic (see 6.2.8. and 6.2.9). The set of all invariants of a finite group G
will sometimes be written in the form of a table, where each row consists of
the invariants corresponding to a prime written in decreasing order. In
certain cases we will insert p? = 1 into a row. For example, if a finite grou p is
decomposed into the direct product of four cyclic groups of orders 2 2 ,2 2 ,
2 5 ,2 6 and three cyclic groups of orders 3,3 3 ,3 4 , then the invariants can
be written in the form of the following table:
2 8 , 2, 2 2 , 2 2
3 4 , 3 3 , 3, 3 0
One should keep in mind the fact that in the set of invariants the same
number may occur several times, i.e., the set of invariants represent a system
rather than a set (see Chapter 1.1).

Finite Abelian Groups
137
6.2.1. Let H be a subgroup of a finite abelian group G, let the index of H in
G be k, and let H' be a subgroup of G of order pi, where p is a prime such that
(k, p) = 1. Prove that H' c H.
6.2.2. Find all primary components of abelian groups of orders (1) 21, (2)
30, (3) 462, and (4) 101.
6.2.3. For a group of order 35, determine:
1) the number of subgroups;
2) the number of proper subgroups;
3) the number of proper characteristic subgroups;
4) the number of proper completely characteristic subgroups.
6.2.4. Decompose a group of order 30,030 into the direct product of cyclic
groups.
6.2.5. Let G be a finite p-primary abelian group and let a be an element of
maximal order pk. Denote by H the maximal subgroup of G for which
H n [a] = e. Suppose x rt [H, a], but x P E [H, a]. Prove that xPa pm E H for
some m.
6.2.6.H. Let G be a finite abelian p-primary group. Let a E G be a maximal
element of order pk. Prove that the subgroup [a] is a direct factor of G.
6.2.7.T.H. Prove that every finite abelian group can be decomposed into a
direct product of primary cyclic groups.
6.2.8.T. Prove that two finite groups having the same set of invariants are
isomorphic.
6.2.9.T.H. Prove that two finite groups having different sets of invariants
are not isomorphic.
6.2.10. Find the invariants of the following finite groups which are defined
by the indicated generating sets and sets of relations:
1) {ah a2}, a; = e, a=e
2) {ab 1 4_ 6)
a2f' a 2 - at, ai=e
3) {a., a2, aa}, aa_ e q
1- , ala2 = e, a 3 =e
4) {ah a2, aa, a4}, a 3 -e a = e, a = e, al a 2 = aaa4
1- ,
5) {ab a2, aa}, ar =e, ai =e, <)
aa=e
6.2.11. Let p be a prime number. What conditions must be imposed on a
finite group G so that the only proper homomorphic images (see 6.1.16) of G
are p-primary groups?
6.2.12. T. Let G be a finite abelian p-primary group with invariants pkt,
pk2, . . . , pk s . Prove that if a subgroup H of G has invariants pll, pi2, . . . pls, then
k i  li (i = 1,..., s).

138
Abelian Groups
6.2.13.T.H. Let G be a finite abelian group with invariants
P a 11 P a un
1 ,..., 1
. . . . . . .
a. a.
P 11 P 1m
i ' ..., i
. . . . . . .
t1kt akm
Pk ' ... , P k
and let H be a subgroup of G with invariants
P 11 P lm
I , ..., 1
. . . . . . .
P ;l P im
i ' ..., l
. . . . . . .
P kl P km
k ' ..., k
Prove that Pij  (Xij for all i = 1,..., k; j = 1,..., m.
6.2.14. Let G be a finite abelian group such that every proper subgroup is
characteristic. What are the invariants of G?
6.2.15. Let G be a finite abelian group such that every proper subgroup is
completely characteristic. What are the invariants of G?
6.2.16. Find all finite abelian groups which have the property that the only
proper homomorphic images (see 6.1.16) are cyclic groups.
6.2.17. What type of finite groups do not have proper endomorph isms, i.e.,
endomorphisms which are neither automorphisms nor the identity homo-
morphism?
6.2.18. Can a finite group G contain an element g =F e for which the equation
x n = g is solvable for every n?
6.2.19. A primary group G is given by its invariants p4, p3, p2. How many
elements in G have order p?
6.2.20. A finite group G is given by its invariants 2 3 , 2; 52, 5, 5. Determine:
1) the order of G;
2) if G has subgroups of orders 31, 25, 10,40, 27, 120;
3) the invariants of a subgroup of order 20.
6.2.21. A finite group G is given by its invariants
5 3 ,5 2 ,5;7;3
and a group G' by its invariants
2 3 ,2 2 ,2; 11
Does there exist a nonidentical homomorphism of G into G'?
6.2.22. A finite group G is given by its invariants 3; 7. How many nonidentical
automorphisms does G have?

Finitely Generated Abelian Groups
139
6.2.23. Prove that if a subset S of a finite abelian group G contains all of the
elements of maximal order in G, then [S] = G.
3. Finitely Generated Abelian Groups
Every finitely generated abelian group can be decomposed into the
direct product of primary cyclic groups, both finite and infinite, where the
number of infinite cyclic factors is finite and is equal to the rank of the group
(see 6.3.4 and 6.3.19). A finitely generated group G has a finite periodic part,
and is completely determined by the set of invariants of the periodic part
and its rank (see 6.3.5 and 6.3.6). The set of invariants of the periodic part
and the rank constitute a complete system of invariants of a finitely generated
group, in the sense that they determine the group up to isomorphism.
6.3.1.T.H. Prove that a finitely generated (abelian) group G cannot have an
infinite number of distinct subgroups Hi (i = 1, 2, . . .) such that
Ht C Hi C... Hi C.. ·
6.3.2.H. Let A = [a l , . . . , an] be an abelian group, and n l , n2, . . . , n k be any
integers whose greatest common divisor is 1. Denote by b 1 the product
al . ai 2 . . . ak. Show that there exist elements b 2 ,..., b k such that A =
[ b 1 , b 2 , . · . , b k]'
6.3.3. Prove that the periodic part of a finitely generated abelian group is a
finite group.
6.3.4.T.H. Prove that every finitely generated abelian group G can be repre-
sented by a direct product of a finite number of cyclic primary groups which
are either finite or infinite.
6.3.5.T. Prove that if two finitely generated abelian groups have the same
rank and the same set of invariants of the periodic part, then they are iso-
morphic.
6.3.6.T. Prove that if two finitely generated abelian groups have either
different ranks or different sets of invariants of their periodic parts, then they
are nonisomorphic.
Remark. Compare the results of these last three problems.
6.3.7.T.H. Let a finitely generated abelian group G have rank r l and set of
invariants
au aim
PI , ... , PI
. . . . .
a. a.
Piu, ... , Pi,m
...... .
P ak1 P akm
k ' ..., k

140
Abelian Groups
and let H be a subgroup of G having rank r 2 and set of invariants
P 11 P 1m
1 ' ..., 1
...... .
P il P rim
i ' ..., i
. . . . .
P kl P Pkm
k ,..., k
Prove that r 1  r 2 and (Xij  Pij for all i = 1,..., k; j = 1,..., m.
6.3.8.H. Let the following groups be defined by the indicated generating
sets and sets of relations. Find the rank and set of invariants of each:
1) A = [ah a'll,
2) A = [ah a'll,
3) A = [ah a, aal,
4) A = [ah a'll,
5) A = [a., a'.b aat,
6) A = [ah a, aa1,
7) A = [a., a2' a31,
8) A = [a., a, a 3 1,
9) A = [ah a, aa),
a ' - e .
1- ,
a s - a ·
J - '!,
a; = e,
a 3 a ' - e
t -2- ,
a 3 a 2 a - e
I  3- ,
a 3 a Oa 6 - e
1 g 3- ,
a:a = e
a:aa = e,
t 3 -'>
a1a.]aa" = e,
1
a] = a
ara = e
a1a2a: = e
a1a4 3 a; = e
9 -3 9
ala aa=e
a:a;2aa3 = e
Remark. From this problem we see that two groups can be isomorphic
even though they have different generating sets and defining sets of relations.
6.3.9. Let a group G be defined by its generators at , a2, a 3 and the relation
ao = e. Find:
1) the rank of G;
2) the periodic part of G ;
3) the factor group of G modulo the periodic part;
4) the number of periodic subgroups of G;
5) all primary components of the periodic part.
6.3.10. Let G be a finitely generated group with rank 1 and set of invariants
52, 5 ; 3. Determine whether there exists:
1) a homomorphism of G onto a cyclic group of order 25;
2) a homomorphism of G onto a finite group G' which is defined by the
invariants 3, 5 ;
3) a homomorphism of G onto a torsion-free group G' which is a direct
product of three infinite cyclic groups;
4) a nonidentical homomorphism of G into a finite group G' which is
given by the invariants 2 3 , 2, 2; 7 2 , 7, 7 0 ; 11, 11, 11.
6.3.11. Let a group G be defined by the generating set {at, a2, a 3 } and the
relation at = a. Does G have any elements. a =1= e of finite order?

Infinite Abelian Groups
141
6.3.12. Let a group G be defined by a generating set {a t ,a 2 ,a 3 } and the
relations a = e, a = e. Find the invariants of the subgroup generated by the
elements a and at.
6.3.13. Suppose that a finitely generated group G has rank 2 and set of
invariants 2 3 , 7. Prove that there exist elements at, a2, a 3 , a 4 E G such that
G = [at, a 2 , a3, a 4 J and the set of relations ai = e, a = e is a defining set
with respect to the generating set {at, a2, a3, a 4 }.
6.3.14. For a finitely generated group G with rank 3 and set of invariants
52, 52 ; 7 3 , determine:
1) the number of periodic characteristic subgroups different from the
identity;
2) the number of completely characteristic, periodic subgroups different
from the identity;
3) whether there exist completely characteristic subgroups of G which
are torsion-free.
6.3.15. Let a group G be defined by the generating set {at, a2} and the re-
lation ai = e. Determine the number of automorphisms of G.
6.3.16. A group is defined by the generating set {at, a2} and the relations
ara = e, aia23 = e. What is the order of the subgroup generated by the
element ara2?
6.3.17. Let a finitely generated group G have rank 5 and set of invariants
3 2 , 3; 53, 5. Determine whether there exists:
1) a proper subgroup of G which is isomorphic to G;
2) an endomorphism of G onto itself which is not an automorphism.
6.3.18. Prove that the set of all nonisomorphic finitely generated abelian
groups is countable.
4. Infinite Abelian Groups
An abelian group G is said to be divisible if for every a E G and every
natural number n, the equation x n = a has at least one solution in G. * The
class of divisible groups has been completely described. The importance of
this class of groups stems from the fact that every abelian group is isomorphic
to a subgroup of a divisible group (see 6.4.25).
Let C be a subgroup of G. If for every C E C the solvability of x n = C
in G implies the solvability of x n = c in C, then C is called a pure (or serving)
subgroup.
Let G be a p-primary group. An element a EGis said to have height
k if Xpk = a is solvable in G but Xpk + 1 = a is not solvable in G. If Xpk = a is
solvable in G for every number k, then we say that a has infinite height in G.
*Thus a divisible group is a complete abelian group (see 5.5.17). [Trans.]

142
Abelian Groups
Let a E G be an element of infinite order, and let PI' P2, . . . , Pn, . .. be the
increasing sequence of all primes. Define the symbol ((Xl' (X2, . . . , (Xn, . . .) by
setting (Xn = 0 if the equation x Pn = a does not have a solution in G, (Xn = kif
the equation xP = a can be solved in G but xP+ 1 cannot, and (Xn = 00 if
all of the equations xP = a are solvable in G (i = 1, 2, . . .). The symbol
((X I , · · . , (Xn, . . .) is called the characteristic of a.
Two characteristics (X = ((Xl" . · , (Xn, . ..) and P = (PI' · . · , Pn, . ..) are
said to be equivalent if (Xn = Pn for all n and both (Xn and Pn are different from
00 for all but a finite number of n. The class of all characteristics can be par-
titioned into disjoint classes having equivalent characteristics (see 6.4.29).
These classes are called types. If (x' and P' are types, we define (x'  P' if there
exist characteristics (X of type (x' and P of type P' such that (Xn  Pn for all n,
where 00 is considered as being greater than any number.
6.4.1. Prove that if in a nonperiodic group G the set of all elements of infinite
order together with the identity forms a subgroup, then G is torsion-free.
6.4.2.T. Prove that the factor group of an abelian group by the periodic part
is torsion-free.
6.4.3. Suppose a group G contains elements of orders k l , k 2 , . . . k n , where the
k i (i = 1,..., n) are relatively prime numbers. Prove that G contains an
element of order k l k 2 . . · k n .
6.4.4.T.H. Prove that if a group G has finite rank, then so does every.sub-
group A and factor group G/A. Moreover, show that the sum of the ranks of
A and G/ A is equal to the rank of G.
6.4.5. Prove that the rank of the direct product of a finite number of groups,
each of which has finite rank, is equal to the sum of the ranks of the factors.
6.4.6. Let P be a prime and G ' be the additive group of all rational numbers
whose denominators are powers of p. Denote by G" the subgroup of G '
consisting of all integers. Prove that G = G'/G" is an infinite periodic group.
Remark. The group G is called a group of type p oo .
6.4.7. Prove that a group of type p oo can be defined by a set of generators
{ai, a2"'" an,"'} and a defining set of relations af = e, a:+ I = an (n = 1,
2, . . .).
6.4.8. Does a group of type poo have any infinite proper subgroups?
6.4.9. Is it possible for a group G of type poo to be mapped by a proper homo-
morphism (see 6.1.16) onto a group which is not isomorphic to G?
6.4.10. Can a mixed group G have the property that every nonidentical
homomorphic image of G is isomorphic to G?
6.4.11. Which of the following groups are divisible: (1) a group of type poo;
(2) the additive group of rational numbers; (3) the multiplicative group of
complex numbers different from zero; (4) the multiplicative group of all roots
of all powers in (1); (5) a direct product of cyclic groups?

Infinite Abelian Groups
143
6.4.12. Prove that if for every prime p the set of pth powers of all elements in a
group G coincides with G, then G is divisible.
6.4.13. Let G be a p-primary group. Prove that if every element of order pin G
has infinite height, then G is divisible.
6.4.14. Can a divisible group be mapped by a homomorphism onto a
finite group of order greater than I?
6.4.15. Prove that in a torsion-free group G, every equation of the form
x n = a (a E G) can have at most one solution.
6.4.16. Which of the following subgroups of G will always be pure subgroups
ofG:
1) the group G itself;
2) the periodic part of G ;
3) a direct factor of G?
6.4.17.T. Let G be a p-primary group and C a subgroup of G. Prove that C
is a pure subgroup of G if and only if every element in C has the same height
in C as it does in G.
6.4.18. Prove that a divisible group D is a pure subgroup of any group G
which contains D as a subgroup.
6.4.19. Let D be a divisible group, and C =1= e a pure subgroup of D. Show
tha t C canno t be finite.
6.4.20.H. Prove that the intersection of any class of pure subgroups of a
torsion-free group is pure.
6.4.21.T.H. Let C be the intersection of all pure subgroups of a torsion-free
group G which contain the set M c G.Prove that C consists of all elements
which are linearly dependent on elements in M.
6.4.22. Prove that a subgroup C of a torsion-free group G is pure if and only
if G/C is torsion-free.
6.4.23. Prove that every infinite cyclic group is isomorphic to a subgroup
of a divisible group.
6.4.24.T. Prove that every group is isomorphic to a factor-group of a direct
product of infinite cyclic gr<,J.lps.
6.4.25.T.H. Prove that every group is isomorphic to a subgroup of a divisible
group.
6.4.26.H. Prove that every torsion-free group of rank 1 is isomorphic to a
subgroup of the group of all rational numbers under addition.
6.4.27. Let m be an arbitrary integer, m =1= O. Find the characteristic of m
in the following groups:
1) rational numbers under addition;
2) integers under addition;
3) rational numbers with denominator 2 under addition.

144
Abelian Groups
6.4.28. Prove that the relation on the set of characteristics which was
defined in the introduction to this section satisfies the definition of an equiva-
lence as given in Chapter 1.3.
6.4.29. Prove that the relation  introduced on the set of types is an order-
Ing.
6.4.30. Prove that all elements different from the identity in a torsion-free
group of rank 1 have equivalent characteristics.
Remark. The class to which all elements of a torsion-free group G of
rank 1 belong is called the type of G.
6.4.31.T. Let G 1 and G 2 be torsion-free groups of rank 1. Prove that in
order for G 1 to be isomorphic to a subgroup ofG 2 it is necessary and sufficient
for the type of G 1 to be less than or equal to the type of G 2 .
6.4.32.T. Let G 1 and G 2 be torsion-free groups of rank 1. Prove that in
order for G 1 and G 2 to be isomorphic it is necessary and sufficient that they
have the same type.

Chapter 7
GROUP REPRESENTATIONS
1. Representations of a General Type
Let JIt be a class of multiplicative sets (cf Chapter 2.3), and let A be any
multiplicative set. A homomorphism (X of A into some set belonging to JIt
is called a representation of A in the class JIt. The set (X(A) is called the image
of the representation. If JIt consists of one multiplicative set M, we say that
(X is a representation of A in M instead of saying "a representation of A in the
class consisting of the one set M." If (X is an isomorphism, the representation is
said to be faithful. A representation in a class of semigroups of transformations
is called a representation by transformations (or by permutations).
Let (Xl be a representation of a multiplicative set M by transformations
of a set Xl' and let (X2 be a representation of M by transformations of X 2'
We say that (Xl is similar to (X2 if there exists a one-to-one mappingf of X I onto
X 2 such that for all a E M, x E X I'
f((X1 (a) . x) = (X2(a) · f(x)
Thus two similar representations (X I and (X2 differ only by the nature of the
elements in the sets X I and X 2'
We associate with every element a in a multiplicative set M the trans-
formation Aa of M defined by
AaX = ax (x E M)
(where the multiplication takes place in M). We call Aa the left translation of
M corresponding to a.
A set of representations R = {(X} of a multiplicative set A in a class JIt
is called a complete system of representations if for every pair x, YEA, x i= y,
there exists (X E R such that (X(x) i= (X(y).
7.1.1. Let a multiplicative set Z= {ZI,Z2,Z3} be defined by its Cayley
table I ZI Z2 Z3
ZI ZI Z2 Z1
Z2 Za z. Z3
Za Z3 Z3 Z1
145

146
Group Representations
Describe all representations of Z in the class .A of multiplicative sets con-
sisting of the single set U = {u I , u 2 }:
I "I ".
U t U t U 1
11 2 U t U 2
7.1.2. Let N be the multiplicative semigroup of natural numbers and let lp
be a mapping of N into the semigroup of all transformations of the set
X = {a, b, c,} for which
( a b C )
cp (1) =
b C a
Is lp a representation of N by transformations?
7.1.3. Let a semigroup S be defined by the generating set {ai, a 2 } (in the
sense of semigroups) and the defining set of relations a I a 2 = a 2 a I , a = a.
Consider the mapping lp of the generating set of S into the semigroup of all
transformations of the set X = {Xl' X 2 , X 3 , X 4 }:
) ( XI Xi Xa X' ) ) ( XI Xg Xa Xi )
cp (at = , cp (ag =
Xi XI Xa X, Xl Xg Xi Xa
Can lp be extended to a representation of S by transformations of X? (See
Chapter 1.2.).
7.1.4.T. Let (Xl and (X2 be representations of a multiplicative set M. Prove
that if (Xl is similar to (X2, then (X2 is similar to (Xl .
7.1.5. Let (X I and (X2 be similar representations of a multiplicative set M.
Prove that if one of them is faithful, then so is the other.
7.1.6. Let X I and X 2 be two sets with the same cardinality. Let (Xl be a
representation of a multiplicative set M by transformations of X I' Prove
that there exists a representation (X2 of M by transformations of X 2 which is
- similar to (X I .
7.1.7. Let (Xl and (X2 be similar representations of a multiplicative set M.
Prove that the images of these representations are isomorphic.
7.1.8. Define an operation on the set M = {ai, a2, a 3 , a 4 , as} byaia j = as
(i, j = 1, 2, 3, 4, 5). Find all representations of M by transformations of the
set {I, 2}. Determine which ones are similar. Are any of these representations
faithful?
7.1.9. Associate with every element a in a multiplicative set M the left
translation Aa of M. Prove that if this mapping of Minto TM (the set of all
transformations of M) is a representation of M by transformations, then the
multiplication in M is associative, i.e., M is a semigroup.

Representations of a General Type
147
7.1.10. Associate with every element a in a semigroup M the left translation
Aa. Prove that such a mapping of Minto TM (see 7.1.9) is a representation of
M by transformations.
Remark. Compare this result with 7.1.9. Such a representation of a
semigroup will subsequently be called a representation by left translations.
7.1.11. Find the representations by left translations of the following semi-
groups:
1) a cyclic semigroup of type (3, 5);
2) the semigroup Sl defined by the generating set {a l , a 2 } and the
defining set of relations
aial = alai = a" ai = ab a; = a,
3) the semigroup S2 = {a l , a2, a 3 } given by the Cayley table
I at as as
at a 1 at at
as at a 2 a 2
as a 1 a 2 aa
Which of the above representations by left translations are faithful?
7.1.12. Let S be a semigroup with an identity. Prove that the representation
of S by left translations is faithful.
7.1.13.T. Assume that the representation ofa semigroup S by left translations
is faithful. Let at , a2 E S. Prove that if at x = a2x for all XES, then a l = a2'
7.1.14.T. Assume that a semigroup S does not contain elements at i= a2
such that alx = a2x for all XES. Prove that the representation of S by left
translations is faithful.
Remark. Problems 7.1.13 and 7.1.14 yield a necessary and sufficient
condition for the representation of a semigroup by left translations to be
fai thf ul.
7.1.15. Define an operation on a set M by xy = x for all x, y E M. Describe
the representation of M by left translations.
7.1.16. Define an operation on a set M by xy = y for all x, y EM. Describe
the representation of M by left translations.
7.1.17. Let ./1/ be a class of semigroups, each of which consists of one-to-one
transformations. Let S be a semigroup with identity having a faithful repre-
sentation in JIt. Prove that S is left cancellative and has no idempotents
different from the identity.
7.1.18.T.H. Prove that every semigroup S has a faithful representation by
transformations.

148
Group Representations
Remark. Thus, up to isomorphism, the class of all semigroups is ex-
hausted by semigroups of transformations.
7.1.19.T. Assume that for every element a in a semigroup S there exists
Za E S such that aZa = Za' Let (X be a faithful representation of S by trans-
formations. Prove that every transformation (X(a) E (X(S) has a fixed point.
7.1.20.T.H. Let S be a semigroup such that for every faithful representation iJ.
of S by transformations, every transformation in the image of (X has a fixed
point. Prove that for every a E S there exists Za E S such that aZa = Za'
Remark. From problems 7.1.19 and 7.1.20 we obtain a necessary and
sufficient condition for a semigroup S to have the property that for every
faithful representation of S by transformations, every transformation has a
fixed point.
7.1.21. Let Q be the multiplicative semigroup of rational numbers. Does Q
have a complete system of representations in the class ./1/ of all groups?
7.1.22. Does there exist a complete system of representations of a cyclic
semigroup of type (5, 7) in the class of all regular commutative semigroups?
7.1.23. Describe all cyclic semigroups which have a complete system of
representations in the class of all groups.
7.1.24. Let C = {O, I} be the multiplicative semigroup under ordinary
multiplication. Describe all finite cyclic semigroups which have a complete
system of representations in C.
7.1.25. Find all semigroups which have a complete system of representations
in the class of all commutative semigroups.
7.1.26. Let./lt 1 ,./It 2 , and ./It 3 be the classes of all left, right, and two-sided
cancellative semigroups, respectively. Find all semigroups which have a
complete system of representations in (1) ./1/ 1 , (2) ./It2' and (3) ./It3'
7.1.27. Let ./It be the class of all regular commutative semigroups. Prove
that every periodic semigroup S having a complete system of representations
in dl" is regular.
7.1.28.". Let all elements in a commutative semigroup S be idempotent.*
Prove that S has a complete system of representations in the semigroup C
in problem 7.1.24.
7.1.29.H. Find all semigroups which have a complete system of representa-
tions in the class of all semilattices.
2. Representations of Groups by Transformations
In this section we will apply to groups the concepts and terminology
introduced in the preceding section for arbitrary multiplicative sets. The
* A semigroup S having the property x 2 = x for all XES is called a band. A commutative band
is called a semi/attice. [Trans.]

Represen tations of Groups by Transformations
149
only representations of groups which we shall consider are representations
by invertible transformations. We shall simply call them representations by
transformations.
A representation C( of a group G is said to be transitive if the image C«G)
is a transitive group.
The importance of transitive representations lies in the fact that every
representation of a group can be written by means of transitive representa-
tions of this group (see 4.9.29 and 7.2.32).
Let G be a group, H a subgroup of G, and X the set of right cosets of H
in G. For each g E G we define the transformation A: of X by A: (xH) = gxH.
The mapping C(H of the group G into the semigroup of transformations
Tx defined by
C(n(g) = A:
is a representation of G by transformations (see 7.2.1 and 7.2.2). We call C(H
the representation of G by H. The importance of this type of representation
lies in the fact that every transitive representation of a group G is similar to a
representation of G by one of its subgroups (see 7.2.22).
We also consider representations of groups by left translations (see
7.1.10). Such representations are always faithful (see 7.2.7). Thus every group
is isomorphic to some group of transformations. In other words, up to iso-
morphism, the class of all groups is exhausted by the class of all groups of
transforma tions.
7.2.1. Let G be a group, H a subgroup of G, and X the set of right cosets of
H in G. Prove that the transformation A: on X defined by A(xH) = gxH is
in verti ble.
7.2.2.T.H. Let G, H, and X be as in 7.2.1. Prove that C(n is a representation of
G by transformations on X.
7.2.3. Prove that the image of the representation of the quaternion group by
the subgroup H = {I, - I} is a direct product of two cyclic groups of order
two.
7.2.4. Let G be the group in problem 5.4818, H = {e, (ab)2}. Find the images
of a and b under the representation C(n. Is this representation faithful?
Remark. Since G = [a, b], knowing the images of a and b under any
representation enables one to easily find the representation itself.
7.2.5. Prove that the image of a representation of a group G by left transla-
tions consists of invertible transformations.
7.2.6. Find the representations by left translations of the following groups:
1) the quaternion group;
2) the group in problem 5.4.18 ;
3) a noncyclic group of order 6;
4) a cyclic group of order 4.

150
Group Representations
7.2.7.T. Prove that the representation of any group G by left translations is
always faithful.
7.2.8. Prove that every group G contains a subgroup H such that lJ.H is a
representation by left translations. What is this subgroup?
7.2.9. Let G be the multiplicative group of matrices of the form
(:  )
where a and b are rational numbers, a #- O. Let H be the subgroup of G
consisting of matrices of the form
( )
Prove that:
1) the representation lJ.H of G by H is similar to the representation lJ. of
G by transformations of the set of nonzero rational numbers defined
by
a( /J = Aa
where AaX = ax for all rationals x ;
2) the representation lJ.H is not faithful.
7.2.10.H. Let G be the group in problem 5.2.21. Prove that:
1) G has a transitive representation by transformations of degree n for
an y posi ti ve integer n ;
2) the group G has at least two similar representations by transforma-
tions of degree 9.
7.2.11.T. Let G be a group, H a subgroup of G. Prove that the image of the
representation of G by H is a transitive group of transformations.
7.2.12. Let G be the group in problem 5.4.18, HI = {e, a}, H 2 = {e, bab}.
Prove that the representations of G by the subgroups HI and H 2 are similar.
7.2.13.H. Let H 1 and H 2 be conjugate subgroups of a group G. Prove that
the representations of G by H 1 and H 2 are similar.
7.2.14.H. Suppose that the representations of a group G by its subgroups
H 1 and H 2 are similar. Prove that HI and H 2 are conjugate subgroups of G.
Remark. Problems 7.2.13 and 7.2.14 yield a necessary and sufficient
condition for two representations of a group G by two subgroups H 1 and H 2
to be similar.
7.2.15. Let G be a group, H a subgroup of G, and r:J.. H the representation of G
by H. Prove that the set K of all elements in G which are mapped onto the

Representations of Groups by Transformations
151
identity by lJ.n is contained in H and forms a normal subgroup of G. Further,
show that K is the maximal normal subgroup of G contained in H.
7.2.16.T. Let G be a group, H a subgroup of G. Prove that if H contains a
normal subgroup N :/:. e of G, then r:J.. H is not faithful.
7.2. 17.T.H. Let G be a group, H a subgroup of G. Prove that if H contains
no normal subgroups of G other than the identity, then lJ.H is faithful.
Remark. Problems 7.2.16 and 7.2.17 yield a necessary and sufficient
condition for a representation of a group by one of its subgroups to be
faithful.
7.2.18. Let H = [(1234), (24)] be a subgroup of th symmetric group S4' Is
the representation lJ.n of S4 by H faithful?
7.2.19.T. Let lJ. be a representation of a group G by transformations on a set
X, a E X. Denote by H the set of elements g E G such that a is a fixed point of
lJ.(g). Prove that H is a subgroup of G.
7.2.20.T. Assume the same conditions and notation as in 7.2.19. Let lJ.(x)(a) =
b for some x E G. Prove that lJ.(y)(a) = b for all y E xH, but lJ.(z)(a) :/:. b for
all z ft xH.
Remark. Compare the results of these last two problems with 4.9.5.
7.2.21.T.H. Assume the same conditions and notation as in 7.2.19, and let
(G) be a transitive group. Define a mapping g on X by g(y) = xH if lJ.(x)a = y
(y E X). Prove that g is a one-to-one mapping of X onto the set of right co sets
ofG by H.
7.2.22.T.H. Assume the same conditions and notation as in 7.2.19, and
let r:J..( G) be a transitive group. Prove that lJ. is similar to the representation of
G by H.
7.2.23.H. Let G be the group defined in 5.4.18. Prove that G has faithful
transitive representations of degree four which are similar.
7.2.24. Prove that the group in the preceding problem has. three similar
faithful transitive representations by transformations. Find these rep-
resen ta tions.
7.2.25.T. Prove that any two faithful transitive representations of an
abelian group are similar.
7.2.26. Determine whether the quaternion group possesses faithful transitive
representations by transformations of degree less than 8.
7.2.27.". Consider HI = [(1234)] and H 2 = [(12), (34)] as subgroups of the
symmetric group S4' Find the images of(12) and (1234) under the representa-
tions ({JHl and ({JH2'
7.2.28.". Prove that the representations ({Jill and ({J1I2 in the preceding
problem are similar faithful representations of S4'

152
Group Representations
7.2.29.H. Prove that the symmetric group S4 has only two similar faithful
transitive representations of degree 6.
7.2.30. Let G be a noncyclic group of order 14 (see 5.4.27). Prove that G
does not have any faithful transitive representations of degree less than 7.
7.2.31.H. Prove that any two transitive representations of degree 7 of the
group G in the preceding problem are similar.
7.2.32. Let C( be a representation of a group G by transformations on a set X
such that C«(G) = G' is intransitive. Let M i be an intransitive system of G',
and denote by Pi the mapping of G into the group of transformations of M i
defined by Pi(g) = lli if ct(g) = ct, where C(i is the transformation defined in
4.9.28. Prove that Pi is a transitive representation of G.
7.2.33.H. Let G be a group, Pg the transformation of G defined by pg(x) =
xg (x E G). Denote by f the mapping of G into the semigroup of all transfor-
mations of the set G defined by f(g) = Pg. For what kind of groups is this
maJJping a representation of G by transformations?
3. Representations of Groups by Matrices
A knowledge of matrix theory is necessary for understanding this
section. All matrices considered in this section will be taken over the field of
complex numbers.
A representation of a group G in the class of all groups of nonsingular
n x n (n = 1,2,...) matrices is called a matrix representation of G. If, for a
specific value of n, all matrices il.1 the image of the representation have degree
n, then we say that the representation is of degree n.
Two matrix representations Tl and T 2 of degree n of a group G are said
to be equivalent if there exists a nonsingular n x n matrix A such that
T2(x) = AT1(x)A- 1
for all x E G.
Let 1i(i = 1,..., k) be matrix representations of degrees m i of a group G.
The mapping T which associates to every element x E G the matrix T(x) in
block form of order n = m 1 + ... + m k with diagonal matrices T1(x), T 2 (x),
. . . , (x) is called a matrix representation of G of degree n = m 1 + m2 +
.. . + m k (see 7.3.13). Further, Tis called the direct sum of the representations
Tl,T2""''
A matrix representation T of G which is equivalent to a direct sum of
representations of G of degrees less than the degree of T is said to be com-
pletely reducible. Sometimes completely reducible representations are
called decomposable.
Let T be a matrix representation of G of degree n. Let P be a nonsingular

Representations of Groups by Matrices
153
matrix such that the matrix S(x) = PT(x)P- 1 has the form
s (x) = ( T. (x) 0 )
A (x) T 2 (x)
where Tl(X) and T 2 (x) are matrices of degrees m 1 and m2 (m 1 + m2 = n),
A(x) is an m2 x m 1 rectangular matrix, and 0 is the ml x m2 zero matrix.
Then T is called a reducible representation of G; otherwise T is called ir-
reducible.
It is obvious that every completely reducible (decomposable) representa-
tion is reducible. In general the converse does not hold (see 7.3.16). However,
every reducible representation of a finite group is completely reducible (see
7.3.20).
7.3.1. Let G be an arbitrary group. Prove that the mapping
To(x) = En
(x E G)
where En is the n x n identity matrix, is a matrix representation of G of
degree n. Further, prove that the identity of G corresponds to En under every
matrix representation of G of degree n.
Remark. To is called the identity matrix representation of G of degree n.
7.3.2. Let Sn be the symmetric group of degree n, S E Sn,
s= (  2 ... n )
It i g .. · in
Let T be the mapping of Sn into the group of nonsingular n x n matrices
defined by T(s) = (a ij ), where alh = a2i2 = . .. = a nin = 1 and all other
aij = O. Prove that T is a matrix representation of Sn.
7.3.3. Let G be a group of rotations about the Oz-axis in three-dimensional
space R3' Let x E G be the rotation through the angle (Xx' Prove that the
mappIng
T(x) =
sIn (X x
- sin (Xx 0\
cos (Xx 0 )
o 1
xEG
cos (Xx
o
is a matrix representation of G.

154
Group Representations
7.3.4. Let a grou p G be defined by the generating set {a I , a 2 } and the defining
set of relations: ai = e, a = e, a2al = aia2' Prove that G = {e, ai' ai,
a2, at a 2, ai a 2}' Define a mapping T by
( 1 0 0\
T (e) = 0 1 0 J
o 0 1/
( ,0 0 1 \
T(ai)= 1 0 0 )
010
( 1 ° 0\
T (ala2) = 0 0 1 )
010
T(al)= (    )
1 0 0
(0 1 O )
T(a2)=: 1 0
\0 0
(0 ° 1 \
T(a:a 2 )= \ 0 1 0 )
1 0 0/
Is T a representation of G?
7.3.5. Let G be a grou p of rotations aoou t a fixed axis in three-dimensional
space R 3 , and for every g E G let qJg be the angle of rotation. Is the mapping
T (g) = ( CPg 0 )
o <fg
a matrix representation of G of degree 3?
7.3.6. Let a grou p G be defined by the generating set {a 1 , a2} and the defining
set of relations ai = e, a = e. Define a mapping T of the generating set
{a l , a2} into the group G' of nonsingular 2 x 2 matrices by
T(a 1 )=G }
T(ai)=( )
Can T be extended to a matrix representation T' of G of degree 2?
7.3.7. Let C be the multiplicative group of nonzero complex numbers.
Prove that C has a faithful representation of degree 2 by matrices over the
real numbers.
7.3.8.H. Prove that every finite group of order n has a nonidentity matrix
representation of degree n.
7.3.9. Let G be a group, T a matrix representation of G of degree n, and A
any nonsingular n x n matrix. Prove that the mapping T A defined by TA(x) =
A T(x)A - 1 (x E G) is a matrix representation of G.

Representations of Groups by Matrices
155
7.3.10. Does the symmetric group Sn have any nonidentity matrix rep-
resentations of degree n which are different from the representation given
in 7.3.2?
7.3.11. Let G be a group of rotations about the Ox-axis of the three-dimen-
sional vector space R3' For every g E G, let qJg be the angle of rotation. Prove
that the mapping T' defined by
( COS fg + 2 sin fg
T' (g) = sin tpg
o
- 5 sin tpg
- 2 sin 'g+ cos 'g
o
3COS'g+ 16Sin,g--3 )
7 sin Cfg- 2 COSfg+2
1
is a matrix representation of G of degree 3.
7.3.12. Prove that all matrix representations of a group G which are equiva-
lent to a given representation T are equivalent.
7.3. 13.T. Let 1; (i = 1,..., k) be matrix representations of degrees mi of a
group G. Let T be the mapping of G into the group of nonsingular matrices of
order n = ml + . . . + mk defined in the following manner: for every x E G,
let T(x) be the matrix with diagonal blocks Tt (x), . . ., Ji(x) (see the introduc-
tion to this section). Prove that T is a matrix representation of G of degree n.
7.3.14.T. Let T't and T be two equivalent matrix representations of degree
m of a group G, and let T'; and T be two equivalent matrix representations
of degree k of G. Let Tt denote the direct sum of T't and T';, and T 2 the direct
sum of T and T'2 . Prove that T 1 and T 2 are equivalent matrix representations
of G of degree m + k.
7.3.15. Let G = [a] be an infinite cyclic group. Consider the mapping
T(an>=G )
(n = 0, + 1, ...)
Is T a completely reducible representation?
7.3.16. Let T be an irreducible matrix representation of degree n of a group
G, and let X be a nonzero column with n entries. Prove that the set of all
linear combinations of columns of the form T(x)X exhausts the entire
space of columns with n entries.
7.3. 17.T.H. Let T be an irreducible matrix representation of degree n of a
group G, and let A be an n x n matrix which commutes with all the matrices
T(x), x E G. Prove that A is a scalar matrix, i.e., A = a.E n for some number a.
(En is the identity matrix of order n).
7.3.18.T.H. Prove that every reducible matrix representation of degree n
of a finite group G is completely reducible.

156
Group Representations
7.3.19.T.H. Let G be a finite group. Prove that every reducible matrix
representation of G is a direct sum of irreducible representations of G.
7.3.20.H. Let G be a finite group, T an irreducible matrix representation of
degree n of G, and A any n x n matrix. Denote by X the matrix L T(x)
A T(x- 1). Prove that X = rxE n for some number rx, where En is thee x n
identity matrix.
7.3.21.T.H. Prove that every matrix representation of degree n  2 of an
abelian group G is reducible.
4. Groups of Homomorphisms of Abelian Groups
Let A and B be abelian groups. We consider the set R(A, B) of all
representations of the group A in the class consisting of the group B, and
introduce an operation on R(A, B) by setting rxlrx2 = rx3 (rx 1 , rx2, rx3 E R(A, B))
if rx 1 (x)' rx 2 (x) = rx 3 (x) for all x E A (see 7.4.1). Then R(A, B) forms an abelian
group under this operation (see 7.4.2).
The group R(A, B) will be called the group of homomorphisms or the group
of representations of A into B. The operation indicated above on the set
R(A, B) can also be considered for arbitrary groups, but in this case the
operation may not be defined on all of R(A, B).
If A = B, then R(A, A) is the set of endomorphisms of the group A.
The set R(A, A) forms a group under the operation indicated above, called the
group of endomorph isms of A. Thus there are two distinct operations on the
set R(A, A): the operation introduced above, and composition (cf. Chapter
3.4). In this section, the only operation which we shall consider on R(A, A) is
the one introduced above, and we shall denote it by a dot.
7.4.1. Let rx I and rx2 be homomorphisms of an abelian group A into an abelian
group B. Prove that the mapping rx 3 defined by rx 3 (rx) = rxl(OC)' rx 2 (rx) (rx E A)
is a homomorphism of A into B.
7.4.2. Prove that the set R(A, B) of all homomorphisms of an abelian group A
into an abelian group B forms a group relative to the operation defined above
on R(A, B).
7.4.3. Find R(A, B), where A and B are infinite cyclic groups.
7.4.4. Let A be an infinite cyclic group. Prove that R(A, B) is isomorphic to
B for any abelian group B.
7.4.5. Let A be a periodic abelian group, and let B be a torsion-free abelian
group. Find R(A, B).
7.4.6. Let A and B be primary cyclic groups of orders P 1 and p2, respectively.
Prove that:
1) if PI = P2, then R(A, B) is a cyclic group of order p;nio(k 1 ;k 2 ) 
2) if PI #- P2, then R(A, B) is the identity group.

Groups of Homomorphisms of Abelian Groups
157
7.4.7. Let A be a cyclic primary group of order pn, B any abelian group.
Determine R(A, B).
7.4.8. Find the group of endomorphisms of an abelian group G of order pq,
where p and q are distinct primes.
7.4.9. Prove that the group of endomorphisms of the additive group of
rational numbers R is isomorphic to R.
7.4.10.T. Prove that if A is the direct product of abelian groups Ai (i = 1,
. . . , n), then R(A, B) is isomorphic to the direct product of the groups R(A i , B)
(i = 1,..., n) for any abelian group B.
7.4.11.8. Let an abelian group A be defined by the generating set {a l , a2, a3}
and the defining set of relations af = e, a = e. Let B be an infinite cyclic
group. Find R(A, B).
7.4.12.8. Let A be a finitely generated abelian group of rank r, and let B be
an infinite cyclic group. Find R(A, B).
7.4.13.8. Let A be a finite abelian group with invariants 3 2 , 3, 2 and let B
be a cyclic group of order 2. Find R(A, B).
7.4.14.8. Let A be a finite abelian group with invariants
53 5 i , 5, 5, 5
,
3 3 , 3 3 , 3, 3
2 8 2 7 2, 2
, ,
and let B be a cyclic group of order 25. Find R(A, B).
7.4.15.8. Let A be the direct product of a cyclic group of order 7 and an
infinite cyclic group. Let B be a cyclic group of order 49. Find R(A, B).
7.4.16.8. Let A be a finite abelian group with invariants 3 2 , 3, 3, 3; 55; 7 3 , 7,
7, 7, and let B be a mixed abelian group whose periodic part is a cyclic group
of order 5. Find R(A, B).
7.4.17.T. Let B be the direct product of the abelian groups B i (i = 1,..., n).
Prove that for any abelian group A, R(A, B) is isomorphic to the direct pro-
duct of the groups R(A, B i ) (i = 1,..., n).
7.4.18.". Find R(A, B) for the following pairs of finite abelian groups A
and B which are given by their respective invariants.
1) A is a cyclic group of order 5, B has invariants 3 2 , 3; 5; 7 3 , 7;
2) A has invariants 2 3 , 2, 2; 3 2 , 3; 5, 5, 5; B has invariants 52, 5; 3 3 , 3 2 ;
7 3 , 7;
3) A has in varian ts 5 3 , 5 2 , 5 ; 3 3 , 3 2 , 3 ; 2 ; B has in variants 2 ; 7 3 , 7 2 , 7 ; 11.
7.4.19.". Find R(A, B) for the following pairs of abelian groups A and B

158
Group Representations
which are defined by the indicated generating sets and defining relations:
I) A=[a., ab aa], at=e, ala2=a')a.
a.as = aaa., aa3 = a3a
B=[b., bi]' b;=e, b 1 b 2 =b 2 b.
2) A=[a., Q21, aia=e, a:a=e, ala=aal
B = [b., b 2 ], b = e, b.b'J = bb.
3) A=[a., a21, aia;7=e, ala=e, ala=a2al
B=[b., b 2 J, bf7=e, b4 7 =e; b 1 b'J=b g b l
7.4.20.H. Find the groups of endomorph isms of the following abelian
groups:
1) a finite group with invariants 2 3 , 2 2 , 2; 3 3 , 3; 52, 5; 7 2 , 7;
2) a finite group with invariants 3, 3, 3 ; 52, 5, 5 ; 7, 7;
3) a finitely generated group of rank 2 with invariants 5, 5, 5 ; 7 3 , 7;
4) a group which is a direct product of seven infinite cyclic groups.
7.4.21.H. Prove that if A and B are finite abelian groups, then R(A, B) is a
finite group.
7.4.22.H. Prove that the group of endomorphisms of a finitely generated
abelian group A is finitely generated.
7.4.23.". Let A be a finite abelian group, B a finitely generated abelian
group. Prove that R(A, B) is a finite group.
7.4.24.H. Let A be a finitely generated abelian group, B a finite abelian
group. Prove that R(A, B) is a finite group.
7.4.25. Let A be an abelian group in which all elements have finite order
less than or equal to k. Prove that for any abelian group B all elements in
R(A, B) also have finite order less than or equal to k.
7.4.26. Let A be a periodic abelian group.
1) Is it possible for the group of endomorphisms of A to be a periodic
group in which the orders of the elements are unbounded?
2) Is R(A, A) a divisible group?
7.4.27. Prove that if the group of endomorphisms of an abelian group A
is divisible, then so is A.
7.4.28.H. When is the group of endomorphisms of a finitely generated
abelian group (1) a finite cyclic group, and (2) an infinite cyclic group?
7.4.29. Let A be a p-primary abelian group and let B be any abelian group.
Does there always exist a nonidentity element in R(A, B) (i.e., different from
(Xo) for which the equation (Xpk = (XO is solvable for every positive integer k?
7.4.30. Let B be a torsion-free abelian group. Prove that R(A, B) is also
torsion-free for any abelian group B.'

Characters of Groups
159
7.4.31. Let B be a divisible torsion-free group. Prove that R(A, B) is a divisible
torsion-free group for any abelian group A.
7.4.32. Is it possible for the group of endomorphisms of a divisible group to
be a periodic abelian group?
7.4.33. Let A be a divisible group. Prove that R(A, B) is a torsion-free group
for any abelian group B.
7.4.34.". Let A be a finitely generated abelian group. Determine when
R(A, A) is isomorphic to A.
5. Characters of Groups
Let T be a matrix representation of degree n of a group G. By the
character of T we mean the mapping X of G into the set of complex numbers
defined by
n
X (a) =  au
i=1
where the au are the diagonal entries of the matrix T(a). It follows from 7.3.1
that every irreducible representation of an abelian group is a homomorphism
into the multiplicative group C of complex numbers different from zero.
Thus the character of every irreducible representation of an abelian group
is a homomorphism into C.
The set of all homomorphisms of an abelian group G into the multi-
plicative group of complex numbers modulo 1 forms a group relative to the
operation defined in the preceding section. This group is called the group
of characters of the abelian group G.
7.5.1. Find the characters of the representations of the groups stated in 7.3.3,
7.3.4, and 7.3.15.
7.5.2. Let T be the representation of Sn indicated in 7.3.2, and let X be the
character of this representation. Prove that X(s) = k for all s E Sn, where k
is the number of elements which are left fixed by the transformation s.
7.5.3. Prove that two characters X 1 and X2 of equivalent matrix representa-
tions Tl and T 2 of a group G are equal.
7.5.4. Let the matrix representation T of a group G be the direct sum of the
representations T 1 , T 2 , · . . , Tm and let X be the character of T and Xi the
character of T;, i = 1,..., m. Prove that
m
X. (a) =  Xi (a)
i=1
for each a E G.

160
Group Representations
7.5.5. Is it possible for a nontrivial group G to have the property that the
character X of each nonfaithful matrix representation T of degree n be of the
form x(a) = n for each a E G?
7.5.6. Let x and y be conjugate elements in a group G, T a matrix representa-
tion of G, and X the character of T. Prove that X(x) = X(y).
7.5.7.H. Find all irreducible representations of the cyclic group [a] of
order n.
7.5.8.H. Let G be a finite abelian group. Prove that the characters of all
irreducible representations of G are mappings whose images consist of roots
of 1.
7.5.9. Find the group of characters for each of the following abelian groups
which are defined by the indicated generating sets and defining relations.
pk h . .
a =e, were p IS a prIme
a: = e, a = e, a1a = aal
a=a, a=e, ala=a2al
1) A = [a],
2) A = [ah a2]'
3) A= [ah a2]'
7.5.10.T.H. Prove that the group of characters of a finite abelian group G is
isomorphic to G.
7.5.11. Determine the group of characters of an infinite cyclic group.
7.5.12. Let G be the direct product of r infinite cyclic groups. Find the group
of characters of G.
7.5.13. Prove that every character of a subgroup H of a finite abelian
group G can be extended to a character on all of G.
7.5.14.H. Let G be a finite abelian group, at, a 2 E G, at "# a2' Show that there
exists a character X of G which separates at and a 2 , Le., x(a 1 ) "# x(a 2 ).
Remark. It follows from this problem that the set of all characters of a
finite abelian group forms a complete system of representations (cf Chapter
7.1) in the class consisting of the multiplicative group of complex numbers
modulo 1. This result also holds for infinite abelian groups.

Chapter 8
TOPOLOGICAL AND ORDERED GROUPS
1. Metric Spaces
Let M be a set and let p be a mapping of the Cartesian product M x M
into the set of nonnegative real numbers [in other words, to every pair
(x, y) of elements in M associate a real number p(x, y)  0]. This mapping is
called a metric, or a distance function (d is often used instead of p) ifit satisfies
the following three conditions:
1) p(x, y) = 0 if and only if x = y;
2) p(x, y) = p(y, x) for all x, y EM;
3) p(x, y)  p(x, z) + p(z, y) for all x, y, Z EM.
The set M together with the metric p defined on M is called a metric space
relative to p. The elements of a metric space are called points, and the number
p(x, y) is called the distance between the points x and y.
A sequence {x n } of points in a metric space is said to converge to a
point XoE M if lim p(x n , xo) = O. In this case we write X n -+ Xo or lim X n =
n-oo
Xo, and say that the limit of the X n is Xo. For A c M, the point a E M is
called a limit point of A if there exists a sequence {an} of distinct points in A
which converges to a.
A sequence {xn} of points in a metric space M is called fundamental if
lirn P (x m , x n ) = 0
n, m-+oo
A metric space M is called complete if every fundamental sequence in M
converges to a point in M.
A metric space M is called separable if it contains a countable subset
A c M such that for every x E M there exists a sequence an E A whose
limit is x. A subset E of M is said to be compact if every sequence {xn} of
points in E has a convergent subsequence {x nk }. The space M is called com-
pact if it is a compact subset of itself.
A subset E of M is said to be bounded if there exists a nonnegative
number k such that p(x, y)  k for all x, y E E. If M itself is a bounded set,
then p is called a bounded metric.
161

162
Topological and Ordered Groups
Let PI and P2 be two metrics on a set M such that the convergence of the
sequence {xn} to Xo relative to P I implies the convergence of {xn} to Xo
relative to P2, and vice versa. We then say that P I and P2 are equivalent
metrics.
As examples of metric spaces we consider the n-dimensional vector
spaces R n (n = 1, 2, . . .), i.e., the Cartesian product of n copies of the real
numbers R in which addition and scalar multiplication are defined by
1) (at, a 2 , ..., an) + (b 1 , b 2 , ..., b n )=
= (at + b h a 2 + bh ..., an + b n )
2) A (a H ..., an)=(AaH...,Aa) (ai,bj,A E R)
and the metric P is defined by
p(x, y)= V (X 1 -Yl)2+ ... +(x n -Yn)2
where x = (Xl' . . . , x n ), Y = (YI' . . . , Yn) [see 8.1.1, (3)].
8.1.1.H. For each of the following sets M i define a mapping Pi from M i x M i
into the set of nonnegative real numbers. Determine which of these are
metric spaces.
1) M I any set
{ 0, if x = y;
Pt (x, y) = . f -I- (x, y E Mt)
1,1 x-r-Y
2) M 2 the set of real numbers
P2 (x, y) ==-1 x - Y I
(x, y E M g )
3) M 3 an n-dimensional vector space,
pa (x, y) = V(Xt - Yt)'J + ... + (X n - Yn)'A
where X = (Xl" . · , x n ), Y = (YI" · . , Yn);
4) M4 = [a] a cyclic group of order n
Pi (a k , al) = I'l - II,
o k, l<n
5) M 5 the group of 2 x 2 nonsingular matrices
Ps(X, y)= I Dx - Dy I
where Dx and Dy are the determinants of X and Y, respectively;

Metric Spaces
163
6) M 6 the symmetric group Sn
P6(X, Y)=lx(I)-y(I)1
(x, y E Sn)
7) M 7 the symmetric group Sn
P'1(x, y)= max Ix(k)-y(k)1
kE{I. 2, ...,n}
8.1.2. Let C[a,b] denote the set of all continuous functions on the interval
[a, b].
1) Prove that the mapping
(x, y E Sn)
p(f, g)= max I/(x)-g(x) I
x E la, b)
(/, g E CIa, b)
is a metric.
2) Is the following mapping PIa metric:
b
PI (f, g)=  I I (x) - g(x) I dx
a
(/, g E C(a, b)
8.1.3. Let X be a metric space with metric p. Determine which of the follow-
ing mappings are metrics :
1) Cf>. (x, y) = pi (x, y)
1
2) Cf>i (x, y) = p (x, y) + 1
3) Cf>a (x, y)== e P (x, y) - 1
4) Cf>,(x, y)= V p(x, y)
8.1.4. Let S be the set of all sequences of real numbers. For
x = (x 1 , X2, . . . , x k , . . .) E S,
Y = (y 1 , Y 2 , · . . , Y k, . . .) E S
set
00 1 IX k - Ykl
p(x,y) = I 2 k 1 + I I
k = 1 Xk - Yk
Prove that S is a metric space relative to p.
8.1.5. Let p be a metric on a space )( and let k be a positive number. Prove
that the mapping p defined by
p(x, y) = min{p(x, y), k}
(x, Y E X)
is a metric.

164
Topological and Ordered Groups
8.1.6. Let P be a metric on a space X. Determine for what real numbers a
and b the mapping lJ. defined by
lJ.(x, y) = a . p(x, y) + b
(X,YEX)
is a metric.
8.1.7. Let PI and P2 be metrics on a set X. Determine which of the following
mappings are metrics :
1) l (x, y) = Pi (x, y) + pg (x, y)
2) 2 (x, y) = e Pt (x. y) + e P2 (x. y)
3 ) "1 ( X y) = Pt (x, y) + 1
'1'3' P 2 (x, y) + 1
8.1.8. Let u, x, Y, z be arbitrary points in a metric space with metric P;
prove that
I P (x, y) - P (z, ll) I  P (x, z) + P (y, ll).
8.1.9. Let X be a metric space with metric P, and let x n , Yn E X (n = 1, 2, . . .).
Prove that if X n converges to Xo E X and Yn converges to Yo E X, then p(x n , Yn)
converges to p(xo, Yo)'
8.1.10. Suppose that in a metric space X the sequence X n converges to Xo
and x. Show that Xo = x.
8.1.11. Define a metric P on C[a,b] (see 8.1.2) by
p(f, g) = maxlf(x) - g(x)1 x E [a, b]
Let f" E C[a,b] be a sequence which converges to f in the sense of the metric
on 'C[a,b]' Does fn converge uniformly to f? Assume now that a sequence
f" E C[a,b] converges uniformly to f E C[afb]; does fn converge to f in the
sense of the metric p?
8.1.12. Prove that every convergent sequence in a metric space is funda-
mental.
8.1.13. Prove that the set of real numbers is a complete metric space with
respect to the metric p(x, y) = Ix - yi.
8.1.14. Which of the following sets are complete metric spaces:
1) the set of rational numbers with metric
PI (x, y) = I x - Y I
2) the set of sequences of real numbers x = (x 1 , . . . , x n , . . .), where X n
converges to zero as n  00, with metric
Pi (x, y) = max I x n - Yn I
n

Metric Spaces
165
3) the set of all bounded continuous real functions with metric
Pa(}, g)=supl/(x)-g(x)1
x
4) the set of all continuous functions defined on the interval [a, b], with
the metric P from 8.1.2, (1)1
8.1.15. Consider the set R of real numbers as a metric space with the metric
P2 from 8.1.1, (2). Prove that R is separable.
8.1.16.H. Let C[a.b] be the metric space of continuous functions on [a, b]
with the metric P from 8.1.2, (1). Prove that C[a,b] is separable.
8.1.17.H. Let M T denote the set of all real bounded functions defined on an
infinite set T. Can a metric be introduced on M T for which it is a nonseparable
metric space
8.1.18. Which of the following sets are compact metric spaces:
1) the set of real numbers relative to the metric
p(x, Y)=lx-YI
2) the n-dimensional vector space Rn relative to the metric
p (x, y)= V(Xt - Yt)'I. +... +(x n - Yn)'1.
where x = (Xl' · . . , x n ), Y = (YI' . . . , Yn); \
3) the space of all continuous functions on the interval [a, b] with the
metric P from 8.1.2, (1)?
8.1.19. Prove that every closed interval is a compact set in the metric space
of real numbers (see 8.1.13).
8.1.20.H. Prove that every metric is equivalent to a bounded metric.
8.1.21.T. Prove that every compact subset of a metric space is bounded.
8.1.22.T. Prove that every compact metric space is complete.
8.1.23.T. Let M and N be metric spaces and let R = M x N be the Cartesian
product of M and N. Define a mapping PIon R x R by
PI [(XI' Yl), (Xi' Yi)] = P (Xb Xi) + P (Yb yJ
where Xl' X 2 EM, Yl, Y2 EN. Prove that R is a metric space relative to PI'
Remark. We call the space R th product of M and N.
8.1.24. Prove that the product of two compact metric spaces is compact.
8.1.25. Prove that the product of two complete metric spaces is complete.
8.1.26. Prove that the product of two separable metric spaces is separable.
8.1.27. Let M be a metric space with metric P and let M' be a subset of M.

166
Topological and Ordered Groups
Define a mapping p' of the set M' x M' into the set of nonnegative real
numbers by
p' (x', y') = p (x', y')
(x', y' EM')
Prove that M' is a metric space relative to p'.
Remark. The space M' is called a subspace of M.
8.1.28. Prove that every subspace of a separable metric space is separable.
8.1.29. Prove that every subspace of a compact metric space is compact.
8.1.30. Let R be the metric space of real numbers with the metric defined in
8.1.13. Show that R contains a compact subspace.
Remark. Compare this result with problem 8.1.18, (1).
8.1.31. Let Z be the set of nonnegative integers, and define a mapping p on
the set Z x Z by
o if i = j
p(i,j) = 1 if i and j are even, i -# j
2 if either i or j is odd, i -# j
Prove that p is a metric on Z, and that p is equivalent to the metric in problem
8.1.1, (1).
8.1.32. Let G = [a] be an infinite cyclic group. Define a mapping p of
G x G into the set of nonnegative real numbers by
p (a k , a') = I k -I)
(I, k=O; + 1; + 2; ...)
Prove that p is a metric, and that it is equivalent to the metric in problem
8.1.1, (1).
2. Groups of Continuous Transformations of a Metric Space
Let M be an arbitrary metric space. A transformation f of M is called
continuous iffor every point Xo E M and every sequence {xn} which converges
to Xo, the sequence {f(x n )} converges to f(xo). An invertible transformation
f of M is called bicontinuous if both f and its inverse f - 1 are continuous.
The set of all invertible bicontinuous transformations of M forms a
group under composition of transformations (see 8.2.6). A group G of
invertible transformations of M in which every fE G is continuous is called a
group of continuous transformations of M.
Let G be a group of continuous transformations of M,f, g E G. If
sup p if (x), g (x»
xEM

Groups of Continuous Transformations of a Metric Space
167
exists, then it is natural to associate this number with the pair (f, g). Denote
p* if, g) = sup p if (x), g (x»
xEM
Then p* will be a metric on G if sup p(f'(x), g(x)) exists for every pair of
xeM
elelnents f, g E G (see 8.2.17). We call p* the natural metric on the group of
continuous transformations of M.
There exist metric spaces such that it is impossible to introduce the
natural metric on the group of continuous transformations of these spaces
(see 8.2.22). However on every metric space M with metric p there exists a
bounded metric Po which is equivalent to p. One can, of course, introduce the
natural metric on the group of continuous transformations of M relative to
Po (see 8.2.23).
An invertible transformation f of a metric space M with metric p is
called an isometry of M if
p (f (x), f (y) = p (x, y)
The three-dimensional Euclidean space R 3 is a metric space relative to
the metric defined in 8.1.1, (3). Consequently we can speak of the isometries
of R 3 . The concept of an isometry of R 3 qua metric space coincides with the
concept of an isometry of R 3 when considered as in Chapter 3.5 (see 8.2.10).
Thus the concept of an isometry of a metric space is a generalization of the
concept of an isometry of the line, plane, or three-dimensional space. Con-
sequently this section, which is devoted to isometries, is a continuation of
Chapter 3.5.
8.2.1. Let M be a metric space and fix Xo E M. Prove that the transforma-
tion f of M defined by f(x) = Xo for all x E M is continuous.
8.2.2. Let M be a metric space with the metric
{ o ifx=y
p(x, y) = 1 if x # y
Prove that every transformation of M is continuous, and every invertible
transformation is bicontinuous.
8.2.3. Which of the following functions are continuous transformations of
the space of real numbers [see 8.1.1, (2)]:
1) ft(x) = x 2 + 5;
2) f2(x) = 1/(x 3 + 2);
{ I if x is rational
3) fJ(x) = 0 if x is irrational
[see 8.1.1, (1)]

168
Topological and Ordered Groups
4) f4(x) = ax + b, where a and b are fixed real numbers;
{ I if Ixl > 2
5) fs(x) = 0 if Ixl  2
8.2.4. Let R be the metric space of real numbers with the metric defined in
8.1.1, (2). Determine what real functions defined on all of R are continuous.
Which functions are bicontinuous?
8.2.5.T. Prove that the product of two continuous transformations of a
metric space is continuous.
8.2.6. T.H. Prove that the set of all bicontinuous invertible transformations
of a metric space forms a group of transformations.
8.2.7. Prove that every isometry of a metric space M is a bicontinuous
transformation of M.
8.2.8. Let R J be the three-dimensional real vector space, and introduce the
metric from 8.1.1, (3) on RJ' Let F denote a figure in RJ (a subset of RJ)
and let G be the group of self-coincidences of F (see Chapter 3.5). Prove that
G is a group of continuous transformations of RJ'
8.2.9.H. Prove that the set of all isometries of a metric space forms a group
of continuous transformations.
8.2.10. Let M be a three-dimensional vector space, and introduce the metric
from 8.1.1, (3) on M. Prove that a transformation f of M qua metric space
is an isometry if and only iff satisfies the definition of an isometry as given
in Chapter 3.5.
8.2.11. Let R be the metric space of real numbers [see 8.1.1, (2»). Consider
the set G of all transformations of R of the form
f(x) = x + a
(x E R)
where a is an arbitrary but fixed real number. Prove that every transforma-
tion in G is an isometry of R. Does G form a group of continuous transforma-
tions?
8.2.12. Let  be an n-dimensional vector space. Introduce the metric from
8.1.1, (3) on . Consider the set G of all transformations h of  defined by
fa. (X., X'l, ..., x n ) = (axb CXX2J ..., a.JC n )
where (Xl' . . . , X n ) E V n and lJ. #- 0 is a fixed real number. Prove that each h
is an invertible bicontinuous transformation of , and that G forms a group
of continuous transformations of .
8.2.13. Let V n be as in 8.2.12. Let G be the set of all nonsingular linear
transformations of  onto itself. Prove that each f EGis bicontinuous and
that G forms a group of continuous transformations of .

Groups of Continuous Transformations of a Metric Space 169
8.2.14. Let f be a continuous transformation of a metric space M and let
M' be a compact subset of M. Prove that the image of M' under f is a compact
subset of M.
8.2.15. Let f be a continuous transformation of a compact metric space
with metric p. Prove that for all e > 0 there exists i5 > 0 such that
p (/(x), fey» < e
for all x, Y E M with p(x, y) < i5.
8.2.16.H. Let M be a compact metric space with metric p. Let f, g be con-
tinuous transformations of M. Show that
max p (/ (x), g(x»
xEM
always exists.
8.2.17. Let M be a metric space with metric p. Let G be a group of continuous
transformations of M such that
sup p (/ (x), g(x»
xEM
exists for all f, g E G. Prove that the mapping p* defined by
.;. (/, g) = sup p (f(x), g(x»
xEM
is a metric on G, i.e., G admits the natural metric.
8.2.18.H. Let G be the group of all isometries of a compact metric space M
(see 8.2.9). Prove that the natural metric can be introduced on G.
8.2.19.H. Let G be the group of all isometries of a compact metric space M
and introduce the natural metric p* on G (see 8.2.18). Prove that G is a compact
metric space.
8.2.20. Let G be a group of bicontinuous transformations of the metric
space R of real numbers [see 8.1.1, (2)]. Can the natural metric be introduced
on G?
8.2.21. Let M denote the unit interval [0, 1]. Consider the mapping p defined
by
p(x, y)=lx-yl
(x, Y E M)
Show that p is a metric. Let G be the group of all invertible bicontinuous
transformations of M (see 8.2.6). Prove that the natural metric can be
introduced on G.

170
Topological and Ordered Groups
8.2.22. Let G be a group of bicontinuous transformations of the metric
space R of real numbers [see 8.1.1, (2)] as defined in 8.2.12. Can the natural
metric be introduced on G?
8.2.23.H. Let M be a metric space with metric p. Prove that there exists a
metric Po which is equivalent to p and such that the natural metric can be
introduced on the group of continuous transformations of M relative to Po.
3. Topological Spaces
Let X be any set and denote by P(X) the set of all subsets of X. A trans-
formation ex of P(X) is called a closure operator of X if it satisfies the following
conditions:
1) ex(A) = A for every subset A c X consisting of one element;
2) ex(A U B) = ex(A) U ex(B) for all A, BE P(X);
3) ex(ex(A» = ex(A) for all A E P(X).
The set ex(A) is called the closure of A and is denoted by A.
If a closure operator ex is defined on a set X, then we say that ex defines a
topology on X or that a topology is introduced on X. In such a case X is
called a tOPQlogical space relative to the closure operator ex.
A subset F of a topological space X is said to be closed if F = F. In
other words, closed sets are those sets which are fixed points under c(. A
subset G c X is called open if X'\.G is closed.
A family of open sets 31 in a topological space X is called a basis if
every open subset of X is a union of sets in 31. It is obvious that every topo-
logical space has a basis.
A mapping f of X into a topological space Y is said to be continuous
if f(A) c f( A) for every A c X. The mapping f is called a homeo morph ism
if it is one-to-one and preserves the closure operator, Le., f(A) = f(A) for
every A c X. The spaces X and Yare then said to be homeomorphic. Thus
homeomorphic spaces differ only by the nature or notation of their elements.
A mapping f of X into Y is called open if f( G) is open in Y for every open
set G in X.
8.3.1.T. Let M be a metric space and P(M) the set of all subsets of M. Set
ex(A) = A (A E P(M), where A consists of all elements in A and all of its
limit points. Prove that M is a topological space relative to the operator ex.
8.3.2. Which of the following transformations exi (i = 1, 2, 3, 4, 5) are closure
operators on the sets Ai given below?
1) Let A 1 be an infinite set. Define ex 1 (M) = M if M is a finite subset of
A 1 and ex 1 (M) = A 1 if M is infinite.
2) Let A 2 = {ao, al"'" an," .}. Set ex2(M) = M if M is a finite subset
and ex 2 (M) = M U ao if M is infinite.

Topological Spaces
171
3) Let A3 = {at, a 2 , a 3 }, and for M = {a2, a3} put cx 3 (M) = {at, a2}.
4) Let A4 = {..., a - 2 , a - 1 , ao, at, a2 , · . .} and set cx 4 ( M) = M U ao
for every M c A4'
5) Let As = al' a2"'" an," .}. If M is infinite set cxs(M) = M and
if M = {ail"'" a ik } set cxs(M) = M U {an, a n + 1'" .}, where n =
max {i t , i 2 , . · . , i k } .
8.3.3. Let X be an arbitrary set. Show that it is always possible to introduce
a topology on X for which every subset of X is closed, and that this topology
. .
IS unIque.
Remark. This topology is called the discrete topology.
8.3.4. Let A = {a o , at, a2,' . .}. Define a closure on the set of all subsets of A
as indcated in 8.3.2., (2). Determine which of the subsets listed below are
open and which are closed.
1) Ml={a., a2, aa, ...}
2) M=A"{ah a2, ..., Q6}
3) Ma = {aa, ai' a};
4) M, = {aa, ah a2, a3, ...}
5) MIS = {aa, a, a" at., ..., aik' ...}
8.3.5. Let X be a topological space, and M and N subsets of X. Prove that
the following relations hold: (1) M c M , (2) if MeN, then M c N ,
(3) M n N c M n N , and (4) if M is finite, then M = M.
8.3.6. Let X be a metric space, Xo an arbitrary but fixed point in X, and k a
positive number. Prove that the set of all elements x E X such that p(x, xo)  k
is closed relative to the topology defined in 8.3.1.
8.3.7. Let R be the set of all real numbers. Introduce a topology on R relative
to the metric p(x, y) = Ix - yl (x, y E R), as stated in 8.3.1. Prove that every
open set in R is a union of a countable number of closed sets in R.
Remark. The topological space R is usually called the real line.
8.3.8. Let A be a set of points on the real line and denote by A' the set of all
limit points of A. Does there exist a subset Ao of the real line such that
A = (A)' is nonempty and A = (A)' is empty?
8.3.9. Find the closure of the set of all rational numbers on the real line (see
8.3.7).
8.3.10.T. Prove that in any topological space:
1) the intersection of any family of closed sets is closed;
2) the union of a finite number of closed sets is closed.
8.3.11. Let M be a metric space on which the topology as indicated in
8.3.1. is introduced and let k be an arbitrary positive number. Prove that for
any Xo E M, the set of all points x E M such that p(x, xo) < k forms an open
set.

172
Topological and Ordered Groups
8.3.12. Prove that every closed subset F of the real line (see 8.3.7) is an
intersection of a countable number of open sets.
8.3.13.T. Prove that in any topological space:
1) the union of any family of open sets is open;
2) the intersection of a finite number of open sets is open.
8.3.14. Let M be a complete metric space with metric p and introduce on M
the topology as indicated in 8.3.1. Prove that every closed subset F of M
is a complete metric space relative to the metric p' considered on F (see
8.1.27).
8.3.15. Let X be a topological space, A a subset of X, and ff the family of all
closed subsets of X containing A. Denote by D the intersection of all sets in ff.
Prove that A = D.
8.3.16. Let R be the real line (see 8.3.7). Does R possess two distinct subsets
F 1 and F 2 for which £1 = £2?
8.3.17. Let M be a metric space with metric p, n a natural number, and
Xo EM. Define
S(xo, n) = {x EM: p(x, xo) < 1/n}
Denote by .rJI the class of all S(xo, n). Prove that PJ forms a basis for the topology
introduced in8.3.1.
8.3.18. T. Let X be a topological space. Prove that in order for a class PJ
of open sets in X to be a basis it is necessary and sufficient that for every
open set G c X and every a E G there exists U E fJI such that a E U c G.
Does every topological space have a basis?
8.3.19. Let X be a topological space, A c X, and fJI a basis for X. Prove
that a E A if and only if UnA -# 0 for every set U E PJ which contains
a o .
8.3.20.T. Let PA be a basis for a topological space X. Prove that:
1) for all a, b E X, a -# b, there exists U E PJ such that a E U, b  U;
2) for all U, V E PJ and for every a E U n V there exists WE PJ such that
aE We U n v.
8.3.21. T.H. Let X be a set and let PJ be a collection of subsets of X satisfying
conditions (1) and (2) of 8.3.20. Let A c X and denote by A the set of all
x E X with the property that A n U -# 0 for every U E PJ containing x.
Consider the transformation II of P(X) defined by ll(A) = A. Prove that II is
a closure operator. Is PJ a basis for the topological space thus obtained?
Remark. Compare the results of this problem with those for 8.3.20.
8.3.22. Let R be the real line (see 8.3.7). Consider the class PJ of all open
intervals having rational endpoints.
1) Does PJ form a basis for R?
2) Prove that fJI satisfies conditions (1) and (2) of 8.3.20.

Topological Spaces
173
3) Introduce a topology on R by means of fJI as indicated in 8.3.21.
Does this topology coincide with the topology introduced in 8.3.7?
8.3.23. Let M be a metric space and f be a continuous transformation of
M as defined in Chapter 8.2. Define the topology on M as indicated in
8.3.1. Is f a continuous transformation of the topological space M?
8.3.24. Let C[Q,bJ denote the metric space of continuous functions defined on
the interval [a, b] with the metric from 8.1.2, (1). Define a topology on
C[a,bJ as indicated in 8.3.1. Let R be the set of real numbers with the topology
in trod uced in 8.3.7.
Which of the following mappings of C[Q,bJ into R are continuous:
1) F 1 (y)=y(a) (yEC[a,b)
2) F i (y) = max Iy (x) I (y E C[a, b)
x E [a, b)
3) Fa (y) = max y (x) (y E C[a, b)
xE [a, b)
b
4) P, (y) = y (x) dx (y E C[a, b)
a
o if f(x o } < 0 for some Xo
5) Fs(f) = t if f(x) = 0 for all x
1 if f(x)  0, and f(xo) -# 0 for at least one Xo
8.3.25. Let g be a one-to-one mapping of a topological space X onto itself
such that g and g- I are continuous. Is g a homeomorphism of X onto itself?
8.3.26.T. Let X and Y be topological spaces and f a mapping of X into Y.
Prove that in order for f to be continuous it is necessary and sufficient that
the inverse image F of every closed set F' in Y be closed in X.
8.3.27.T. Let X and Y be topological spaces and f a mapping of X into Y.
Prove that in order for fto be continuous it is necessary and sufficient that the
inverse image G of every open set G' in Y be open in X.
8.3.28.T.H. Let X and Y be topological spaces and f a mapping of X into
Y. Prove that f is continuous if and only if for every x E X and every open
G' c Ywith f(x) = Y E G' there exists an open G c X such that f(G) c G'.
8.3.29.T.H. Let X and Ybe topological spaces and f a mapping of X into Y.
Prove that a necessary and sufficient condition for f to be open is that for
every x E X and every open G c X containing x there exists an open V c Y
containing f(x) such that V c f( G).
8.3.30. T.H. Lt X be the Cartesian product of the topological spaces X I'
X 2 ,...,X n . For any open sets VI' V 2 ,..., Vn(V i C Xi) denote by (VI'

174
Topological and Ordered Groups
V 2' . · · , V n) the set of all elements (x I , X 2' . . . , x n ) in X such that Xi E Vi'
Denote by 11 the class of all sets (V I , V 2' . . . , V n), where the Vi are arbitrary
open sets in Xi' Introduce the transformation of P(X) as indicated in
8.3.21. Prove that X is a topological space relative to this transformation.
Remark. The space X is called the direct product of the topological
s paces X I , X 2 , . · · , X n .
8.3.31. Let M be a metric space with metric p. Let M' be the metric space
consisting of all elements of M, but with the metric PI defined by
PI (x, y) = kp (x, y) (x, y EM')
where k is a fixed positive number (see 8.1.6). Define the topologies on M
and M' beginning with the metrics p and PI as indicated in 8.3.1. Do these
topologies coincide?
8.3.32. Let C be the set of all continuous functions defined on the interval
[a, b]. Introduce two metrics on C as indicated in 8.1.2, (1) and (2). Define
topologies on C starting with these metrics as in 8.3.1. Do these topologies
coincide?
8.3.33. Let X be the direct product of n copies of the real line (see 8.3.7).
Define the metric P on X by
X=(Xb ...J xn)EX; Y=(Yb ..., Yn)E X
p (x, y)= V (XI - Yt)i +.. .+(x,. - YnYJ
Introduce the topology on X starting with P (see 8.3.1). Prove that the topol-
ogy on X from 8.3.30 coincides with the topology introduced by means of p.
8.3.34.T. Introduce distinct topologies on a set X with bases 31 1 and fJl 2
(see 8.3.18). Assume that for each V E fJl 1 and every a E V there exists V' E 31 2
such that a E V' c V, and for each V' E 31 2 and every a' E V' there exists
V E fJl 1 such that a' EVe V'. Prove that these topologies coincide.
8.3.35. State and prove the converse of 8.3.34.
8.3.36. Let there be given two equivalent metrics PI and P2 on a set X.
Prove that the topologies introduced on X starting with PI and P2 (see
8.3.1) coincide.
4. Topological Groups
Let G be a set with an operation (which we shall call multiplication)
under which G is a group and a closure operator under which G is a topologi-
cal space. Then G is called a topological group relative to the given operation

Topological Groups
17S
and topology if it satisfies the following condition: for all a, bEG and for
every open set W containing ab - 1 there exist open sets U and V such that
a E U, b E V, and U V-I C W, where V-I is the set of all inverses of elements
in V.
This latter condition connects the group operation and the topology in
the following sense:
1) each left translation of G is a continuous transformation of G;
2) the transformation of G which takes each element onto its inverse is
continuous (see 8.4.5 and 8.4.40).
The set of elements of a topological group G when considered with
respect to the group operation only is called an algebraic group. This defini-
tion conforms with the definition given in Chapter 2.6, since it is possible to
introduce a topology on every group G such that G is a topological group
with respect to this topology (see 8.4.15). In this section we will usually add
the term "algebraic" to the terms group, subgroup, homomorphism, etc.,
when we consider these concepts with respect to the group operation only.
Thus we speak of algebraic subgroups, algebraic homomorphisms, algebraic
isomorphisms, etc. We will use the same letter to denote both the algebraic
group and the topological group.
A subset H of a topological group G is called a subgroup of G if:
1) H is a subgroup of the algebraic group G;
2) H is a closed subset of the topological space G.
A subgroup N of a topological group G is called a normal subgroup of
G if N is a normal subgroup of the algebraic group G.
Let G and G' be topological groups. A mapping f of G onto G' is called a
topological isomorphism, or simply an isomorphism, if:
1) f is an isomorphism of the algebraic group G onto the algebraic
group G' ;
2) f is a topological mapping (homeomorphism) of the topological
space G onto the topological space G'.
A mapping g of G into G' is c(\lled a homomorphism if:
1) g is a homomorphism of the algebraic group G into the algebraic
group G' ;
2) g is a continuous mapping of the topological space G into the
topological space G'.
A homomorphism f of G into G' is said to be open iff is an open mapping
of the topological space G into the topological space G'.
Let Tbe a topological space and let G be a topological group of invertible
transformations of T(cf. Chapter 3.2). Then G is called a continuous group of
transformations of T if for every (X E G, x E T and for every open set WeT
containing ex(x) there exist open sets U c G containing ex and VeT con-
taining x such that (X'(x') E W for all (x' E U, x' E V. Every transformation
(X EGis a homeomorphism (see 8.4.30).

176
Topological and Ordered Groups
A continuous group of transformations of a space T is called transitive
if it is. a transitive algebraic group of transformations of T (cf. Chapter 4.9).
8.4.1. Let G 1 and G 2 be the groups defined below and let (Xi (i = 1, 2) be
transformations of P(G i ). Determine which of the G i are topological groups:
1) G 1 any group, (Xl (A) = A for all A c G l ;
2) G 2 = [a]g an infinite cyclic group, (Xl (A) = A if A is finite and (X2(A) =
A U e if A is infinite.
8.4.2. Does the real line (see 8.3.7) form a topological group relative to
ordinary addition?
8.4.3. Let  be an n-dimensional vector space and introduce the metric p
on :
P [(Xh ..., x n ); (y., ..., Yn)] = V (XI - Yl) + ... + (x n - Yn)i
Define the topology on  as indicated in 8.3.1. Does Jt:. form a topological
group relative to ordinary addition of vectors?
8.4.4. Let G be the set of n x n real matrices a = (aij) with determinants
different from zero, and let k be a positive integer. Consider the class PJ of all
sets U ak, where U ak consists of all matrices x = (x ij ) for which
IXij-aijl <  (i,j=l,...,n)
Show that it is possible to define a topology on G such that fJI is a basis for
that topology (see 8.3.21). Does G form a topological group relative to this
topology and ordinary matrix multiplication?
8.4.5.H. Let G be a topological group, a E G. Prove that each of the mappings
below is a homeomorphism of the topological space G onto itself:
1) /1 (x)=xa (x E 0)
2) f (x) = ax (x E 0)
3) fa (x) = X-I (x E 0)
8.4.6. Let G be a topological group, F a closed set in G, U an open set in G,
P any subset of G, and a E G. Which of the following sets are closed: Fa,
aF, F- 1 ? Which of the following are open: UP, PU, U- 1 ?
8.4.7.T. Let G be a topological group and a, bEG. Show that there exists a
homeomorphism f of the space G for which f(a) = b.
8.4.8.T. Let 11* be the set of all open sets in a topological group G which
contain the identity. Let fJI consist of all sets of the form aU, where a E G,
U E fJI*. Prove that PJ is a basis for the topological space G.

Topological Groups
177
8.4.9. Let G be the additive group of integers, a E G, and p a prime. Let U ka
denote the set of all integers of the form a + bpk (b E G) and let fJI p consist of
all sets U ka (k is a natural number, a E G). Show that f4 p satisfies the conditions
in 8.3.21. Does G form a topological group relative to ordinary addition
and the topology determined by PJ p ?
Let p :F q be two primes. Let fJl p and PJ q be bases for topologies as defined
above. Are these two topologies equal?
8.4.10. Let R be the real line. Then R is a topological group under addition
(see 8.4.2). Denote by Z the algebraic subgroup of integers. Is Z a topological
subgroup of R?
8.4.11.H. Let G be the topological group of all points in the plane relative to
the operation and topology introduced in 8.4.3. Let N be a line with slope
ex and H the set of all points with integral coordinates.
1) Is either H or N a subgroup of the topological group G?
2) For which ex is the set P = H + N closed?
8.4. 12.T.H. Let.G be a topological group and H an algebraic subgroup of the
algebraic group G. Is H a subgroup of the topological group G?
8.4. 13.T.H. Let G be a topological group. Prove that for every algebraic
normal subgroup N of G, N is a normal subgroup of G.
8.4.14. Let G and G' be topological groups and let f be an algebraic iso-
morphism of G onto G' which is continuous. Is f necessarily a topological
isomorphism?
8.4.15. Let fJI be the family of all subsets of an algebraic group G. Prove that
(1) there exists a unique topology on G for which PJ is a basis, (2) G forms a
topological group relative to this topology, and (3) this topology is discrete.
Is every algebraic subgroup of G a topological subgroup? Is every algebraic
isomorphism of G onto a topological group G' continuous? Is every algebraic
isomorphism of G onto a topological group G' a topological isomorphism?
Remark. A group with the discrete topology is called a discrete group.
8.4.16. Let R be the topological group of real numbers under addition (see
8.4.2). Let R' be the set of all positive real numbers. Define a mapping f
from P(R') into itself by f(A) = A (A c R') where A is the set of all a E R'
for which there exists a sequence an E A with lim an = a. Prove that:
n-+ 00
1) f defines a topology on R' ;
2) R' forms a topological group relative to multiplication and this
topology. Are the topological groups Rand R' isomorphic?
8.4. 17.T.H. Let H be an algebraic subgroup of a topological group G. Prove
that in order for H to be a topological subgroup of G it is necessary and suffi-
cient that there exist an open set U such that U n H = U n H -# 0.

178
Topological and Ordered Groups
.8.4.18. Let G be the topological group of nonsingular n x n real matrices
(see 8.4.4). Which of the following subsets of G are subgroups, and which of
these are normal:
1) the set of all nonsingular matrices whose determinants are rational;
2) the set of all nonsingular diagonal matrices;
3) the set of all nonsingular matrices of the form
au 0 0
ail aii 0
.. . 0
. . . 0
. . . .
. . .
ani anC) a n 3 ... ann
4) the set of all matrices with determinant I?
8.4.19.T.H. Let G be a topological group and H a subgroup of G. Let GIH
denote the set of all right cosets of H in G, and let 11 be a basis for G. For each
U e 31, denote by U* the set of all cosets xH where x e U. Let fJI* be the set of
all U*, where UefJI. Prove that a topology can be introduced on GIH for
which 31* is a basis.
8.4.20.T.H. Let N be a normal subgroup of a topological group G and let
GIN be the factor group of the algebraic group G by N. Introduce a topology
on GIN as indicated in 8.4.19. Prove that GIN is a topological group.
Remark. The topological group GIN is called the .factor group
of G by N.
8.4.21.H. Let G be a topological group, N a normal subgroup of G, and GIN
the factor group of G by N. Consider the natural homomorphism cp (see
2.4.18) of the algebraic group G onto the algebraic group GIN. Is cp a continu-
ous mapping of the space G onto the space GIN? Is cp an open mapping of G
onto GIN? Find conditions which insure that cp will be a topological iso-
morphism.
8.4.22. Let G and G' be topological groups, f a homomorphism of G onto G',
and N the inverse image of the identity e' e G' (the kernel of f). Prove that N
is a normal subgroup of the topological group G.
8.4.23.H. Let G and G' be topological groups, g an open homomorphism of
G onto G', and N the kernel of g (see 8.4.22). Prove that GIN is topologically
isomorphic to G'.
8.4.24.T. Let G 1 , G 2 , . . . , G n be topological groups, and let G be the Cartesian
product G 1 x G 2 X ... x G n , with componentwise multiplication (see
5.6.29). Introduce a topology on G as indicated in 8.3.30. Prove that G is a
topological group.
Remark. The group G is called the direct product of the topological
groups G 1 , G 2 ,..., G n .

Topological Groups
179
8.4.25.H. Let G be the direct product of the topologicalgroupsG l ,G 2 ,..., G n .
For each x j E G j define f(x j ) = (e t ,. . ., ej-t, Xj' e j + 1" . ., en) E G, where ei
is the identity of G i (i = 1,..., n). Prove that f is an isomorphism of the
topological group G j into the topological group G.
8.4.26. Let C be the topological group of complex numbers under addition
(see 8.4.3) and let C' = R x R be the direct product of two copies of the
topological group of real numbers (see 8.4.2). Are C and C' isomorphic
topological groups?
8.4.27. Let C be the topological group of complex numbers under addition
(see 8.4.3) and let A be the set of all pairs of real numbers (x, y) such that x  0
and
1
Oyl- X+l
Denote by B the set of all pairs of real numbers of the form (x, 0).
1) Is either A or B a closed set?
2) Is A +. B closed?
8.4.28. T.H. Let G be a topological group. Prove that for every open set U
and for every x E U there exists an open set V such that x E Vand V c U.
8.4.29.T.H. Let G be a topological group and let A be the center of the
algebraic group G. Prove that A is a subgroup of the topological group G.
8.4.30. Let G be a continuous group of transformations of a topological
space X. Prove that every g EGis a homeomorphism of X onto itself.
8.4.31. Let G' be the set of all invertible topological transformations of a
topological space X. Prove that G' forms an algebraic group under composi-
tion of transformations.
8.4.32. Is the product of two continuous transformations of a topological
space continuous?
8.4.33. Let G be a continuous group of transformations of a topological
space X. (1) Is every subgroup H of G a continuous group of transformations
of X? (2) Let G be a transitive group of transformations of X. Is every sub-
group H of G a transitive group of transformations of X?
8.4.34. Let G be the group of all rotations of the plane D about a point O.
Introduce a topology on D as indicated in 8.4.3. Prove that each x EGis a
homeomorphism of D. Each rotation x E G determines an angle of rotation
a, so we can write x = Xa' For every A c G, define f(A) = A, where Xo =
xao E A if and only if there exists a sequence of angles {an} such that
lim an = ao and Xa E A (n = 1, 2, . . .). Prove that G is a topological group
n- 00 "
relative to f. Is G a continuous group of transformations of D?
8.4.35.H. Let D be the set of all points in the plane. Introduce a topology on

180
Topological and Ordered Groups
D as indicated in 8.4.3. Denote by G the group of all affine transformations of
D.
1) Prove that every affine transformation is a homeomorphism of D.
2) Is it possible to introduce a nondiscrete topology on G such that G
is a continuous group of transformations of D?
8.4.36. T.H. Let G be a topological group and H a subgroup of G. Introduce
a topology on the set GIH of right cosets as indicated in 8.4.19. Define the
transformation a by a(A) = aA, where a E G, A E GIH. Prove that G is a
transitive continuous group of transformations of the topological space GIR.
Remark. Compare this result with the introductory remarks to Chapter
7.2.
8.4.37. T. Let G be a transitive continuous group of transformations of a
topological space X. Fix a E X and denote by At the set of all a E G for which
a(a) = t (t E X). Prove that Aa is a subgroup of the topological group G.
8.4.38.T.H. Let G be a continuous group of transformations of a topological
space X and define At as in 8.4.37. Prove that the mapping f defined by
f(t) = At is a one-to-one mapping of X onto the set GIAa of right cosets of Aa
in G. Is f - I a continuous mapping of the topological space GI Aa (see 8.4.19)
onto X?
8.4.39.H. Let M be a metric space with a bounded metric p. Prove that it is
possible to introduce the natural metric p* on the group of transformations
G of M. Introduce the topologies on M and G relative to the metrics p and p*,
respectively, as indicated in 8.3.1. Prove that G is a continuous group of
transformations of the topological space M.
8.4.40. Let G be a set with an operation under .which G is a group and a
closure operator under which G is a topological space. Prove that if the
following two conditions are satisfied then G is a topological group:
1) each left translation of G is continuous;
2) the transformation f defined by f(x) = X-I (X E G) is continuous.
5. Ordered Groups
Let G be a group and define an ordering relation on the set of elements in
G. If this relation is two-sided compatible, then G is called an ordered group
relative to this ordering. In particular we speak of a linear[ y ordered group
if the given ordering is also linear (in the lite-rature the term "ordered group"
is sometimes used in place of what we have called a "linearly ordered group").
An element X of an ordered group G is said to be positive if x  e and
negative if x  e. The set of all positive elements of G is called the positive part
ofG and is denoted by G+. (It is also called the semigroup of positive elements,
which is implied by 8.5.6.) The set of all negative elements of G is called the
negative part of G (semigroup of negative elements) and is denoted by G -.

Ordered Groups
IS I
I t is natural to use additive notation to denote the operation of the
group. In this case the identity element is called the zero, denoted by 0, and the
inverse of an element x is denoted by - x. The positive part then becomes all
elements x such that x  o.
8.5.1. Let G be an ordered group. Prove that x  y if and only if X-I Y  e.
Remark. Hence it follows that the positive part of an ordered group G
completely determines the ordering.
8.5.2. Prove that in an ordered group G, x  y if and only if y-l x  e.
Remark. Hence it follows that the negative part also determines the
ordering completely.
8.5.3. Let C be the additive group of complex numbers. Determine which
of the following orderings on C are two-sided compatible, and find their
positive and negative parts:
1) Zl  Z2 if a l > a2 or a l = a2 and b l  b 2 (Zl = al + b1i, Z2 =
a 2 + b 2 i) ;
2) Zl  Z2 if arg (Z2 - Zl) e [a, P], where a and P are fixed angles satis-
fying 0  P - a < 1t;
3) Z I  Z 2 if a 1  a 2 and b 1  b 2 ;
4) Zl  Z2 if arg Zl  arg Z2 and IZII  I Z 21.
8.5.4. Prove that an element x -# e of an ordered group G is negative if and
only if X-I is positive.
8.5.5. Prove that if an ordered group G contains an element x -# e of finite
order, then x is neither positive nor negative.
8.5.6.T. Prove that the positive part G+ of an ordered group G satisfies the
following properties:
1) ifx,yeG+ thenxyeG+;
2) e e G+ ;
3) ifxeG+, where x -# e, then X-I G+;
4) ifxeG+ andgeG,theng-1xgeG+.
Remark. Property 1 implies that G+ is a semigroup.
8.5.7. T.H. Let H be a subgroup of a group G satisfying the following
conditions:
1) if x, ye H, then xye H;
2) e e H ;
3) if x e H, where x -# e, then X-I ft H;
4) ifxeHandgeG,theng-1xgeH.
Prove that it is possible to introduce a two-sided compatible ordering on G
such that H is the positive part of G relative to this ordering, i.e., H = G + .
Remark. Compare this with 8.5.6.

182
Topological and Ordered Groups
8.5.8. Let {Pa} be a family of two-sided compatible orderings on a group G,
and denote the positive part of G relative to Pa by G:. Prove that n G:
a
is the positive part of some two-sided compatible ordering t on G. How is t
related to the Pa?
8.5.9. Prove that the negative part G- of an ordered group G has the
properties:
1) if x, y e G - , then x y e G - ;
2) e e G - ;
3) if x e G - , where x -# e, then X-I  G - ;
4) ifxeG- andgeG,theng-1xgeG-.
Remark. Compare this result with 8.5.6.
8.5.10. Prove that every subset of a group G which is the positive part for
some two-sided compatible ordering P is the negative part of G for some other
two-sided compatible ordering t. How are these two orderings related?
8.5.11. Prove that an ordered group G is linearly ordered if and only if
G+ U G- = G.
8.5.12.H. Describe all possible two-sided compatible orderings on cyclic
groups.
8.5. 13.H. Let G be the additive group of all polynomials with real coeffi-
cients. Let H c G consist of all polynomials which are nonnegative on the
interval [0, 1]. Prove that H is the positive part for some two-sided compatible
ordering p. Determine how the following elements are related under P :
fl(x) = x 2 + 1, f2(x) = x 3 + 2
f3(x) = x 3 + x 2 + 1, f4(X) = x 8 - 1
8.5.14.H. Let G be the set of all pairs of real numbers. Introduce an operation
on G by setting
(X., Yt) (Xi' Yi) = (XI + Xi' e X2 V1 + Yi)
Prove that G is a group with respect to this operation. Prove that the subset
H of all elements (x, y) such that x > 0 or x = 0 and y  0 is the positive part
for some two-sided compatible linear ordering.
8.5.15. Let G = n G a be a direct product of groups G a , each of which has a
a
two-sided compatible ordering defined on it. Put x  y in G if and only if
Xa  Ya in G a for every factor Ga.
1) Determine whether this ordering is two-sided compatible.
2) If all of the orderings on the G a are linear, under what conditions will
the ordering on G be linear?

Ordered Groups
183
8.5.16. Let G = n*G be a free product of groups G, each of which has a
oc
two-sided compatible ordering defined on it. Let x, Y E G be written in the
form
x = xtltx tli · · · x tln
Y = YtllYtli.. .Ytl n
where XOCi,YociEGOCi' Set x  Y if and only if X OCi  YOCi (i = 1,2,...,n). Does
such a rule define an ordering on G? If so, is it two-sided compatible?
8.5.17. Let G be a free product of two (finite or infinite) cyclic groups,
G = [x] * [y]. Let H be the subset of G consisting of the identity and all
elements z which can be written in the form
z = xa1yb1xtJaybJ.. . xGnyb"
al +ai+.. .+a,,>O
Determine under what conditions there exists a two-sided compatible order-
ing on G for which H is the positive part of G, i.e., H = G +.
8.5.18. Let G be a commutative group, x E G, x # e. Determine when there
exists a two-sided compatible ordering on G such that x  e.
8.5.19. Prove that the ordering on the infinite cyclic group G = [x] defined
by x n  x m if and only if n  m is two-sided compatible.
8.5.20. Let there be given a two-sided compatible ordering on the infinite
cyclic group G = [x] under which x  x 2 . Prove that x n  x m if and only if
n < m.
Remark. Thus the relation x  x 2 completely determines the ordering
considered in 8.5.19.
8.5.21. Assume that any two elements in an ordered group have an upper
bound. Prove that any two elements will then also have a lower bound.
Remark. An ordered group 'having this property is called a directed
group.
8.5.22.T. Let G be a directed group. Prove that for every x E G there exist
u,vEG+ such that x = uv- t .
8.5.23.T. Assume that for every element x in an ordered group G there exist
u, v E G + such that x = uv - 1 . Prove that G is a directed group.
Remark. Compare this result with 8.5.22.
8.5.24. Determine which of the ordered groups stated in problems 8.5.3, (1)
and (2), and 8.5.13 are directed.
8.5.25. Let G be the set of real matrices of the form
( 1 a C )
M= 0 1 b
001

184
Topological and Ordered Groups
Prove that:
1) G is a group under ordinary matrix multiplication;
2) G has a nontrivial (different from the diagonal) two-sided compatible
linear ordering.
8.5.26.H. Prove that if an abelian group G has at least two distinct two-sided
compatible orderings, then there is an infinite number of all such orderings on
G.
8.5.27. Prove that a nontrivial (see 8.5.25) two-sided compatible ordering
can be defined on every group G which contains elements of infinite order in
the center of G.
8.5.28. Prove that if a nontrivial (see 8.5.25) two-sided compatible ordering
can be defined on the center of a group G, then a two-sided compatible order-
ing can also be defined on all of G.

HINTS
CHAPTER 1
1.2.3. Use the well-known fact that the set of all primes is infinite.
1.2.18. Take a one-to-one mapping CPk of M k into M k for each k = 1,2,....
Select subsets N I ,N 2 ,N 3 ,... in U M k for which xEN k if xEM, and
k
l + cp,(x) = k for some l. Each N k is finite (or empty) and U M k = U N k .
k k
Apply 1.2.17.
1.2.19. Write the set of all rational numbers as a union of subsets M k
(k = 1,2,.. .), where M k consists of all rational numbers p/q (p and q are in-
tegers) such that Ipl + Iql = k. Apply 1.2.17.
1.2.20. Use 1.2.2.
1.2.21. Write the set of all polynomials with rational coefficients as a union
of subsets M k (k = 1, 2, 3, . . .), where the polynomial
Po xn+ Pt xn-1 + ...+ Pn-l x+ Pn
qo ql qn-l qn,
(Po, PI"'" Pn, qo, ql,"" qn are integers) lies in M k if
n, n
n+  IPil+  Iqjf=k
i=O j=O
Apply 1.2.17.
1.2.23. Suppose that there exists a mapping cP of the set of all natural numbers
onto the set of real numbers r such that 0  r  1. Write each of these num-
bers in the form of an infinite decimal fraction:
,1 = 0, £11£J2£13 ...
,2 = 0, £2J£22£23 ...
<f3 = 0, £ 31£32£83 ...
............
185

186
Hin ts
(heretheGijarethesymbolsO, 1,2,..., 9). Definethenumberr = 0,'11'12'13""
where '1k = 1 if Gkk -# 1 and '1k = 2 if Gkk = 1 (k = 1, 2, 3, . . .). Show that
cpm -# r for all m = 1, 2, 3, . . . , which contradicts the definition of cpo
1.2.24. Use 1.2.16 and 1.2.23.
1.2.25. First prove that the set of all algebraic numbers is countable by using
1.2.17 and 1.2.21, and then show that each polynomial has a finite number of
roots. Next prove that the sets of all real numbers and all complex numbers
are uncountable by using 1.2.16 and 1.2.23. Finally prove that the sets of all
real transcendental numbers and all transcendental numbers are uncountable
by using 1.2.16 and 1.2.18.
1.2.26. First prove that any two intervals are equivalent. Then represent
each of the given sets as a disjoint union of a countable number of intervals.
1.2.27. Prove the existence of a one-to-one mapping of A into M. Suppose
there exists a one-to-one mapping cp of A onto M. Fix two elements b, b' in
B, b -# b'. Consider the mapping t/J of A into B defined by t/Ja = b if cpa -# b
and t/Ja = b' if cpa = b. Prove that t/Ja -# cpa for all a EA.
1.2.28. For each subset N of M define a mapping CPN of M into the set
{O, I} by CPN(X) = 1 if x EN, and CPN(X) = 0 if x  N. Prove that the set of all
cP N coincides wi th the set of all ma ppings of M in to {O, 1 }. Use 1.2.15 and 1.2.27.
1.3.18. Prove by induction on the number of elements in the set. Select one
of the maximal elements y E X with respect to p. By using the induction
assumption, establish a linear ordering a on X,," {y} such that a '" b(p)
implies a '" b(a) for all a, bE X,," {y}. Starting with a construct the desired
ordering p' on X.
CHAPTER 2
2.1.11. Consider the mappingofG onto H which takes rEG onto In r E H.
2.2.9. Use 1.4.1 and 1.4.4.
2.3.12. Take x E R, y ELand consider their product xy.
2.3.13. Use 2.3.12.
2.3.18. First prove that for any two matrices A and B, the rank of the
product AB is less than or equal to the minimum of the ranks of A and B.
Prove next that if A and B have the same rank, then there exist matrices P
and Q such that A = PBQ. Finally, the proof follows by using elementary
transformations, and the fact that each elementary transformation can be
brought about by multiplying matrices on the left or right by certain non-
singular matrices.
2.4.22. Use 2.4.20 and 2.4.21.

Hints
187
2.5.1. The proof proceeds by induction on n, the length of the word
XIX2 . .. X n . For a given word consider two processes for reducing the length
of the word. These reductions are achieved by a sequence of substitutions of
any two adjacent terms in the product.
Let the first process begin with the substitution of a pair XiX i + 1 in the
product, and the second process with the substitution of a pair x jX j + 1 .
Then consider two cases separately: (1) i + 1 < j, and (2) i + 1 = j.
2.5.16. Take any element a' such that aa'a = a. Then consider the element
a = a' aa' .
2.5.19. Consider the relation PIon S defined by x '" y(p 1) (x, yeS) if there
exist elements a, be S, k, k' e K such that one of the four,following conditions
holds: (1) x = akb, y = ak'b; (2) x = ak, y = ak'; (3) x = kb, y = k'b; and
(4) x = y.
Consider the relation P2 on S defined by x '" Y(P2) if there exist elements
t 1 = x,t 2 ,...,t n - 1 , t n = yeS such that t i '" ti+1(Pl) (i = 1,2,...,n - 1).
Prove that P2 is a congruence. Consider Sf P2 and the canonical mapping on
S onto Sf P2 which takes each xeS onto the p2-class containing x.
2.5.23. Take any element x of a finite semigroup. Consider [x]. Use 2.5.10.
2.6.16. Choose the appropriate powers of x as the desired elements Yl'
Y2, . . . , Ym' Use the well-known result from number theory that if d is the
greatest common divisor of the integers k 1 , k 2 , . . . , k m , there exist integers
x 1 ,x 2 ,...,x m such thatx1k 1 + X2 k 2 + ... + xmk m = d.
2.6.21. Let G' be an arbitrary subgroup of G = [x]g. Select the element Xk
in G' having the smallest exponent k. Next use 2.6.7. Then compare G' with
Hk.
2.6.22. Use 2.6.8 and the hint to 2.6.21.
2.6.23. Use 2.6.21 and 2.6.22.
2.6.24. Use 2.6.21 and 2.6.23.
CHAPTER 3
3.1.26. Use 3.1.22 and 3.1.25.
3.1.31. If rp  n, then choose permutations y and  such that ycxt> = p.
3.1.34. Use 3.1.25,3.1.26, and the existence of an identity in a group.
3.1.35. Consider the set G of all transformations p satisfying (1) px = py
if and only if cxx = cxy, and (2) PX = cxX. Prove that G is a group.
3.1.37. Use 3.1.18.
3.1.38. Prove by induction on the rank, beginning with n - 2.

188
Hints
3.1.39. Use the fact that a transformation of rank n cannot be written as a
product of transformations of smaller rank. Prove that by multiplying a
permutation of rank n - 1 by a suitably chosen permutation of rank n it
is possible to obtain any other permutation of rank n - 1.
3.1.40. Use the hint to the preceding problem.
3.2.6. Use 3.2.5.
3.2.8. Consider the binary relation p on the set X defined by (x,y) E p if
exx = y. Prove that there exists P E Tx such that p = Pp, and that p is the
inverse of ex.
3.2.9. Use 3.2.7.
3.3.14. Prove that (a l a2 . . . ak) = (alak)(alak-l)' . . (ala2)'
3.3.16. Use 3.2.23 and 3.2.24. Prove that if
( I...n )
ex = a I . . . an
is an even permutation, then j(a I . . . an) > 0, and if ex is odd then
fn( a I · . . an) < O.
3.3.17. Use 3.2.25. Multiply each even permutation by the transposition
( 12).
3.3.19. Use 3.3.5, 3.3.9, and 3.3.14.
3.3.21. Use 3.3.9 and 3.3.19. Show that the order of the group [(34), (123),
(456)] is less than 31.
3.3.23. Use 3.3.19.
3.3.24. Use 3.3.7.
3.3.26. Use 3.3.24 and prove that if two numbers appear in a cycle in the
decomposition of ex into disjoint cycles then they appear in the same cycle in
the decomposition of ex', and vice versa.
3.4.13. Use 3.4.4.
3.4.23. Use 3.4.4. In the set of integers consider the ordering whereby the
natural numbers are ordered by the usual ordering and the remaining num-
bers are not comparable.
3.4.27. Introduce a system of coordinates and prove that if the abscissas of
PI' P2, and P 3 satisfy Xl < X2 < X3, then so do the abscissas of their images.
3.4.30. Use 3.4.6.
3.4.32. (2) Prove the assertion for transformations of rank n - 1 and pro-
ceed by induction on the rank.
3.4.34. Prove that
ex = ( a bed f )
a a b b c

Hints
189
is an endomorphism of X which is not a regular element.
3.5.25. Use the preceding problem and the fact that for each pair of vertices
there exists a self-coincidence of the tetrahedron mapping one vertex onto
the other.
3.5.28. Use the hint to 3.5.25.
3.5.29. Use the preceding problem. Pove that the identity transformation is
the only permutation which maps each of the four diagonals of a cube onto
itself. Consider the set of transformations of the diagonals of the cube which
correspond to all rotations of the cube.
3.5.32. Use the hint to 3.5.25.
3.6.8. Use the theorem concerning the existence of roots of a polynomial.
3.6.22. Use 3.1.40 and 3.3.19.
3.6.24. See the preceding problem.
3.6.28. Prove that eM is the identity of A, and use 3.6.5.
3.6.29. Use invertibility and the definitions of the domain and range of the
product of two transformations. Prove that drx = dp and rrx = rp for any
rx, p in the given group. Then use 2.6.1 and 3.6.12.
CHAPTER 4
4.1.1. Use 2.6.19.
4.1.2. Use 4.1.1.
4.1.4. By virtue of 4.1.3 the union of all right cosets is equal to G. Eliminate
the repeated cosets from this union, i.e., select one member from each col-
lection of equal cosets. Apply 4.1.2.
4.1.9. Use 2.6.21.
4.1.16. Prove that Hy-l = Hz- 1 if and only if yH = zH. Consider the fact
thatg = xhimpliesg-l = h-lXl.
4.1.20. Use 4.1.4 and 4.1.19. The right decomposition of H in G gives a
partition of the n elements of G into k subsets, each of which consists of
m elemen ts.
4.1.21. Use 4.1.20 and 2.6.9.
4.1.23. Consider the subset of G of all elements of the form u-1v, where
u, v E K, and the subset of all elements of the form uv - 1, where u, v E K.
4.1.26. Develop an argument in a manner similar to the one exhibiting the
existence of a right decomposition of a subgroup in a group (see 4.1.2,
4.1.3, and 4.1.4).
4.1.30. Consider the right decomposition of D in HI' where D = HI n H 2 .
Multiply both parts of the obtained equality by H 2' Use 2.6.19.

190
Hints
4.2.10. Use 4.1.20 and 4.2.9.
4.2.24. For x e Kl denote by Sx the number of pairs (y, z) (ye K 2 , z e K 3 ) for
which x = yz. Prove that Sx = Sx' for all x, x' E Kl' Show that k2k3 = kts,
where s = Sx for all x e K 1 .
4.2.26. Consider (x- 1 H lx)H 2' and use 4.1.30 and 4.2.25.
4.2.27. Use 4.1.19, 4.1.30, and 4.2.26.
4.3.5. Use the result concerning the determinant of the product of two
matrices.
4.3.12. Use the preceding problem.
4.3.17. Use 3.5.5. Prove that each coset of N in G' different from N consists
of rotations about all points through an angle cpo
4.3.25. Consider the decomposition of S4 by the Klein group. With all
elements in a class associate the permutation in that class which has 4 fixed
points.
4.3.26. Use 4.3.21 and 2.4.18.
4.3.27. If f(gl) = f(g2), then find f(gll g2)'
4.3.28. Use 4.3.27. Consider the following mapping t/J of G' into G/ N: if
g' e G', then t/J(g') = gN is the set of elements mapped onto g' by the given
homomorphism.
4.4.1. Consider the equality stated at the end of the introduction to Chapter
4.2, and take 4.2.10 into account.
4.4.2. Use 4.3.29 and 4.4.1 to show that the group is commutative. Consider
the product of two distinct cyclic subgroups of the group, and apply 4.1.30.
4.4.3. Prove by induction on the order of G. Take the product of all cyclic
subgroups of G which are generated by elements in the center of G. If the
order of one of these subgroups is divisible by p, then this subgroup has an
element of order p. If no such supgroup exists, then according to 4.1.30 the
order of the center of G is not divisible by p. In this case, by using the equation
stated at the end of the introduction to Chapter 4.2, we get that some k i is
not divisible by p. But by 4.2.10 k i is the index of some proper subgroup Hi
of G. Since the index of Hi in G is not divisible by p, then it follows from 4.1.20
that the order of Hi is divisible by p. By the induction hypothesis, Hi contains
an element of order p.
4.4.4. Use 4.4.3.
4.4.5. Use 4.1.30 and 4.4.3.
4.4.6. Proceed by induction on the order of G. Consider the equation given
at the end of the introduction to Chapter 4.2. By 4.2.10 each k i > 1 is the
index of some subgroup of G. Apply the result of 4.4.3 to the center of G.
4.4.7. Note that P is a normal subgroup of its normalizer. By taking 4.1.30
into account, show that the order of [x]P is a power of p for all x e N.

Hints
191
4.4.10. Let PI and P 2 be two p-Sylow subgroups ofG. Consider the decom-
position of G by the pair (Pi' P 2 ), and use the equation given in 4.2.27.
4.4.11. Let P be one of the p-Sylow subgroups of G, and let N be the nor-
malizer of P. Consider the decomposition ofG by the pair (N, P)and the equa-
tion in 4.2.27 associated with this decomposition. By virtue of 4.2.10 and
4.4.10, the number of p-Sylow subgroups is the index of N in G. By using
4.4.7, show that in the decomposition of G by (N, P), X-I Nx n P = P for
exactly one of the classes N x P.
4.4.12. Use 4.4.5 and 4.4.11.
4.4.14. Use 4.4.11.
4.4.18. Consider the decomposition of G by the pair (P, H), where P is a
p-Sylow subgroup of G, and the equation in 4.2.27.
4.4.19. Prove that for such a group G the elements in distinct p-Sylow
subgroups of G commute. Use 4.4.1. Consider the set of all elements of G
which commute with elements in the center of one of the p-Sylow subgroups.
4.4.20. Prove by induction on the order of G, and use 4.3.33, 4.4.1, and 4.4.3.
4.5.14. Prove that every 3-cycle is a commutator in Sn.
4.5.17. Use 4.4.20 and the fact that a group of order p is abelian.
4.5.18. Prove that in this case the commutator subgroup lies in the center.
Prove that besides elements in the commutator subgroup, the group has
other elements of order 4.
4.5.19. Use 2.2.15.
4.5.22. Use the result concerning the deterolinant of the product of two
matrices. Represent matrices whose determinant is 1 as products of matrices
of the form
(::). (::).
( a  ) , ( 0 -1 ) , ( -1 0 )
o - 1 0 0 -1
a
4.6.3. Use 4.4.13.
4.6.5. Use 4.5.14 and 4.3.4.
4.6.6. Suppose that Sn is solvable for n  5. Take a sequence of subgroups
as in 4.6.4. Select the last term H k of this sequence which contains all of the
3-cycles. By using 4.5.10, show that this implies that Hk+ 1 also contains all
of the 3-cycles.
4.6.7. Take a sequence of subgroups as in 4.6.4. For a subgroup F c G
consider the sequence
F n Htt F n H2' ... , F n H m - u F n Hm.

192
Hints
4.6.8. Use 4.5.20 and 4.6.4.
4.6.9. Construct a sequence of subgroups of G as in 4.6.4, starting with the
corresponding sequences of subgroups of N and GIN. Use 4.5.20.
4.6.13. Use 2.6.24 and 4.5.12.
4.6.14. Use 4.5.17.
4.6.15. Use 4.3.7, 4.6.5, 4.6.6, 4.6.7, and 4.6.9.
4.6.18. Use 4.4.11 and 4.6.3.
4.6.19. Use 4.4.11 and 4.6.3.
4.6.20. Use 4.3.17.
4.7.4. Use 4.6.12 and 4.6.13.
4.7.8. Use 4.7.7; prove by induction on k such that Hk C Zn-k.
4.7.9. Prove by induction on k such that Hn-k c Zk'
4.7.10. Use 4.7.8 and 4.7.9.
4.7.14. Use 4.4.1.
4.7.15. (1) Use 4.7.12; (2) use 4.7.11; (3) use 4.7.12; (4) use 4.7.14 and 4.7.4.
4.7.16. Use 4.7.14.
4.7.21. Use 4.7.4.
4.8.10. Use 4.8.8.
4.8.11. Use 4.8.8.
4.8.20. Compute the number of elements in the conjugate classes of all
elements of order 2. Use 4.2.19.
4.8.21. Use the preceding problem.
4.8.22. Use the preceding problem. If cp is an automorphism of S4 and
cp(12) = (ab), cp(13) = (ac), cp(14) = (ad), then consider t(% where
 = ( 1 2 3 4 )
a bed
4.8.28. Use 3.3.20 and 4.8.3. Find the number of generating sets of A4 which
consist of 2 elements.
4.8.31. Use 4.3.29 and 4.8.18.
4.9.6. Use the preceding problem.
4.9.8. If  is a regular permutation, then k maps at least one number onto
itself for some k.
4.9.20. If G x c G' c G and G' is a right coset of G' in G, then prove that
M = {giX, gix, . . .}, where gi, gj,. . . E G' is an imprimitive system.
4.9.21. Let (ij 1), (ij2)"'" (ijk) be all transpositions in G containing the
number i, and let H be the group generated by the group G i and these trans-
positions. Prove that k < n - 1 and that H -# G. Use 3.3.19 and 4.9.20.

Hin ts
193
4.9.22. Use 4.9.13 and 4.9.21.
4.9.26. Letting H be an intransitive normal subgroup of a transitive group G,
prove that the set of intransitive systems of H is an imprimitive series of G
(see 4..3).
4.9.27. Use the hint to the preceding problem.
4.9.34. Use 4.9.6.
CHAPTER 5
5.1.5. Prove that the last three relations are consequences of the previous
ones.
5.1.6. Show that the relation a 2 = c 2 (see 5.1.15) is not a consequence of the
given set of relations.
5.1.18. Use the preceding problem.
5.1.19. Use the preceding problem.
5.1.20. Determine the form of the direct consequences of the given set of
relations.
5.1.24. Use 5.1.10.
5.1.27. Use 5.1.26.
5.2.4. Use 5.1.4.
5.2.7. Use 5.1.19.
5.2.8. Use 5.1.19.
5.2.13. Use 5.2.9, 5.2.11, and 5.2.12.
5.2.21. Prove that every word in G can be reduced to the form cmd n , where
m and n are integers.
5.2.22. Use 5.2.3.
5.2.23. Consider the fact that if r is a solution of the congruence x P = 1
(mod q), then the remaining solutions are r 2 , r 3 , . . . , r P - 1 .
5.3.4. Prove that K can be decomposed into two disjoint subsets K 1 and K 2
of the same cardinality such that the set of all classes containing k E K 1 is a
free generating --set of F K .
5.3.8. (3) Use the preceding problem.
5.3.12. If K = {a, b, . . .} is a free generating set, then for every integer n
the set K' = {ab n , b, . . .} is also.
5.3.13. Prove the statement for the group F K in 5.3.4.
5.3.17. Use 5.3.15 and 5.3.16.
5.3.19. Use 5.3.11 and 5.3.14.
5.4.1. Construct the free group over K and use 5.3.15.

194
Hints
5.4.3. To show that G 2 and G 3 are not commutative, use 5.1.18 and 4.3.35
and find a noncommutative group in which the stated relations do not
hold
5.4.4. Use the hint to the preceding problem.
5.4.5. Use the hint to 5.4.3.
5.4.6. Use 5.4.2 and 4.3.35.
5.4.7. Use 5.1.18 and the hint to 5.4.3.
5.4.8. Use the hint to the preceding problem.
5.4.10. Construct the group of ordered pairs (,:yj).
5.4.14. Use 5.4.12 and 5.4.13.
5.4.15. Use the hint to the preceding problem.
5.4.16. Use 5.4.13.
5.4.18. Use 5.2.18 and show that every automorphism maps (ab)2 onto
itself.
5.4.20. Use 5.4.19, 5.2.16, and 5.2.17.
5.4.22. Use 5.4.21, 5.2.15, and 5.2.19.
5.4.23. Use 5.4.20 and 5.4.22.
'"
5.4.24. Use 4.4.6 4.4.11, and 4.4.14.
5.4.25. For a, b elements of orders p, q, respectively, show that all of the
elements a, b-1ab, b- 2 ab 2 ,..., b-(q-l)ab q - 1 are distinct. Then verify that
a- jba j = b rj for every natural number j. Finally, use 4.4.5.
5.4.26. Use the preceding problem.
5.4.27. Use 5.4.26, 5.2.22, 5.2.23, and 4.4.14.
5.4.28. Use the preceding problem.
5.4.29. Use 5.4.27, 5.4.23, and 4.4.3.
5.5.20. Denote by f the partial transformation f(z) = -l/z (z =F 0) and
by g the partial transformation g(z) = z + 1.
Prove that {J: g} is a generating set of G. Let h be any element in G;
then h can be realized as a transformation of the form
h(z) = az + b
ez + d
First consider the case d = O. Then for the case d =F 0, let Ibl  Idl > O.
Prove that gnh can be realized as a partial transformation
(z =F die)
"h(z) = (a + nc)z + (b + nd)
g ez + d
(z =F - die)
If 0  Ibl  d, consider the element fh. Further, by multiplying h on the
left by the corresponding power of g and by J: show that h can be expressed

Hin ts
195
in terms of f and g. Consider the elements u = gf and f. Prove that G is
generated by u and f. Show that u 3 = f2 = e and hence every element in G
can be written as a product in which the elements f and either u or u 2 alter-
nate. Assume that this representation is not single valued for some element in
G. Show that in this case a relation of the form fu k 1u k2 . . . fu kn = e holds,
where the k i assume the values 1 or 2. Proceed by induction on n to obtain a
contradiction.
5.5.24. Consider a family of groups G a (E A) such that G a is isomorphic to
G a and G a n G a = e a , where e a acts as the identity of both G and Ga. Con-
sider the set of words X t X2 . . . X n over M = U G a in which no two adjacent
aeA
elements Xi and x i + 1 lie in the same Ga. Define an operation on M which
maps a pair of words over M onto the successive multiplication of the factors
in the same G a , if such exist.
5.6.5. (4) Use 4.4.14.
5.6.26. Use 5.6.25.
5.6.28. Use 5.6.26.
5.6.29. Use 5.6.14.
5.6.31. Prove that there exists a homomorphism of G onto the direct product
of n infinite cyclic groups (see 5.6.29).
5.6.32. Compare the cardinality (Chapter 1.2) of G with the cardinality of
the set of all sequences (at, . . . , ai' . . .) (ai E G i ), where only a finite number of
ai are different from the identity of G i .
CHAPTER 6
6.1.9. Consider the set G of all elements in G whose orders are powers of
the prime p. Show that G is a subgroup of G, and that Gl n G2 = e if
Pt=FP2'
6.1.27. Use 6.1.26.
6.1.28. Use 6.1.24 and 6.1.27.
6.2.6. Let H be the subgroup introduced in the preceding problem. Prove
that [H, a] = [a] x H = G.
6.2.7. Use 6.1.9 and 6.2.6.
6.2.9. Do the problem first for fjnite primary groups.
6.2.13. Use 6.2.12.
6.3.1. Proceed by induction on the number of generators.
6.3.2. Proceed by induction on the number n = Inti + .. . + Inkl.
6.3.4. By using 6.3.2, find a generating set {g t , . . . , gk} of G such that the
orders of g t , . . . , gk do not decrease, and such that there does not exist a

196
Hints
generating set {g'l , . . . , g} for which the order of g 1 is equal to the order of g'l
and the order of gi is greater than the order of g for some i = 2, . . . , k.
6.3.7. Use 6.2.13.
6.3.8. (6) Choose {a l aa, a 2 , a3} as a new generating set; (7) choose
{ata2' a2} as a new generating set; (8) choose {alaa, a2, a3} as a new
generating set; (9) choose {ala2a3 l, a2, a3} as a new generating set.
6.4.4. Consider the set consisting of all elements in the maximal linearly
independent set A and the representatives of all classes in the maximal
linearly independent set G/ A.
6.4.20. Use 6.4.15.
6.4.21. Consider the set H of all elements in G which are linearly dependent
on elements in M. Prove that H is a pure subgroup of G. Use the result of
6.4.20.
6.4.25. Use 6.4.23 and 6.4.24.
6.4.26. Use 6.4.21 and 6.4.25.
CHAPTER 7
7.1.18. Consider the set S' = S U z (z rt S). Prove that the mapping  defined
by
ff (a) = ( z, aa" at... , aE,.. . )
a, aaa" aa, ..., aaE, ...
(a E S)
is an isomorphism of S into the semigroup of all transformations of S'.
7.1.20. Use the representation defined in the hint to 7.1.18.
7.1.28. Consider the following mapping of S into C: let ao be a fixed element
in S, then set
_ { 0 if aao # ao
fao(a) - .
1 If aao = ao
Prove thatfa o is a homomorphism for every ao E S.
7.1.29. Use the preceding problem.
7.2.2. Use the preceding problem.
7.2.10. Consider the subgroups HI = {(a 9a , b P )} and H 2 = {(a 3a , b 3P )},
where , p are any integers.
7.2.13. If H 2 = a-I HI a, then consider the mappingf of the set of right cosets
of HI in G into the set of right cosets of H 2 in G defined by f(xHl) = xaH 2 .
7.2.14. Let f be a one-to-one mapping of the set of right cosets of HI onto
the set of right cosets of H 2 satisfyingf(A.:t(xH 1» = A.2(f(xH 1»' Prove that
(a E S)

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197
if f(H 1 ) = aH 2 then f(xH 1 ) = xaH 2 and H 2 = a- 1 H 1 a. In this proof use
the fact that gH 1 = H 1 if and only if g E H 1 .
7.2.17. Use 7.2.15.
7.2.21. Use the preceding problem.
7.2.22. Use 7.2.21.
7.2.23. Use 7.2.12.
7.2.27. Use 7.2.13 and 7.2.14 and prove that every faithful transitive rep-
resentation of S4 by a subgroup of order four is similar to one of those in
7.2.13.
7.2.28. Use 7.2.14.
7.2.29. Use 7.2.13, 7.2.14, 7.2.27, and 7.2.28. Prove that every faithful
transitive representation of 54 by a subgroup of order four is similar to one
of those in 7.2.27.
7.2.31. (1) Consider the mapping g of the set of right cosets over H 1 into
the set of right cosets over H 2 defined by g(xH 1) = f(x)H 2 and (2) consider the
mapping t of the image of the representation rxH 1 into the image of the
representation rxH 2 defined by
e ( HI ) - 82
U g - UcIJ (g)
7.2.33. Let t be an isomorphism between the images of the representa-
tions rxH 1 and rxH2' Let g be a one-to-one correspondence between the right
cosets over H 1 and the right cosets over H 2' If rx is the automorphism of G
determined by t (see 7.2.32) and if g(H 1) = xH 2' then prove that rx(H 1) =
xH 2X - 1. Thus deduce that the desired automorphism f is defined by f(a) =
x- 1 rx(a)x.
7.3.17. Use 7.3.16.
7.3.18. Let
T (x) = ( Tl (x) 0 )
A (x) T 2 (x)
where T 1 (x) is an m 1 x m 1 matrix and T 2 (x) is an m 2 x m2 matrix. Suppose
that the order of G is n. Consider the matrix
p= ( E m1 0 )
F Em!
where E mi is the mi x m i (i = 1,2) identity matrix, and
F=  I A (x) T 1 1 (x)
xEO

198
Hin ts
7.3.19. Use the preceding problem.
7.3.20 and 7.3.21. Use 7.3.17.
7.4.11 and 7.4.12. Use 7.4.10.
7 .4.1 7 .4.16. Use 7.4.6 and 7.4.10.
7.4.18-7.4.20. Use 7.4.6, 7.4.10, and 7.4.17.
7.4.21-7.4.24, 7.4.28, and 7.4.34. Use 7.4.10 and 7.4.17.
7.5.7 and 7.5.8. Use 7.3.21.
7.5.10. Use 7.4.10 and 7.5.7.
7.5.14. Use 7.5.13.
CHAPTER 8
8.1.1. (3) Use the Cauchy inequality
n n n
( a j h i )2  (  al) · (  hi)
1=1 i=1 1=1
8.1.16. Consider the set of polynomials with rational coefficients and use
the result of problem 1.2.21.
8.1.17. Introduce the following metric on M T:
p (x, y) = sup I x (t) - Y (t) I
(x, yEM T )
8.1.20. Consider the mapping
, ( { p (x, y), if p {x, y)  1
p x, y) = .
1, If p (x, y) > 1
8.2.6. Use 8.2.5.
,
8.2.9. Use 8.2.7.
8.2.16. As a preliminary, prove that the set {p(f(x), g(x»} (x E M) is bounded
in the metric space of real numbers (8.1.1 (2». Then use the well-known
theorem that the least upper bound of a bounded set of real numbers always
exists, and that this bound can be reached.
8.2.18. Use 8.2.7 and 8.2.16.
8.2.19. Use 8.2.15.
8.2.23. Use 8.1.20.
8.3.21. Use 8.3.18.
8.3.28 and 8.3.29. Use 8.3.19.
8.3.30. Use 8.3.21.

Hin ts
199
8.4.5. Use 8.3.28.
8.4.11. Suppose ex is an irrational number. It is well-known from number
theory that for any positive number e and any real number d whatsoever,
there exist integers m and n such that Imex - d - nl < e. Make use of this
result to solve the second part of the problem.
8.4.12 and 8.4.13. Use 8.3.19.
8.4.17. Suppose that x E U n H. Prove that there exists an open set V
containing e such that xV c U. Let a E H . Prove that there exists be H such
that xba- 1 E x V c: U. Thus xba- 1 E H. Prove that a E H.
8.4.19. Use 8.3.21, 8.3.19, and 8.4.6.
8.4.20. Use 8.4.19.
8.4.21. Use 8.4.6, 8.3.25, 8.3.27, and 8.3.29.
8.4.23. Use 8.4.21, 8.4.20, 8.3.28, and 8.3.29.
8.4.25. Use 8.3.28 and 8.3.29.
8.4.28. Use 8.4.5, 8.4.7, and 8.3.19.
8.4.29. Use the preceding problem and 8.4.13.
8.4.35. Introduce a coordinate system on D and use the well-known theorem
that all formulas which express affine transformations in these coordinates
are linear.
8.4.36. Use 8.4.19 and 8.3.28.
8.4.38. Let g be the composition of the mappings cp and f - 1, where cp is
the canonical mapping of G onto the set of right cosets G/ Aa. Prove that cp
is open. Prove that since g is continuous, it follows that f - 1 is continuous.
8.4.39. Use 8.2.23.
8.5.7. Consider the relation p defined by x '" y(p) if and only if X-I Y E H.
Prove that p is a two-sided compatible ordering.
8.5.12. Use 8.5.5.
8.5.13 and 8.5.14. Use 8.5.7.
8.5.26. Use 8.5.1, 8.5.6, and 8.5.7.

ANSWERS
CHAPTER 1
313
1.1.1. 1 EMt; - 2, 0, 1 E M:a; - 2, 0, 1, 5' 7 E Ma; 1, 5' V2: 'It,
1 3 1 1 1
7 EM,; 0, 1 ( M,; 5' V2: -2, 1t, 7' i, -i, Y2 + V2 i, 2+iE Me.
1.1.2. x, z.
1.1.3. Three elements.
1.1.4. M 1 c M 2 c M 3 ; M 1 c M 4 ; Ms c M 2 C M3'
1.1.6. Zero.
1.1.7. (1) all real n umbers; (2) 0.
1.1.8. (1) The set of all natural numbers; (2) 0; (3) M k , where k is the least
common multiple of m and n; and (4) the set of all natural numbers different
from 1.
1.1.10. (1) M; (2) 0.
1.1.11. (1) A; (2) M; (3) 0; (4) An B; (5) AU B; (6) A; (7) AU B; (8)
A U B;(9) 0; (10) M.
1.1.12. 2 4 ; 14.
1.1.13. There are three partitions: Mo U M 2 U M 3 ; M4 U Ms; Mo U
M 1 U M3 U M6 U M7'
1.1.14. Yes, it forms a partition.
1.1.15. 0..
1.1.17. 243 = 3 s .
1.1.18. (-1, a, a), ( -1, b, a), ( -1, c, a), (1, a, a), (1, b, a), (1, c, a).
1.1.19. k 1 . k 2 . k3'
1.1.20. Let M have m elements. If m is even, then both classes must contain
tm elements. If m is odd, then one class will have ¥m + 1) elements, and
the other ¥m - 1).

202
Answers
1.2.1. The largest number of mappings is m n ; mappings of A onto B exist
when n  m, one-to-one mappings of A into B when n  m, and one-to-one
mappings of A onto B when n = m; there are n(n - 1)...3.2.1 = n!
mappings of A onto B.
1.2.3. Mappings of C onto A of the desired type do not exist; there are
mappings of C onto B of the desired type.
1.2.4. 175.
1.2.5. Yes, cp is one-to-one; cp(M) consists of all natural numbers except
for the n integers n, n + 1,..., 2n - 1.
1.2.6. Yes, cp is one-to-one; cp(N) # N, Le., cp maps N into N but not onto N.
1.2.7. It is necessary and sufficient that h(x) # 0 for all x.
1.2.8. The mapping is one-to-one, but not onto. All equilateral triangles
are fixed points of the mapping.
1.2.18. The set U M k is finite if and only if all of the M k are finite and there
k
exists a natural number n such that each M i is equal to one of the sets
M 1 ,M 2 ,...,M n .
1.2.31. The mapping cp must be one-to-one, and N"",cp(N ' ) must be infinite.
1.3.3. (1) pr 1 Pk = {k + 1, k + 2,...} if k # 0; pr l Po = N; (2) pr 2 Pk =
{I, 2,...}; (3) n  m(O') if and only if n  m; (4) n  m( O'* ) if and only if
m < n; (5) n  m(t) if and only if n  m; and (6) n  m(t*) if and only if
m  n.
1.3.5. (ai' a2, . . . , an)  (b 1 , b 2 , · . . , bn)(-r 1) if a 1 = b 1 , a2 = a3 = · . · = an =
0; t 2 = ; (a 1 , a 2 , . · . , an)  (b 1 , b 2 , · · · , b n) ( t 3) if a 1 = b 1 , a 2 = a 3 = · · · =
an = b 2 = b 3 = · . · = b n = O.
1.3.6. (1) The relation PI is symmetric; (2) P2 is reflexive, transitive, and
antisymmetric; (3)P3 is antisymmetric; (4) P4 is transitive and antisymmetric;
(5) Ps is reflective, antisymmetric and transitive; and (6) 0'0 is reflexive,
transitive, symmetric, and antisymmetric, whereas O'k for k # 0 is only
antisymmetric.
1.3.11. The real numbers a and b are in the same equivalence class of the
transitive closure of P if and only if b  a. The transitive closure of P U p* is
equal to (OR'
1.3.12. (1) z 1 -- Z 2 (PI n pT) if and only if Iz 11 = I Z 21 ; (2) (OK; (3) ; (4) (OK'
1.3.13. p' = P U p*.
1.3.14. p' = P U .
1.3.15. PI is an ordering, but P2, P3, P4, and Ps are not orderings.
1.3.16. PI is an ordering, P2 is an equivalence, P3 is neither an ordering nor
an equivalence, and P4 is an ordering.

Answers
203
1.3.20. All finite linearly ordered sets and infinite linearly ordered sets are
of the following three forms:
a! < a! < ... < an < ..., a 1 > a! >... > an > ...
... < a_ n < ... < a_ 2 < CX -1 < CX o < a 1 <CX 2 < ... < a,a < ...
1.3.21. The sets given in (2) and (6) are well-ordered. The others are not
well-ordered.
1.3.22. M is a finite set.
1.4.1. p2 = P; pa = a; a 2 is a binary relation such that n ,......, m(a 2 ) for
n - m < 1; t 2 = w; PAk is a binary relation such that n ,......, m(pAk) if n divides
either m + k or m - k; AkP is a binary relation such that n ,......, m(AkP) if m
is divided by either n - k or n + k; aA k = Aka is a binary relation such that
n ,......, m(aAk) if n < m + k; AkA, is a binary relation such that n ,......, m(AkA,) if
In - ml = k + I or In - ml = Ik - II.
1.4.6. pR = Z; Rp = Z; aR = R; Rt = R; pC = C; Ca = Z; Ct = C;
pP is the set of all complex numbers z such that Izi = 1; Pa is the set of all
numbers of the form bi, where b is a positive real number; Pt = { - i}.
1.4.9. W(PI n P2) = {(a, a), (b, a)} ; WPI n WP2 = w.
1.4.10. w 2 = w; wL\ = L\w = w if X - contains more than one element;
wL\ = L\w = 0 if X consists of one element;  2 = w if X contains at least
three elements;  2 =  if X consists of two elements;  2 = 0 if X consists
of one element.
1.4.11. PiPj = w if i :1= j and P; = Pi'
1.4.12. p2 = P; pp is a relation such that P ,......, Q(pp) if Q # M; PP is a
relation such that P ,......, Q(pp) if P :1= 0; pp* = w; pp* = p.
1.4.13. P = P = P; wp is a relation such that a ,......, b(wp) if be pr 2 P; pw
is a relation such that a ,......, b(pw) if a E pr IP.
1.4.14. (1) pr 2 P = X; (2) prlP = X; (3) prlP = pr 2 P = X; (4) P # 0; and
(5) ap n bp # 0 for all a, be X I .
1.4.15. af = a I , a = a 2 ; a = w; f(x) ,......, g(x) (a I a 2) if and only if f(a) 
g(a) and f(b)  g(b); a2a3 is a relation such that f(x) ,......, g(x)(a2a3) if f(a) #
g(a) and f(b) # g(b); a 3 a 2 = a 2 a 3'
1.4.20. If PI and P2 are reflexive, then so is PIP2' This result does not hold
for transitivity, symmetry, and antisymmetry.
1.4.22. PIP2 is a linear ordering only when PI = P2'
1.4.24. p2 and p 3 are reflexive, symmetric, and transitive; a 2 and a 3 are
reflexive, anti symmetric, and transitive; pa, ap, apa, and pap are reflexive.

204
Answers
CHAPTER 2
2.1.1. 45.
2.1.2. 1 2 3 4 5
1
2 1
3 2 1
4 3 2 1
5 4 3 2 I
2.1.3.. X2,X3,X4,X 6 .
2.1.4. , Z. Z2 Za Z..
Z.
Z2 Z. . .
Zs Z. .
Z.. . Zl
2.1.5. (1) The result t)f the operation is defined in N for all pairs (a, b) where
a and b have the same parity, i.e., they are both even or both odd; (2) the
result is defined for all pairs of elements in N; and (3) the result is defined for
all pairs (a, b) for which a # b.
2.1.8. M is isomorphic to M 1 , but M 2 is neither isomorphic to M nor MI'
2.1.12. 5.
2.1.13. Two subsets {a, b, c} and {b, c, e}.
2.1.15. M is isomorphic to every infinite subset of itself.
See the following tables for 2.2.1-2.2.3, and 2.2.6.
Operation
2.2.1
Q.)
Q.) Q.) :::
::c ::: .....
Q.) Q.) ..... 
.::: ;9 .- 
..... Q)
..... Q.) .... Q) u
cd > ..... Q.)
.- .... > u s::
..... ..... Q.)
::s cd > .5 s:: cd 
"0 e .- .5 cd u .
Q.) u ..... u ..... .....
C/} e 0 c::: ..s:: c::: ..c: s:: 0
0 C/} 00 00 Q.) ....
0 0 C/} Q.) .- Q.)  "0 Q.)
U < -J  -J """"'I N
Yes Yes Yes Yes Yes Yes Yes Yes No
Yes Yes Yes No No No No Yes Yes
Yes No No Yes Yes Yes Yes No No
No No No No No No Yes No No
Addition
M u1 ti plica tion
Subtraction
Division

Answers 205
2.2.2
(1)
(1) >
(1) .-
:0 > -
(1) (1) .- ..:g
=9 -
 - ..:g Q)
-
(1) '-'
- - (1) Q) u
Set Operation cd  '-' s::
- (1) > u
 - 5 s:: cd >.
cd >
"0 e '(3 .5 cd u .
(1) - u - -
C/} E 0 c:: ..c c:: ..c s:: 0
0 0 C/} 00 00 (1) '-'
0 C/} (1) .- (1) Q2 "0 (1)
U  -J  -J  N
Natural Addition Yes Yes Yes No No Yes Yes No No
n um bers Multiplication Yes Yes Yes No No Yes Yes Yes No
Subtraction No No No Yes No Yes Yes No No
Division No No No Yes No Yes Yes No No
Integers Addition Yes Yes Yes Yes Yes Yes Yes Yes No
Multiplication Yes Yes Yes No No No No Yes Yes
Subtraction Yes No No Yes Yes Yes Yes No No
Division No No No No No No Yes No No
Rational Addition Yes Yes Yes Yes Yes Yes Yes Yes No
numbers Multiplication Yes Yes Yes No No No No Yes Yes
Subtraction Yes No No Yes Yes Yes Yes No No
Division No No No No No No Yes No No
Positive Addition Yes Yes Yes No No Yes Yes No No
numbers Multiplication Yes Yes Yes Yes Yes Yes Yes Yes No
Subtraction No No No Yes No Yes Yes No No
Division Yes No No Yes Yes Yes Yes No No
Negative Addition Yes Yes Yes No No Yes Yes No No
numbers Multiplication No Yes Yes No No Yes Yes No No
Subtraction No No No Yes No Yes Yes No No
Division No Yes Yes No No Yes Yes No No
2.2.3
(1)
(1) (1) .
:0 . -
(1) (1) - ..:g
. :0 .- ..:g
- Q)
- (1) .- '-' Q) u
cd .:: - (1)
Operation '-' s::
- (1) > u
 - s:: s:: cd
cd > u >.
"0 e .- .5 .- cd .
(1) u - u - -
C/} e 0 c:: ..c c:: ..c s:: 0
0 0 C/} 00 00 (1) '-'
0 C/} (1) Q2 (1) Q2 "0 (1)
U  -J -J  N
Addition Yes Yes Yes Yes Yes Yes Yes Yes No
Multiplication Yes Yes Yes No No No No Yes Yes
Subtraction Yes No No Yes Yes Yes Yes No No
Division No No No No No No Yes No No

206 Answers
2.2.6
(1) >.
(1)  .
(1) .....
:i5 . ..... eI} c= 0
(1) (1)  eI} .2 (1) ....
:i5 ..... 
. .-  Q) .2 . :-g
(1) .....
Operation ..... .- .... Q) u ..... ..... eI}
 . ..... (1) .- c= "0 0 "0
.... > c= ..... eI}
..... ..... (1) u c= (1) (1) 0 .... (1)
  > .5 c=  (1) :-g "0 .... (1) "0
"0 e '(3 c=  u :-g 'Ci) (1) N .-
eI}
(1) e 0 .- ..... u ..... ..... . N ..... .
eI}  ..c  ..c  ..c 0 ..c 0
0 0 eI} 00 00 .   00 
0 eI} (1) Qa (1) Qa (1) (1) Qa
u  -J -J -J  f- -1 f-
1 Yes Yes Yes No No No No No No No Yes Yes Yes
2 Yes Yes Yes No No No No Yes Yes Yes No No No
3 Yes No No No No Yes Yes No Yes No Yes No No
4 No Yes No No No Yes Yes No No No No No No
2.2.7. Closure and associativity. It has an identity and a zero. Nonsingular
matrices have inverses.
2.2.8. It is closed for all p, q, r; commutative for p = q; associative for p =
q = 1, p = q = 0, p = 1 and q = r = 0, and p = r = 0 and q = 1; left
invertible for p # 0; right invertible for q # 0; left cancellative for q # 0;
right cancellative for p # O.
2.2.9. Closure and associativity. There is an identity and a zero.
2.2.10. Closure, commutativity, left and right invertibility, and left and right
cancellation. Neither an identity nor a zero exists. All elements are idempo-
ten ts.
2.2.11. Closure, left and right invertibility, and left and right cancellation.
2.2.12. The operation corresponds to addition of vectors. It is associative
and left and right cancellative.
2.2.13. (1) All entries are filled in; (2) the table is symmetric with respect to
the diagonal; (3) all of the elements occur in each column; (4) all of the ele-
ments occur in each row; (5) no element occurs in any row more than once;
and (6) no element occurs in any column more than once.
2.2.14. For commutativity, associativity, and left and right cancellation.
2.2.16. The operation of union is closed, commutative, and associative. The
operation of intersection satisfies these same properties.
2.2.17. All of the basic properties except associativity.
2.3.1. RR = RZ = ZR = R;ZZ = Z;PP = P;PN = N;NN = P.
2.3.2. (MM)M = {a}, M(MM) = {a, b}.
2.3.7. A is an ideal; B is closed but is not an ideal; C 1 is closed; C n is not
closed for n > 1; Dn is an ideal; C n n Dm is empty if n < m, is not closed if
n  m, n # 1, and is closed but not an ideal if n = m = 1.

Answers
207
2.3.8. The set of all real numbers is closed but is not an ideal; the set of all
pure imaginary numbers is not closed.
2.3.9. The set of all nonsingular matrices is closed but is not an ideal; the
set of all singular matrices is an ideal.
2.3.16. T 1 and T 2 are ideals; r. T 2 is not an ideal.
2.3.17. {z} is closed if and only if z is an idempotent; {z} is a left ideal if z
is a right zero of M  {z} is a right ideal if z is a left zero of M  {z} is a two-
sided ideal if z is a zero of M.
2.3.18. n + 1.
2.4.1. lp 1 and lp3 are homomorphisms; lp2 and lp4 are not homomorphisms.
2.4.2. (1) The homomorphism lpo which maps all of R onto 0; (2) the
homomorphism lp 1 defined by lp 1 (0) = 0, lp 1 (r) = 1 (r # 0); and (3) the
remaining homomorphisms are constructed in the following manner. Let
R' be the set consisting of - 1 and all prime natural numbers, and P any
nonempty subset of R'. The homomorphism lpp is defined by lpp(O) = 0,
lpp(l) = 1, lpp(r) = (_1)k 1 +...+kn, where r = pl ... pnq'l... qm (PI"" Pn E P,
q 1 , . . . , qm E R',""P; k l' . . . , k n , 11, . . . , 1m are integers).
2.4.3. lp 1 and lp 3 are homomorphisms; lp2 is not a homomorphism.
2.4.7. For each k = 1, 2, 3, . .. the mapping lpk defined by lpk(5 n ) = 5 kn
(n = 1, 2, 3, . . .) is a homomorphism. There are no other homomorphisms.
All of the lpk are isomorphisms.
2.4.10. It is necessary and sufficient that M 0 have at least one idempotent.
2.4.12. a '" b(a) but aa '" bb(a) does not hold.
2.4.13. (1) lpl is a homomorphism; each class of the corresponding partition
consists of all polynomials which have the same leading coefficient; (2) lp2
is a homomorphism; each class consists of all polynomials with the same
leading coefficient; (3) lp 3 is a homomorphism; each class consists of all
polynomials whose values are equal at x = 1; (4) lp4 is a not a homomor-
phism; (5) lps is a homomerphism; each class consists of all polynomials
whose constant terms have the same absolute value; and (6) lp6 is a homomor-
phism; for c # + 1 each class consists of all polynomials of the same degree,
for c = 1 there is one class which consists of all polynomials, and for c = - 1
there are two classes, one consisting of all polynomials of even order and the
other all polynomials of odd order.
2.4.15.
/R+ R- RO
R+ R+ R- RO
R- R- R+ RO
RO RO RO RO

208 Answers
2.4.17.
MITt I 1 T 1 M/ T 2 1 1 2 T 2 MITal 1 2 4 Ta
1 1 T 1 1 1 2 T 2 1 1 2 4 Ta
T 1 T 1 T 1 2 2 T 2 T 2 2 2 4 Ta Ta
Ts Ts T 2 T 2 4 4 Ta Ta Ta
Ta Ta Ts Ta Ta
2.4.18. qJ is an isomorphism only when p is the equality relation.
2.4.21. Every equivalence in which a '" b, and every equivalence in which
a and b are not equivalent and are not equivalent to any elements other than
themselves, is a congruence. Altogether there are 7.
2.4.22. 5.
2.5.3. The set consists of all primes and the number 1.
2.5.4. [x] is the set of all matrices of the form
( ':)
where n is any natural number; [x, t] is the set of all matrices of one of the
following forms:
( 2k 2 k n )
o 0 '
( 2k 0 )
o 0 '
( )
where k and n are any natural numbers; [y, z] is the set of all matrices of one
of the following forms:
(k l ), (k ), (  l )
where k and n are any natural numbers; [y, t] is the set of all matrices of one
of the following forms:
( 1 0 ) ( 0 2n 0 0 )
o 2 n
where n is any natural number.
2.5.12. Ten subsemigroups, of which five are ideals.
2.5.13. (1, p), where p is a prime, and (2, 1).
2.5.15. (1, 1), (2, 1), and (3, 1). Any subset which contains 0 and is such that
if it contains a k then it also contains all elements b ik is a left ideal. Any subset
which contains 0 and is such that if it contains ak then it also contains all
elements b ki is a right ideal. Any subset which contains 0 and is such that if it
contains a k then it also contains all b ik and b ki is a two-sided ideal.

Answers
209
2.5.17. It is a regular semigroup in which every element is an inverse of
every other element.
2.5.20. It is a normal subset which is a subsemigroup.
2.5.21. If S = [x] then besides the four subsets consisting of one element each,
the normal subsets are
{ x, x 8 }, {x 9 , x.}, {x., x.}, {Xl, x 8 , x.}, { x, x', x 8 , x 4 }
2.6.3. The groups are (2), (4), (5), and (6).
2.6.4. M is the set of all matrices of the form
( 000 )
o a 0
000
(a ;l: 0)
2.6.6. The orders, respectively, are 2, infinite, 4, infinite, and infinite.
2.6.11. 8.
2.6.12. 16.
2.6.24. (1) The group consisting of the identity; (2) cyclic groups of prime
order; and (3) cyclic groups whose orders are the form p2, where p is a prime.
2.6.25. (1) {I, -1, i, - i} ; (2) {I, -t + tiJJ, -t - t i J3}; (3) {I, -tJ2 +
tiJ2, -t i J2, tJ2 - tiJ2, tJ2 + tiJ2, i, - i, -I}; (4) {2 4k , 2 4k + Ii,
_2 4k + 2 , _2 4k + 3 i} (k = 0, + 1, + 2,...,); (5) {( -1)"2 k 5"} (k, n = 0, + 1,
+ 2,.. .); (6) {I, -1}, {I}, {1, -1}, {2 4k , _2 4k + 2 } (k = 0, + 1, + 2,.. .),
{( -1)"2 k 5"} (k, n = 0, + 1, + 2,. . .).
2.6.29. All periodic groups.
2.6.33. Abelian groups.
2.6.38. H 2 is not a normal subgroup of G.
2.6.39. {I}, {I, -I}, {I, -1, i, -i}, {I, -1,j, -j}, {I, -1, k, -k}, Q.
CHAPTER 3
( 1 2 3 4 5 6 7 8 )
3.1.2. Cly =
873 1 724 5
2Cl = ( 1 2 3 4 5 6 7 8 ) Cly3 = ( 1 2 3 4 5 6 7 8 )
P 1 8 1 2 228 2 527 8 235 1
( 1 2 3 4 5 6 7 8 )
Clf3}' =
37887 1 3 7
( 1 2 3 4 5 6 7 8 )
YCl = 1 6 2 8 3 8 4 5

210
Answers
3.1.3. The types of Cl, {3, Cl{3, and {3Cl are (1, 3), (2, 2), (3, 1), and (2, 1), respec-
tively.
3.1.4. (1) closure and associativity; (2) closure and associativity; (3) closure,
associativity, and (left and right) cancellation; (4) closure, associativity,
(left and right) cancellation, and (left and right) invertibility; (5) all seven
properties; and (6) closure, associativity, commutativity, and (left and right)
cancellation.
3.1.5. (1) All constant functions are left zeros, there are no right, and there-
fore no two-sided, zeros; (2) all functions whose values lie in the interval
[0, 1] are right zeros, there are no left zeros; (3) the same as in (1) ; and (4) no
zeros.
3.1.6. The left zeros are the transformations Cla(a E X) defined by Cla(X) =
a(x EX).
3.1.7. (1) All of the sets in 3.1.4 are semigroups, and 3.1.4 (4) and (5) are
groups; (2) both sets are semigroups; (3), (4), and (5) are semigroups; and
(6) is not a semigroup. None of the semigroups in (2H5) are groups.
3.1.8. The type of Cl is (1, 1), the type of {3 is (1,2).
3.1.9. The vectors of the translation are perpendicular to the given line,
have length twice the distance between the given lines, and have opposite
sIgns.
3.1.11. All Yc commute with each other and with {3; Cl commutes only with
Yo and Yt .
3.1.12. (1) {31 = 1, {33 = 3, Cl({3n) = Cln (n E X, n # 1,3); and (2) {31 = 3,
{33 = 1, Cl({3n) # Cln (n E X, n # 1, 3).
3.1.13. (1) The semigroup [Cl, {3]s consists of the transformations of the form
{ k + 1,
Yk,,(n) = n - I + k,
ifnI+1
ifn>l+1
where k, I are any positive integers. (2) The semigroup has a unique irreducible
generating set {Cl, {3}.
3.1.14. (1) The inverses of Cl are
(k l 2 3 ... n
12... n-l
. . . )
. . .
(k E N)
the inverses of {3 are
( 1 2 3 ... n . . . )
Cl and
134... n+l...
(2) Cl and {3 are inverses of each other.

Answers
211
3.1.19. (3) The identity transformation is the only idempotent.
3.1.20. M 3 ,M 4 , and Ms are subsemigroups, M4 and Ms are right ideals,
M 4 is a two-sided ideal, and M 1 , M 4' and M s are normal subsets.
3.1.21. The equation OC2Y = P is not solvable; OC 1 Y = f3 has a unique solution;
OC 3 Y = P has four solutions.
3.1.23. f3X c ocX and for each y E f3X there exists a unique x E X such that
ocx = y.
3.1.24. The equations YOC 1 = P and Y OC 2 = f3 are solvable, YOC 3 = f3 is not
solvable. The first has a unique solution.
3.1.25. The equation yoc = P is solvable if and only if ocx = ocy implies
f3x = py (x, Y E X). There is a unique solution if and only if ocX = X holds
in addition to the above condition.
3.1.26. (1) oc is a one-to-one mapping of X into itself; (2) ocX = X; and (3)
ocx = x for all x E ocX.
3.1.27. (2) All elements of T' are idempotents; (3) all subsets containing
OC o and PI are left ideals of T', and {oc o }, {f31}' {oc o , f31}' and T' are the right
ideals of T'.
3.1.28. The normal subsets are (1), (3), and (4).
3.1.32. M 1 and M 3 are groups.
3.2.1. (1) In problem 3.1.2, oc is an invertible transformation, and
oc- 1 = ( 1 2 3 4 5 6 7 8 )
341 567 8 2
f3 and yare neither left nor right divisors of the identity; (2) in problem 3.1.4,
oc is a right divisor of the identity, and P is a left divisor; neither transformation
is in verti ble.
3.2.2. fl(x) is invertible when n is odd, f3(x) is invertible, f2(x) and f4(x)
are not invertible.
3.2.3. pr l Pa = pr 2 Pa = X, each cut consists of one element.
3.2.5. (ocP) - 1 = f3 - lOC - 1 .
3.2.6. The identity is the identity transformation. The inverse element of C(
is its inverse transformation oc -1.
_ 1 ( a be.. . )
3.2.9. oc = ' " .
a be...
3.2.10. (1) The Yc (c # 0) satisfy the first condition, f3 and Yc (c # 0) satisfy
the second; (2) both conditions are satisfied by all of the transformations in
3.1.19; (3) in 3.1.36, the first condition is satisfied by oc and y, the second by
oc, P, and y.

212
Answers
3.2.11. (1) Not invertible; (2) not invertible; (3) not invertible; (4) Ye (c # O)
(1m.n and (1, Y (of 3.1.36) are invertible, the rest are not,
-1
Ye = Ylle
-1
(1m,n = (1.. 11m, - nlm
(1..-1=(1
,
y- 1 = Y
( nt - I )
3.2.12. (1..-1 [ f(t) ] = f -mt + k
g(t) g ( nt - I \
-mt + kJ
3.2.13. (2) It is a group; (3) the transformations of order two are (1 and p,
defined by
( kt + l )
(1 [ f(t) ] = f mt - k
g(t)  ) ,
g\mt=kj
P [ f(t) ] = f( - t)
g(t) g( - t)
3.2.14 (1) (11 (12 (13 (1..4 (15 (1..6
(11 (1..1 (12 (1..3 (14 (15 (1..6
(12 (12 (11 (16 (15 (14 (1..3
(13 (13 (15 (11 (16 (12 (1..4
(14 (14 (16 (15 (11 (13 (12
(15 (15 (1..3 (14 (12 (16 (11
(16 (16 (14 (1..2 (13 (11 (15
(2) (11 is the identity; the inverses of (12' (13' and (14 are (12' (13' and (14' re-
spectively.
3.2.16. H is normal and H" is a subgroup of G.
3.2.18. (1), (3), and (4) are invertible.
3.2.19. The sets (1), (2), and (4) are groups.
3.2.20. 24.
( Xl X2 X 3 ) ( Xl X2 X 3 )
3.2.21. e = , (1 =
Xl X2 X 3 Xl X3 X2

Answers
213
3.3.1. a 3 = (4865), /32a = (235846), yb't = (16824)(375), y 4 b 2 = (2487356),
and 'tby = (186732)(45).
3.3.2. a has order 12, /3 has order 5.
3.3.4. a = (123 )(4568), f3 = (1 9 2 12)(3 8)(4 11 10)( 5 6 7), and }' =
(194)(267)(358).
k { Xi + k , if i + k  n
3.3.7. a Xi = . .
Xi + k - n , If l + k > n
where 0  k  n. If k > nand k = pn + r (0 < r  n), then a k = a r .
3.3.10. aba- 1 = (43251)(23)(26), 't- 2 Y't 2 = (847)(7523)(8641), /3- s bf3s =
(82134)(12)(15).
3.3.12. Only the powers of the given cycle.
3.3.13. All possible products of powers of a l and a2 commute with a, where
a l = (X I X2 X 3 X 4 X S) and a2 = (X6 X 7 X 8 X 9 X IO).
3.3.18. Letting a = (12)(34), /3 = (13)(24), y = (14)(23), we obtain the table:
e a /3y
e e af3y
a ae y /3
/3 /3y e a
y y /3ae
3.3.27. m = 1, 2, 3,4.
3.4.1. Monotonic increasing functions (not necessarily strictly monotonic).
3.4.2. (1) Neither is an endomorphism; (2) t is an endomorphism of N;
(3) y is not an endomorphism; and (4) b m is an endomorphism of N.
3.4.3. The transformation b m preserves addition, 't and b I preserve multipli-
cation.
3.4.4. No.
3.4.13. No.
3.4.14. a( ) = ( ).
3.4.15. G' = {I, -I}, the identity subgroup, and the group itself.
3.4.16. 51.
3.4.17. The group of automorphisms consists of permutations of the form
( a b c d f )
a = abc' d' f'
where (c', d', f') is a permutation of the numbers c, d, f.

214
Answers
3.4.18. The following transformations are endomorphisms: (1) all trans-
formations of rank 1; (2) all transformations oc of rank greater than one
satisfying oca = a.
3.4.24. No.
3.4.27 . Yes.
3.4.28. oc()2) = + )2, oc() = + .
3.4.32. ( It ..., i-I, i, i + 1, i + 2, ..., n )
1, ..., i-I, i, i, i + 2, ..., n
( 1, ..., i-I, i, i + 1, i + 2, ..., n )
I, ... J i-I, I + I, i + 1, i + 2, ..., n
i = 1, 2, ..., n - 1
3.5.3. The elements of finite order are all rotations through an angle r1t
where r is rational, reflections, and the products of reflections and parallel
translations by vectors which are perpendicular to an axis of symmetry.
The elements of order 2 are reflections, rotations through 180°, and the pro-
ducts stated above.
3.5.7. (1) Infinite, (2) 2, and (3) 1.
3.5.8. oc - 1 E A if oc(F) = F.
3.5.9. A is a group for (1) and (3) but not for (2).
3.5.10. The group of all parallel translations of the plane onto vectors
lying on the given plane.
3.5.12. (1) 1 ; (2) 2; (3) infinite.
3.5.13. G consists of parallel translations of the plane by vectors which are
parallel to the given line, rotations of the plane about a point on the given
line through an angle of 1t, and reflections of the plane about the given line
and lines perpendicular to it.
3.5.14. (1) The group consists of all rotations of the rhombus about its
center through the angles 0 and 1t, and reflections about its diagonals; (2)
the group is of order 8 and consists of the rotations of the square about its
center through the angles 0, 1t/2, 1t and 31t/2, and reflections about its diagonals
and the lines which join the midpoints of opposite sides; and (3) the group
consists of the identity transformation and the reflection about the altitude
of the triangle.
3.5.15. n.
3.5.16. The group is of order 2n and consists of all rotations about the center
through the angles 0, 21t/n, . . . ,2(n - 1)1t/n, and reflections about the axes of
symmetry of the polygon.
3.5.17. The group consists of rotations about the common vertex through the
angles 0 and 1t, and reflections about both the line through the diagonals of

Answers
215
the squares and the line perpendicular to it and passing through the common
vertex.
3.5.20. It is a cyclic group of order n.
3.5.21. For n a prime number.
3.5.22. 4n.
3.5.25. The group consists of rotations of the tetrahedron about its altitudes
through the angles 0, 2re13, 4re/3, rotations about the lines joining the mid-
points of opposite edges through the angle re, and reflections about the 12
planes of symmetry of the tetrahedron.
3.5.26. Rotations about the axes of symmetry of the terahedron.
3.5.27. The group is of order 6, and consists of rotations of the cube about the
diagonals which pass through A through the angles 0, 2re13, and 4re/3, and
reflections about the planes of symmetry of the cube which pass through A.
3.5.28. 48.
3.5.30. The alternating group A4'
3.5.31. The group is of order 10, and consists of 5 rotations of the icosahedron
about the axis of symmetry which passes through P through the angles 0,
2re/5, 4re/5, 6re15, and 8re15, and 5 reflections about the planes of symmetry
which pass through P.
3.5.33. (1) Rotations through the angles 2re/3 and 4rel3 about the centers of
the triangles which form adjacent vertices of F and rotations through the
angle re about the centers of the rhombi which form adjacent vertices.
3.5.34. (1) The group consists of rotations through the angles 0, re12, re and
3rel2 about all points with coordinates (tk, tl) where k, I are integers of the
same parity, rotations through the angle re about all points with coordinates
(tk, tl), where k, I are integers of different parity, reflections about the axes of
symmetry of all squares made up of adjacent points in F, and parallel trans-
lations by vectors joining any 2 points of F ; and (2) the subgroup consists of
rotations about a point through the angles 0, re12, re, and 3rel2 or is generated
by 2 reflections about mutually perpendicular axes.
3.6.2. (1) oc = ( 9 10 ) , OC 1 OC 2 = ( 1 7 10 ) , OC2OCI = ( 2 8 ) , p = ( 10 ) ; and (2)
3 10 4 4 5 3 3 2
the types of oc 1 , /31 , and /32 are (2, 1), (4, 1), and (4, 2), respectively.
3.6.3. (1) The type of OC 1 is (1, 1) if x = y and (2, 1) if x # y; (2) the type of
OC3 is (1, 1); and (3) the type of OC3 is (n, 1).
-1 ( 1234578 ) -1 ( 1234578 ) -1 ( 1345678 )
3.6.4. oc = , ococ = , oc oc =
7583641 1234578 1345678
3.6.5. ococ - 1 is the partial identity on roc and oc - lOC is the partial identity on t!;J..
The equality holds if and only if doc = roc.

216
Answers
3.6.6. (l)r[fl(x)] = (- 00, (0),d[f2f3(x)] = {knj2} wherekisanyoddinteger,
r[f2f3(x)] = {O}; (2) d[fi(x)] = [t, ej(l + e)], where
1n x
1 0 ( ) 1 1 - ..t"
i x = n e (I-x)
1n
x
d[fsf4(x)] = { -1, t} where fsf4(x) = 0, and ff4(x) = 0; and (3) d[fi(x)] =
( - 00, - In] U [In, (0), fi(x) = J(x 2 - n).
3.6.7. (1) Those that are one-to-one are fl(x), f4(x), and f6(x); (2) dfl 1 (x) =
( - 00, (0), f 1 1 (x) = eX j(1 + eX), df 4 l(X) = [ -nj2, nj2], f 4 l(X) = sin x,
dfil(x) = (- 00, aje) U (aje, (0) if e # 0, dfil(x) = (- 00, (0) if e = 0,
f6 l (x) = (dx - b)j(a - ex); and (3)J;fi- l (x) = ed/r 1 ,fi- l J;(X) = ed/i-
3.6.9. el 2 -: 0 if and only if del n rel = 0, el 3 = 0 if and only if elX rt del for
all x E del n rel.
3.6.10. When rel2 c: dell'
3.6.12. (1) When rel c: del and el 2 X = elX for all x E rel; and (2) if and only if
it is a partial identity transformation.
3.6.13. (1) If rel = X; and (2) if del = X and el is one-to-one.
3.6.14. (1) If del # X; and (2) if rel # X.
3.6.17. The right cut of p over each x E X must contain at most one element.
3.6.18. (1) el is the identity transformation; (2) del = rel = M and el 2 = eM'
where M is a subset of X ; and (3) if x E del and elX # x then x rt rel.
3.6.23. If X is the set of all real numbers, then
df2 = X" {I, 2j3}, f2(X) = (x - 9)j(3x - 2)
df3 = X" { -1, 2j3, Ilj4}, f3(x) = (-7x - 12)j(4x - 11)
3.6.24. No.
CHAPTER 4
4.1.6. S3 = {e, (12)} U {(13), (123)} U {(23), (132)}.
4.1.7 A4 = {e,(123),(132)} U {(124),(13)(24),(324)} U {142),(143),(14)(23)}
U {(234), (134), (12)(34)}.
4.1.8. K = {I, -I} U {i, - i} U {j, -j} U {k, - k}. The right and left de-
compositions coincide since { -1, I} is a normal subgroup of K.

Answers
217
4.1.9. a = (e} U {. \,} U {2} U {x 3 } U {.4} U r.X 5 } U {X G } U
U {. \. 7 } LJ {y 8 } U {X  }
0= Ie, x 5 } U {.x, tft} U {x 2 , t7} U {.t 3 , ..".S} U {t4, XU}
a = {e, X.2, .4, xO, x 8 } U {.t", x 3 , x 5 , .x 7 , x 9 }
0= {e, x, x 2 , x 3 , x\ x 5 , x G , X7, x 8 , xf)}
4.1.10. G = Ko U Kl U K1,whereKoisthesetofallxnwherenisdivisible
by 3, K 1 the set of all x n where n divided by 3 leaves a remainder of 1, and Kl
the set of all x n where n divided by 3 leaves a remainder of 2. The decomposi-
tion can be given by a set of representatives, for example
0= [x 3 ]g U x [x3lg U x 2 [x 3 ]g
4.1.11. The decomposition of G by e is a representation of G as a union of
its one element subsets. The decomposition of G by G consists of one coset,
which is equal to G.
4.1.14. (1) f[f(z)] = f(z); (2) f(z)z-l E H; and (3) f(hz) = f(z).
4.1.18. 81.
4.1.24. K l' K 3 , and Ks are cosets, K 2 and K4 are not.
4.1.25. The stated set is both a left and a right coset by the subgroup of all
matrices with determinant 1.
4.1.27. 8 4 = Ie, (123), (132), (12)(34), (134), (234)} LJ
U {(12), (23), (13), (34), (1342), (1234)} U
U {(14), (1423), (1432), (1243), (1324), (24)} U
U {(124), (13) (24), (243), (143), (14) (23), (142)}
4.1.28. S3 = {e, (12)} U {(13), (23), (132), (123)}.
4.2.4. S3 = {e} U {(12), (13), (23)} U {(123), (132)}.
4.2.5. K = {I} U {-I} U {i, -i} U {j, - j} U {k, -k}.
4.2.6. 6.
4.2.8. The normalizer of x is the set of all matrices of the form
( ) (a:f= 0, b:f= 0)
The normalizer of y is the whole group G. Te normalizer of z is the set of all
matrices of the form
( o a a b )
(a *- 0)
4.2.13. The cyclic group of order 2.

218
Answers
4.2.14. The normalizer of an element of the form (2k + 1, m) is the set of all
elements of the form (2a, 0) and (2a + 1, b), where a and b are integers; it has
an infinite number of conjugates. The normalizer of an element of the form
(2k, m), where m =F 0, is the set of all (2a, b), where a and b are any integers; it
has 2 conjugates. The normalizer of (2k, 0) is the entire group; it has one
conjugate, the identity element.
4.2.20. 8 4 = Ie} U {(12), (13), (14), (23), (24), (34)} U
U {(123), (124), (132), (134), (142), (143), (234), (243)} U
U {(1234), (1243), (1324), (1342), (1423), (1432)} U
U {(12)(34), (13)(24), (14)(23)}
4.2.21. A4 = {e} U {(123), (134), (142), (243)} U
U {(124), (132), (143), (234)} U {(12)(34), (13)(24), (14)(2)}
4.2.22. The only conjugates are x 1 and x 3
4.3.1. e, {e, (123), (132)}, S3'
4.3.2. M I generates S4, M 2 generates A 4 , and M 3 is a normal subgroup.
4.3.3. Yes.
4.3.12. N is the identity of the group, (XN)-l = X-I N.
_4.3.14. GjG is the identity group, GjE coincides with G.
4.3.32. H I = GjN, H 2 = {N,xN}, where detx = -1,xeG.
4.3.34. The alternating group A 4 , the Klein group, and the subgroups
{e, (12)(34), (13)(24), (14)(23), (12), (34), (1423), (1324)}
{e, (12)(34), (13)(24), (14)(23), (13), (24), (1234), (1432)}
{e, (12)(34), (13)(24), (14) (23), (23), (14), (1342), (1243)}
The Klein group and A4 are normal subgroups of S4'
4.4.8. (1) The subgroups which contain the Klein group (see 4.3.34)  and
(2) the four cyclic subgroups generated by 3-cycles.
4.4.9. (Xl, yO)], (Xl, y), (X O , y2)], (x, y), (x 6 , y)]
[(Xl, y), (Xl, y)], [(x 7 , y), (x 2 , y)], [(Xl, y), (x 4 , y)]
4.5.2. Each commutator is equal to the identity.
4.5.6. (Xl' X 2 ) = (132), (Xl' X3) = (142), (Xl' Y) = (12)(34), (X2, Xl) = (123),
(X3' Xl) = (124), (y, Xl) = (12)(34).
4.5.7. -1 and 1.
( 7 4 ) ( 5 -3 ) ( 1 2 )
4.5.8. (x, y) = -2 -1 ,(y, z) = -3 2 ' (z, x) = 2 5

Answers
219
4.5.9. (u, {I) =
1 -1 -1
1
o 1 -2
o 0 1
({I, w) =
1
-- -1 0
2
3
- - -1 -1
2
821
1. 0 0
3
(w, u) = 0 1 0
o 0 3
.
4.5.10. (x, y) = (a, b, c)
4.5.14. The alternating group An.
4.5.21. The group of all parallel translations.
4.5.22. The subgroups of matrices whose determinant is 1.
4.6.12. Cyclic groups of prime order.
-
4.6.15. For n = 1,2, 3,4.
4.6.16. Yes.
4.6.17. Yes.
4.7.2. 0 for the identity group, 1 for every other abelian group.
4.7.3. (1) Sn is not nilpotent for n > 2, the nil potency class for S 1 is 0, the
nilpotency class for S 2 is 1 ; (2) is nilpotent of class 2; (3) is nilpotent of class
2; (4) is not nilpotent; and (5) is nilpotent of class 2.
4.7.6. Lower central series: Ho = H, HI = {(8,1), (8, -1), (8, k), (8, -k)},
H 2 = {(8,1), (8, -I)}, H3 = e. Upper central series: Zo = e, ZI = {(8,1),
(8, -I)}, Z2 = {(8, 1), (8, -1), (8, k), (8, - k)}, Z3 = H.
4.7.11. 2.
4.7.15. (1) and (3) are not nilpotent; (2) has nilpotency class 2; and (4) has
nilpotency class 3.
4.7.17. The set {k 1 ,k 2 ,k 3 ,...} must have a maximum. This number is
the nil potency class for H.
4.7.20. If and only if the set {k 1 , k 2 , k 3 ,. . .} has a maximum.
4.8.1. 3 is the only automorphism.
4.8.2. 1 and 2 are automorphisms.
4.8.4. 3 and s are automorphisms.
4.8.5. The group of automorphisms is isomorphic to the symmetric group S 3'
4.8.6. A cyclic group of order 2.
4.8.11. (1) Isomorphic to the Klein group (see 3.3.18); and (2) a cyclic group
of order 6.

220
Answers
4.8.14. Letting U3 = (13), U 4 = (23), U 5 = (123),
t = ( e lit U 2 1l:J llt II  ) t = ( e llt U 2 U a II i U 5 )
U t ' fl.)
e U 1 II G lltl lla ll  e lla ll';l lI1 llt 115
4.8.15. t -1 = e, t -1 = (j, - j)(k, - k), and t j = (i, - i)(k, - k).
4.8.16. Abelian groups.
4.8.17. 4.
4.8.22. 24.
4.8.23. q>(x + yi) = (xa + ye) + (xb + yd)i, where (a, b, c, d) is any 4-tuple
of integers satisfying ad - be = + 1.
4.8.27. Letting U 1 = (12)(34), U z = (13)(24), and U3 = (14)(23),
- - - ( e U I 112 U a ) - ( e llt ll';l U a )
t (12) = t l31) = t(142a) = t t (132) =
e U I Ua U 2 e U 2 lla U t
4.8.28. 24.
4.9.4. G 1 is transitive; M 1 = {I, 2, 3, 4} and M 2 = {5, 6} are intransitive
systems of G 2 ; M 3 = {I, 2, 3, 4} and M 4 = {5, 6, 7} are intransitive systems
of G 3 .
4.9.14. M 1 = {a, e}, M 2 = {b, d}.
4.9.15. (1) A4 and S4 are primitive; and (2) the cyclic subgroups of order 4,
the Klein group K, and all subgroups of order 8 containing K are imprimitive
(see 4.3.34).
4.9.17. When n is a prime number.
4.9.18. (1) and (2) are transitive, and (1) is primitive.
4.9.25. The set H in the preceding problem.
4.9.31. (1) is doubly transitive.
4.9.32. n  4.
4.9.33. 2.
CHAPTER 5
5.1.1. The relations are t 1 = t 2 , t 1 = t 4 , t 2 = t 4 , t 3 = t 5 .
5.1.2. The relations which are consequences of the others are a 5 = a 3 ,
a7eb5a3bll = a 3 e,bea 3 = eba 2 ,a 3 ca = a 3 e,be = eb.
5.1.6. No.

Answers
221
5.1.15. II! - 1.
5.1.16. (1) II fJ UV fJ II (2) yes.
u u UV UV II
f) Vlt V V VU
llV II UV UV II
VU Vtl V fJ VU
5.1.20. Only when it consists of one element.
5.1.22. (1) a is the only idempotent; and (2) a and arcSbft where r = 0,1 and
s is any natural number.
5.1.23. (2) (1, 1), (1, 5), (2, 5), (3, 5), (5, 5), infinite; and (3) a and acmbft where
m = 1, 2, 3, 4 and n is any nonnegative integer.
5.1.27. (1) The semigroup S is regular; (2) the inverse of arbs is aSb r (all ele-
ments in S different from the identity can be written in the form arbs for some
nonnegative integers r, s); and (3) the only automorphism of S is the identity.
5.3.3. The class containing kq>(k) is the identity, and the inverse of the class
containing k is the class containing q>(k).
5.3.8. (1) 2; (2) infinite; and (3) 4; HI and H 3 are normal subgroups, H 2 is
not.
5.3.9. A free group of rank 1, Le., the infinite cyclic group.
5.3.17. 2 ft .
5.4.3. G 1 .
5.4.5. Infinite.
5.4.6. Infinite.
5.4.7. U2 = Us, U3 = U6, the others are distinct.
5.4.9. 2 ft , where n is the number of elements in K.
5.4.10. (2) xhy.i1 = xhyh if i 1 = i 2 and jl = j2; xhyh = (Xilyh)-1 if i 1 =
i 2 = 0 and m2 = - m 1 or if i 1 = i 2 = 1 and m 2 = m 1 - 2.
5.4.14. There are an infinite number of elements of finite and of infinite
order.
5.4.15. The identity, and the element (X 1 X 2 . . . Xft) and all of its conjugates
have finite order.
5.4.18. (1) 12; (2) {e, (ab)2}; and (3) the group of automorphisms is iso-
morphic to G.
5.4.20. 3.
5.4.23. 5.

222
Answers
5.4.27. If q  1 (mod p) the group is cyclic. If q = 1 (mod p) the group is
noncyclic and is of the form Gq (see 5.2.22).
5.4.28. One group of order 15, two groups of order 14.
5.4.29.
n
1 2 3 4 5 6 7 8 9 10
a (n) 1 1 1 2 1 2 1 5 2 2
5.5.1. Yes.
5.5.2. A free group G is decomposable into a free product if and only if G
is not an infinite cyclic group.
5.5.4. G = G 1 * G 4 .
5.5.6. No.
5.5.8. I t is never finite.
5.5.9. Only the identity.
5.5.14. Yes.
5.5.15. No.
5.5.17. A free product of complete groups is never complete.
5.5.18. The center is equal to e.
5.5.19. No.
5.5.21. Yes.
5.6.1. G is the direct product of four cyclic groups of orders 3, 5, 7, and 13.
5.6.5. The only decomposable group is (4).
5.6.8. No.
5.6.9. One.
5.6.15. Yes.
5.6.16. ml m 2'" m n
5.6.27. When the orders of the G i are distinct.
5.6.32. No.
CHAPTER 6
6.1.2. The subgroup of all roots of the identity.
6.1.12. No.
6.1.14. There is always a unique solution.
6.1.16. Cyclic groups of prime orders.

Answers
223
6.1.17. No.
6.1.19. (1) A cyclic group of order 15; (2) 4; (3) the direct product of the four
infinite cyclic groups; and (4) the direct product of the cyclic group of order
3 and the four infinite cyclic groups.
6.2.2. (1) Cyclic groups of orders 7 and 3; (2) cyclic groups of orders 5, 3, and
2; (3) cyclic groups of orders 11, 7, 3, and 2; and (4) a cyclic group of order
101.
6.2.3. (1) 4; (2) 2; (3) 2; and (4) 2.
6.2.4. The direct product of cyclic groups of orders 2, 3, 5, 7, 11, and 13.
6.2.10. (1) 5, 7; (2) 2 3 ; (3) 2, 3; (4) 2 2 , 3, 5; and (5) 2 3 , 2, 2.
6.2.11. A necessary and sufficient condition is for G to be a primary group
relative to p.
6.2.14. pl,..., p".
6.2.15. pl,..., p".
6.2.16. Cyclic groups and abelian groups with invariants p, p.
6.2.17. Cyclic groups of prime order.
6.2.18. No.
6.2.19. p3 - 1.
6.2.20. (1) 10,000; (2) there are subgroups of orders 25, 10, and 40; and (3)
2, 2, 5 and 2 2 , 5.
6.2.21. No.
6.2.22. 11.
6.3.8. (1) Rank 1, set of invariants 3; (2) infinite cyclic group; (3) rank 1,
set of invariants 5; (4) cyclic group of order 11; (5) torsion-free group of
rank 2; (6) rank 1, set of invariants 3; (7) torsion-free group of rank 2; (8)
rank 1, invariants 3, 3, 2; and (9) infinite cyclic group.
6.3.9. (1) 2 ; (2) cyclic group of order 10; (3) the direct product of two infinite
cyclic groups; (4) 4; and (5) [ai], [a].
6.3.10. (1) Yes; (2) yes; (3) no; and (4) no.
6.3.11. No.
6.3.12. 3, 5.
6.3.14. (1) 11 ; (2) 11 ; and (3) yes.
6.3.15. 40.
6.3.16. 31.
6.3.17. (1) Yes; and (2) no.
6.4.9. No.
6.4.10. No.
6.4.11. (IH4).

224
Answers
6.4.14. No.
6.4.16. All three.
6.4.27. Let m = p1 . . . pn, where the Pi are primes listed in increasing order.
Then (1) (00,00,...,00,...), (2) (k1,...,kn,O,...); and (3) (00,k 2 ,...,k n ,
0, . . .).
CHAPTER 7
7.1.1. 1 (Zi) = u h
2 (Zi) = Us
(i= 1,2,3)
there are no other representatiol1 of X in ..II.
7.1.2. No.
7.1.3. It is impossible for lJ. to be extended to a representation of S.
7.1.8. da;)=(  ). 'Ps (a;)=(  ). 'Pd a ;)=(   )
i = 1, 2, 3, 4, 5.
All of the <P j (j = 1, 2, 3) are similar, none are faithful.
7.1.11. (1) Let <p be the representation of the cyclic semigroup S = [a] by
left translations. Then
(a a 2 a 8 a 4 a 5 a 8 a 7 ) (a as a 8 a 4 a 5 a 8 a 7 )
Cf (a) = a2 a3 a4 a5 a6 a 7 a 8 , <p (a 2 ) = a 3 a" a 5 a 8 a 7 a 8 a 4
(a a 2 a 8 a 4 a 5 a 8 a 7 ) (a a 2 a 8 at a 5 a 8 a 7 )
<P (a 8 ) = at a 5 a 8 a 7 a 3 a 4 aI'S , <P (a 4 ) = a 5 a O a 7 as at a':) a 6
( a as a 3 at as a 6 a 7 ) ( a a2 a 3 a 4 as a 6 a 7 )
 (as) = a 8 a 7 a 8 a 4 as a6 a7'  (a 8 ) = a7 a3 a4 as a 8 a 7 a3
( a a2 a 3 a 4 as a 6 a 7 )
<P (a 7 ) = a 3 a 4 a 5 a O a 7 a 3 a4
The representation <p is not faithful. (2) The representation <p 1 of S 1 by left
translations is given by
) ( aJ as ) ( ) ( a1 as )
1 (01 = , <Pt a:J =
a 1 as a 2 a 2
It is faithful. (3) The representation ({J2 of 8 2 by left translations is given by
 (a 1 ) = ( a J a2 a 3 )
a1 a 1 a 1 '
( ) _ ( a 1 a2 a 3 )
<p a 2 - ,
at a 2 a 2
( ) _ ( a 1 a2 a 3 )
<P °3 -
at a 2 a 3
I t is faithful.

Answers
225
7.1.15. oc(x)(y) = X (x, Y EM).
7.1.16. The identity transformation.
7.1.21. No.
7.1.22. No.
7.1.23. Infinite cyclic semigroups and cyclic semigroups of type (1, d).
7.1.24. The identity group only.
7.1.25. Only commutative semigroups.
7.1.26. (1) Left cancellative semigroups; (2) right cancellative semigroups;
and (3) two-sided cancellative semigroups.
7.1.29. Only semilattices.
7.2.4.
( ) _ ( Ht aH, bH, abH ) CD H ( b ) = ( Ht aH, blf, ablf )
cP H a - aH H abH bH ' T bH abl 'J H aH
" , , I, ,
lfJ H is not faithful.
7.2.6. Denote the representations by lfJ 1 , lfJ2 , <P 3' and <p 4' respectively. Then
1) CPt (1) = e, <PI (- 1) = (1, - 1) (i, - i) (j, - j) (k, - k)
<PI (i) = (1, i, - 1, -I) (j, k, -j, - k)
CPI (- i) = (1, -i, - 1, i) (j, - k, - j, k)
cP 1 U) = (1, j, - 1, - j) (i, - k, - i, k)
<PI (- j) = (1, - j, - 1, j) (i, k, -i, - k)
,. (k) = (1, k, - 1, - k) (i, j, -i, - j)
1 (- k) = (1, - k, -1, k) (i, - j, - i, j)
(2) lfJ2(e) is the identity permutation,
ff2 (a) = (e, a) (b, ab) (ba, aba) (bab, (ab)2)
'2 (b) = (e, b) (a, ba) (ab, bab) (aba, (ab)2)
<P2 (ab) = (e, ab, (ab)2, ba) (b, a, aba, bab)
2 (ba) = (e, ba, (ab)2, ab) (a, b, bab, aba)
CP2 (aba) = (e, aba) (a, ab) (b, (ab)2) (bab, ba)
CP2 (bab) = (e, bab) (a, (Ob)2) (b, ba) (ab, aba)
q>f [(ab)2] = (e, (ab)2) (ab, ba) (b, aba) (a, bab)
(3) a noncyclic group G of order six can be written in the form G = [a, bJ,
where a 2 = b 3 = e, a-1ba = b 2 , so that G = {e, a, b, b 2 , ab, ab 2 }; lfJ3(e) is the
identity permutation,
a (a) = (e, a) (b, ab) (b 2 , ab 2 ); 3 (b) = (e, b, b 2 ) (a, ab 2 , ab)
CPa (b 2 ) = (e, b 2 , b) (a, ab, ab 2 ); <fa (ab) = (e, ab) (0, ab 2 ) (a, b 2 )
<fa (ab 2 ) = (e, ab 2 ) (a, b) (ab, b 2 )

226
Answers
(4) if G = {e, a, a 2 , a 3 }, then lp4(e) is the identity permutation,
ff4 (a) = (e, a, a 2 , a 8 ); 4 (a 2 ) = (e, a 2 ) (a, a 8 )
4 (a 3 ) = (e, a 8 , a 2 , a)
7.2.8. The identity subgroup.
7.2.18. No.
7.2.24. The representations by the subgroups {e, a} and {e, b} and the
representation by left translations.
7.2.26. It does not.
7.2.27.
1) I? _ ( Hu (12) Hu (13) Hu (14) H 1 , (23) Hu (34) H J )
'fHj ( -) - (12) Hit H1' (34) HH (23) HH (14) Ht, (13) HJ
(1234) _ ( HH (12) Hit (13) Hit (14) f/ h (23) HH (34) Ht )
Cl( H l - HI, (23) Hu (13) H., (12) HH (34) Hu (14) Ht
2) 'fH (12) = ( H2' (13) H 2 , (14) H 2 , (2) H 2 , (13) (24) Hs, (24) 112 )
2 H, (23) Hs, (24) H2' (13) H2' (13) (24) H, (14) H 2
H2 (1234) =
_ ( H2' (13) H, (14) H2' (23) H2' (24) H;), (13) (24) Hf}, )
- (13) H2' (13) (24) 11 2 , (23) Hs, (14) H2' Hs, (24) HI
7.2.33. For commutative groups.
7.3.4. Yes.
7.3.5. No.
7.3.6. No.
7.3.10. Yes.
7.3.15. No.
7.4.3. An infinite cyclic group.
7.4.5. The identity group.
7.4.7. R(A, B) is isomorphic to the subgroup of all elements in B whose
orders divide pn.
7.4.8. The group of endomorphisms is isomorphic to G.
7.4.11. An infinite cyclic group.
7.4.12. A finitely generated, torsion-free, abelian group of rank r
7.4.13. A cyclic group of order 2.
7.4.14. A finite abelian group with invariants 52, 52, 5, 5, 5.
7.4.15. A finite group with invariants 7 2 , 7.
7.4.16. A cyclic group of order 5.

Answers
227
7.4.18. (1) A cyclic group of order 5  (2) a finite abelian group with invariants
5, 5, 5, 5, 5, 5; 3 2 , 3 2 , 3, 3, 3; and (3) a cyclic group of order 2.
7.4.19. (1) The direct product of two infinite cyclic groups and two cyclic
groups of order 3 ; (2) a cyclic group of order 3 ; and (3) a finite abelian group
with invariants 17, 17.
7.4.20. (1) A finite abelian group with invariants 53, 52, 52, 5 2 ,5,5,5,5,5;
7 2 , 7, 7, 7; 3 3 , 3, 3, 3; 2 3 , 2 2 , 2 2 , 2 2 , 2, 2, 2, 2, 2; (2) a finite abelian group with
invariants 3, 3, 3, 3, 3, 3, 3, 3, 3; 52, 5, 5, 5, 5, 5, 5, 5, 5; 7, 7, 7, 7; (3) a finitely
generated abelian group of rank 4 with invariants 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,
5,5,5,5,5;7 3 ,7 3 ,7 3 ,7,7,7, 7, 7; and (4) a direct product of49 infinite cyclic
grou ps.
7.4.26. (1) No; (2) no.
7.4.28. (1) A necessary and sufficient condition is for A to be a finite cyclic
group; and (2) a necessary and sufficient condition is for A to be an infinite
cyclic group. I
7.4.29. No.
7.4.32. No.
7.4.34. A necessary and sufficient condition is for A to be either an infinite
cyclic group or a finite group all of whose primary components are cyclic
grou ps.
7.5.1. (1) X(x) = 2 cos Llx + 1; (2) x(e) = 3, x(at) = 0, x(af) = 0, x(a2) = 1,
x(ata2) = 1, x(afa2) = 1; and (3) x(a k ) = 2, (k = 0, + 1, + 2,. . .).
7 .5.5. Yes; for example, a cyclic group of prime order.
7.5.7. There are n characters Xi (i = 1, 2, . . . 'I n) of the group [a]. These can
be written in the form
Xi( a k ) = t ki (i, k = 1,..., n)
where t is an nth root of unity.
7.5.9. (1) A cyclic group of order pk; (2) a finite abelian group with invariants
3, 3 ; and (3) a cyclic group of order 10.
7.5.11. The group of characters is isomorphic to the multiplicative group of
all complex numbers modulo 1.
7.5.12. The group of characters is isomorphic to the direct product of r
groups, each of which is isomorphic to the multiplicative group of all complex
numbers modulo 1.
CHAPTER 8
8.1.1. M t ,M 2 ,M 3 ,M 4 , and M7 are metrics.
8.1.2. (2) Yes.

228
Answers
8.1.3. Only <p 4'
8.1.6. a > 0, b = O.
8.1.7. Only P3'
8.1.11. A necessary and sufficient condition for f" to converge to f in the
sense of the metric on C[a, bJ is for f" to converge uniformly to f
8.1.14. (2), (3), and (4).
8.1.17 . Yes.
8.1.18. None.
8.2.3. (1) and (4).
8.2.4. The only continuous transformations of R are the continuous func-
tions on R. The only bicontinuous transformations are the monotonic
increasing or monotonic decreasing continuous functions whose domains
are all of R.
8.2.11. Yes.
8.2.20. Yes.
8.2.22. No.
8.3.2. oc I and (X2'
8.3.4. (1) Open, (2) open and closed, (3) open and closed, (4) closed, and
(5) closed.
8.3.8. Yes.
8.3.9. All real numbers.
8.3.16. Yes.
8.3.18. Every topological space has a basis.
8.3.21. Yes.
8.3.22. (1) Yes, (3) yes.
8.3.23. Yes.
8.3.24. All but F 5'
8.3.25. Yes.
8.3.31. Yes.
8.3.32. No.
8.4.1. G 1 and G 2 .
8.4.2. Yes.
8.4.3. Yes.
8.4.4. Yes.
8.4.6. The sets Fa, aF, and F- I are closed, UP, PU, and U- I are open.
8.4.9. G forms a topological group. The topologies are not equal.
8.4.1 o. Yes.
8.4.11. Hand N are both normal subgroups of G ; P is closed if and only if oc
is rational.

Answers
229
8.4.12. Yes.
8.4.14. No.
8.4.15. All algebraic subgroups are topological subgroups. Every algebraic
isomorphism is continuous. Not every algebraic isomorphism is a topological
isomorphism.
8.4.16. Yes.
8.4.18. (1) Not a subgroup  (2) subgroup, not normal  (3) subgroup, not
normal  and (4) normal subgroup.
8.4.21. qJ is continuous and open. A necessary and sufficient condition is
N = e.
8.4.26. Yes.
8.4.27. A and B are closed, but A + B is not.
8.4.32. Yes.
8.4.33. (1) Yes, (2) no.
8.4.34. Yes.
8.4.35. Yes.
8.4.38. Yes.
8.5.3. (1) The ordering is two-sided compatible, the positive part consists of
all z = a + bi such that a > 0 or a = 0 and b  0, the negative part consists
of all z = a + bi such that a < 0 or a = 0 and b  O. (2) The ordering is
two-sided compatible, C+ consists of all z such that arg z E [C(, {J], C- con-
sists of all z such that 1C + arg z E [C(, (J]. (3) The ordering is two-sided com-
patible, C + consists of all z = a + bi such that a  0 and b  0, C - consists
of all z such that a  0 and b  0 and (4) The ordering is not two-sided
compatible.
8.5.8. ! = n P  .
a
8.5.10. P = !-l.
8.5.12. The diagonal (equality) is the only two-sided compatible ordering
on finite cyclic groups. Let G = [x] be an infinite cyclic group. For i = 1,
2, . . . , define the relations Pi on G by x' "" Xk{Pi) if and only if k - 1  0 and
is divisible by i. Then all of the Pi and Pi- 1 are two-sided compatible orderings
on G, and every such ordering must be of the form Pi' Pi- 1, or the diagonal.
8.5.13. .f(x) > .f{x) > .1'1 (x) > .f{x).
8.5.15. The ordering is two-sided compatible. It is linear if and only if no
more than one of the factors G a is different from the identity.
8.5.16. It is a two-sided compatible ordering.
8.5.17. The cyclic group [x] must be infinite.
8.5.18. When x has infinite order.
8.5.24. All of the groups are directed.

Appendix
MULTIPLICATION TABLE FOR S4
In this section we give the multiplication table for the symmetric group on
four elements S4 and list its proper subgroups, 28 in all. We use the following
notation:
e = identity transformation
VI = (12)(34)
V 2 = (13)(24)
V 3 = (1)(23)
A = (234)
B = (243)
C = (123)
D = (124)
E = (132)
F = (134)
G = (142)
H = (143)
a = (34)
b = (23y
c = (24)
d = (12)
f = (13)
g = (14)
h = (1234)
j = (1243)
k = (1342)
l = (1324)
m = (1432)
n = (1423)
The subgroups of S4, listed according to their order, are given below.
Order 12: A4 = {e, VI' V 2 , V 3 ,A,B,C,D,E,F,G,H}
Order 8 : {e, VI' V 2 , V 3 , a, d, I, n }, {e, VI' V 2 , V 3 , c,.t: h, m} {e, VI' V 2 , V 3 , b,
\:1'
g,j, k}
Order 6: {e, a, b, c, A, B}, {e, b, d,.t: C, E} {e, c, d, g, D, G}, {e, a,.t: g, F, H}
"
Order 4: {e, VI' V 2 , V 3 }, {e, a, d, VI}' {e, c,.t: V 2 }, {e, g, b, V 3 }, {e, l, n, VI}'
,...,-- { e, h, m, V 2 }, {e, j, k, V 3 }
Order 3: {e,A,B}, {e,C,E}, {e,D,G}, {e,F,H}
Order 2: {e, VI}' {e, V 2 }, {e, V 3 }, {e, a}, {e, b}, {e, c}, {e, d}, {e,f}, {e, g}
231

232
Appendix
e V 1 V 2 V 3 ABCDEFGHa b cd fg h j k 1 mn
e e VI V 2 V 3 ABC D E F 'G '11,1 a b c d f g h j k 1 m n
VI VI e V 3 V 2 E G FHA C B iJ ,! d k m a h j f g b n c
V 2 V 2 V 3 e VI H C BED G .j. A n j f c k m b g d h a
V 3 V 3 V 2 VI e D F G A H B C 1' g h n m b c k j a f d
A A D E H B e VI C F Yz V 3 G, cab h k n j d 1 f g m
B B C F G e A D V t V 2 E H  3'1 b c a j 1 m d h f k n g
C C B G F V:z H E V 3 e D A VI: j f n b d him c g a k
D 1) A H E F V 3 V z G. B V t > e', ';J h I g c j d k n a m b f
E E HAD G VI e' 'F C F3 VI 8} m d k f big a n h j c
F F G B C V 3 D A V 2 VI H:E e' g h I k a f n c m j d b
G G F C B V t E H e V 3 A Ii Y:z; k m d g n c a f h b I j
H H E D A C V 2 V 3 . B (j e, Jl t ' ,;'1 f n j m gab 1 d c k h
a a din b .c h 'Wj'>it'N'j,rti=gV e A B VI F H C D E V 2 G V 3
b b j k g cad h fin m B e ACE V 3 D VI V 2 F H G
c c h f m a b j d I kg nAB e D V 2 G VIC FE V 3 H
ddanl kmfgbhcj V I EGeCDFHAV 3 BV 2
f f m c h n j bid g k a H C V 2 E e F V 3 B G D VI A
ggkjbhl ncmadfFV 3 DGHeAV 2 V I BEC
h h c m fig k n a j b d D F V 3 A V 1 C V 2 G B He E
j j b g k f n 1 m cd a h C V 2 H B D VIE V 3 e G A F
k kg b j m d a f h n I c G VtE F A V 2 He V 3 C D B
1 I n a d g h c k j mfb V 3 DF V 2 BEGAHV I Ce
m m f h c d kg an b j 1 E G VIH V 3 B e F C A V 2 D
n n I d a j f m b g c h k V 2 H C VJG A BED e F VI

BffiLIOGRAPHy t
1. The books in this section are devoted entirely to group theory and
are suitable for an initial acquaintance.
1. Alexandrov, P. S., An Introduction to the Theory of Groups (translated
by Hazel Perfect and G. M. Peterson), Hafner, 1959.
2. Baumgartner, L., Gruppentheorie, de Gruyter, 1921.
*3. Baumslag, B. and Chandler, G., Theory and Problems of Group Theory,
Schaum's Outline Series, 1968.
*4. Burrow, M., Representation Theory of Finite Groups, Academic Press,
1965.
*5. Dixon, J., Problems in Group Theory, Blaisdell, 1967.
*6. Fuchs, L., Abelian Groups, Pergamon, 1960.
*7. Gorenstein, D., Finite Groups, Harper and Row, 1968.
8. Hall, Jr., M., The Theory of Groups, Macmillan, 1959.
*9. Huppert, B., Endliche Gruppen, 2 vols., Springer Verlag, 1967.
*10. Kaplansky, II, Infinite Abelian Groups, University of Michigan Press,
1954.
11. Kurosh, A. G., The Theory of Groups, 2 vols. (translated by K. A.
Hirsch), Chelsea, 1960.
12. Kurosh, A. G., Lectures on General Algebra (translated by K. A.
Hirsch), Chelsea, 1963.
*13. Lederman, W., Introduction to the Theory of Finite Groups, Oliver and
Boyd, 1957.
* 14. Rotman, J. J., The Theory of Groups : An Introduction, Allyn and Bacon,
1965.
15. Schmidt, 0., Abstract Theory of Groups (translated by Fred Holling
and J. B. Roberts), Freeman" 1966.
* 16. Scorza, G., Gruppi Astratti, Rome, 1942.
*17. Scott, W., Group Theory, Prentice-Hall, 1964.
*18. Sono, S., Group Theory, Tokyo, 1928 (in Japanese).
* 19. Wielandt, H., Finite Permutation Groups (translated by R. Bercov),
Academic Press, 1964.
t The references preceded by an asterisk have been added by the translator.
233

234
Bibliography
2. The books in this section do not deal entirely with group theory
but have chapters which introduce the reader to the basic elements of group
theory.
*20. Albert, A., Modern Higher Algebra, University of Chicago Press, 1937
*21. Barnes, W., Introduction to Abstract Algebra, D. C. Heath, 1963.
*22. Birkhoff, G. and MacLane, S., A Survey of Modern Algebra, Mac-
millan, 1953.
23. Bourbaki, N., Algebre (Structures Algebriques. Algebre Lineaire.
Algebre Multilineaire), Hermann, 1964.
24. Chebotarev, N. G., Basic Galois Theory, GTTI, 1934 (in Russian).
25. Chebotarev, N. G., Lie Group Theory, Gostekhizdat, 1940 (in Russian).
*26. Herstein, I., Topics in Algebra, Blaisdell, 1964.
*27. Jacobson, H., Lectures in Abstract Algebra, 3 vols. (Basic Concepts,
Linear Algebra, Theory of Fields and Galois Theory), Van Nostrand,
vol. I 1951, vol. II 1953, vol. III 1964.
28. Kurosh, A. G., Higher Algebra, Fizmatigiz, 1962 (in Russian).
*29. Lang, S., Algebra, Addison-Wesley, 1965.
30. Lyubarski, G., Group Theory and its Applications to Physics, Gostekhiz-
dat, 1958 (in Russian).
*31. McCoy, N. H., Introduction to Modern Algebra, Allyn and Bacon,
1960.
32. Okunev, L., Foundations of Modern Algebra, Uchpedgiz, 1941 (in
Russian).
33. Postnikov, M. M., Galois Theory, Fizmatgiz, 1963 (in Russian).
34. Proskuryakov, I. V., A Collection of Problems on Linear Algebra,
Gostekhizdat, 1957 (in Russian).
35. Smirnov, V. I., A Course of Higher Mathematics (translated by D. E.
Brown), Addison-Wesley, 1964.
*36. Van der Waerden, B. L., Modern Algebra (translated by F. Blum),
Ungar, 1953.
*37. Zariski.. O. and Samuel, P., Commutative Algebra, vol. I, Van Nostrand
1958.
3. The present book also includes notions which do not belong properly
to group theory. The references listed below are suitable for an introduction
to these notions.
38. Alexandrov, P. S., Introduction to the General Theory of Sets and
Functions, Gostekhizdat, 1948 (in Russian).
39. Birkhoff, G., Lattice Theory, Amer. Math. Soc. Colloq. Publ., vol. 25,
1948.
40. Bourbaki, N., Theorie des Ensembles, Hermann; 1964.

Bibliography
235
41. Bourbaki, N., Topologie Generale (Structures Topologiques), Hermann,
1965.
*42. Bruck, R. H., A Survey 0.( Binary Systems, Springer, 1958.
*43. Clifford, A. H. and Preston, G. B., The Algebraic Theory of Semigroups,
Math. Surveys No.7, Amer. Math. Soc. vol. I 1961, vol. II 1967.
44. Dieudonne, J., Foundations of Modern Analysis, Academic Press, 1960.
*45. Halmos, P. R., Naive Set Theory, Van Nostrand, 1960.
*46. Hoffman, K. and Kunze, R., Linear Algebra, Prentice-Hall, 1961.
*47. Husain, T., Introduction to Topological Groups, Saunders, 1966.
*48. Kamke, E., Theory of Sets (translated by F. Bagemihl), Dover, 1950.
*49. Kelley, J., General Topology, Van Nostrand, 1955.
50. Kolmogorov, A. N. and Fomin, S. V., Elements o.f the Theory of
Functions and Functional Analysis, vol. I (translated by Leo F.
Boron), Graylock, 1957.
51. Lyapin, E. S., Semigroups, 2nd edition (translated by Amer. Math.
Soc.), 1968.
52. Maltsev, A. I., Foundations of Linear Algebra (translated by Thomas
Craig Brown), Freeman, 1963.
53. Natanson, I. P., Theory of Functions of a Real Variable (translated by
Leo F. Boron), Ungar" 1955.
54. Pontriagin, L. S., Topological Groups (2nd edition translated by Arlen
Brown), Gordon and Breach, 1966.
55. Shilov, G. E., Mathematical Analysis, A Special Course (translated by
J. D. Davis), Pergamon, 1965.
*56. Taylor, A. E., Introduction to Functional Analysis, Wiley, 1964.
57. Vulikh, B. Z., Introduction to Functional Analvsis for Scientists and
Technologists (translated by I. N. Sneddon), Addison-Wesley, 1963.

,
INDEX
Abelian group, 6.1
Algebraic group, 2.5, 8.4
Algebraic number, 1.2
Algebraic operation, 2.1
Alternating group, 3.3
Antisymmetric relation, 1.3
Aperiodic group, 6.1
Associative, 2.2
Automorphism, 3.4, 4.8
inner automorphism, 4.8
Compatible relation, 2.4
Complete group, 5.5
Complete metric space, 8.1
Complete ordering, 1.3
Complete system of representations, 7.1
Completely characteristic subgroup, 3.4
Completely reducible representation, 7.3
Component, 5.6
Composition series, 4.6
Congruence, 2.4
Conjugate class, 4.2
Conjugate elements, 4.2
Conjugate relation, 1.3
Connected set, 3.4
Consequence, 5.1
Continuous group, 8.4
Continuous mapping, 8.3
Continuous transformation, 8.2
Converge, 8.1
Corollary, immediate, 5.1
Corresponding relation, 5.3
Coset, 4.1
Countable set, 1.2
Crystal groups, 3.5
Cut, 1.4
Cycle, 3.3
Cyclic group, 2.6
Cyclic semigroup, 2.5
Band, 7.1
Basis, 8.3
,
Bicontinuous transformation, 8.2
Binary relation, 1.3
Bounded metric, 8.1
Bounded set, 8.1
Cancellative, 2.2
Canonical homomorphism, 2.4
Cardinality, 1.2
Cartesian product, 1.1
Center, 2.6
Chain, 1.3
Character, 7.5
Characteristic, 4.8, 6.4
Class, 1.3
Closed operation, 2.2
Closed set, 2.3, 8.3
Closure, 8.3
Closure operator, 8.3
Commutative, 2.2
Commutator, 4.5
Commutator series, 4.6
Commutator subgroup, 4.5
Compact space, 8.1
Decomposable group, 5.6
Decomposable representation, 7.3
Decomposition, 4.1, 4.1, 5.6
Defining relation, 5.1
Defining set, 5.1, 5.1, 6.1
Derived group, 4.5
Derived series, 4.6
Direct consequence, 5.1
237

238
Direct factor, 5.6
Direct product, 5.6, 8.3, 8.4
Direct sum, 7.3
Directed group, 8.5
Discrete group, 8.4
Discrete topology, 8.3
Disjoint cycles, 3.3
Disjoint sets, 1.1
Distance, 8.1
Distance function, 8.1
Divisible group, 6.4
Disjoint sets, 1.1
Distance, 8.1
Distance function, 8.1
Divisible group, 6.4
Double coset, 4.1
Doubly transitive group, 4.9
Element, 1.1
Empty set, 1.1
Endomorphism, 3.4
Epimorphism, 2.4
Equivalence, 1.3
Equivalent characteristics, 6.4
Equivalent metrics, 8.1
Equivalent representations, 7.3
Equivalent sets, 1.2
Even permutation, 3.3
Extension, I .2, 5.6
Factor group, 4.3, 8.4
Factors, 2.1
Factor-set, 2.4
Faithful representation, 7.1
Federov group, 3.5
Finitely generated group, 2.6
Fixed point, 1.2
Four-group, 3.3
Free generating set, 5.1
Free group, 5.3
Free product, 5.5
Free semigroup, 5.1
Fundamental sequence, 8.1
Generating set, 2.5, 2.6
Group, 2.5, 8.4
Group of characters, 7.5
Group of continuous transformations, 8.2
Index
Group of endomorphisms, 7.4
Group of homomorphisms, 7.4
Group of isometries, 3.5
Group of representations, 7.4
Group of transformations, 3.2
Groupoid, 2.1
Height, 6.4
Homeomorphic spaces, 8.3
Homeomorphism, 8.3
Homomorphism, 2.4, 8.4
Ideal, 2.3, 2.4
Idempotent, 2.2.4
Identity element, 2.2
Identity representation, 7.3
Identity transformation, 3.1
Image, I .2, 3.1
Immediate corollary, 5.1
Imprimitive group, 4.9
Imprimitive series, 4.9
Imprimitive system, 4.9
Improper subset, 1.1
Independent cycles, 3.3
Index, 4.1
Induced transformation, 4.8
Infinite height, 6.4
Infinite rank, 6.1
Inner automorphism, 4.8
Intersection, 1.1
Intransitive group, 4.9
Intransitive system, 4.9
Invariants, 6.2
Inverse element, 2.2
Inverse image, 1.2
Inverse mapping, 1.2
Inverse relation, 1.3
Inverse transformation, 3.2
Inverses, 2.5
Invertible endomorphism, 3.4
Invertible operation, 2.2
Invertible transformation, 3.2
Irreducible generating set, 2.5, 2.6
Irreducible representation, 7.3
Isometry, 3.5, 8.2
Isomorphic decompositions, 5.6
Isomorphic sets, 2.1
Isomorphism, 2.4, 8.4

Index
239
Kernel, 4.3
Klein group, 3.3
Order, 2.6, 2.6
Ordered group. 8.5
Ordering, 1.3
lagrange's Theorem, 4.1
Left translation, 7.1
Length of cycle, 3.3
Limit point, 8.1 /
Linear ordering, 1.3
Linearly dependent set, 6.1
Linearly independent set, 6.1
Linearly ordered group, 8.5
Lower central series, 4.7
Lowest layer, 6.1
p-Component, 6.1
p-Primary group, 6.1
Partial identity. 3.6
Partial transformation, 3.6
Partition, 1.1
Periodic group, 2.6, 6.1
Periodic part, 6.1
Permutation, 3.1, 3.3
Point, 8.1
Positive element, 8.5
Positive part, 8.5
Precedes, 1.3
Primary component, 6.1
Primary group, 6.1
Primitive group, 4.9
Product, 2.1, 8.1
Proper homomorphism, 6.1
Proper subgroup, 2.6
Proper subset, 1.1
Pure subgroup, 6.4
Mapping, 1.2
Matrix representation, 7.3
Maximal element, 1.3
Maximal periodic subgroup, 6.1
Metric, 8.1
Metric space, 8.1
Minimal element, 1.3
Minimal normal subgroup, 4.3
Mixed group, 6.1
Monogenic semigroup, 2.5
Monomorphism, 2.4
Multiplicative set, 2.3
Multiply transitive group, 4.9
Quotient-set, 2.4
n-Angular dihedron, 3.5
Natural homomorphism, 2.4
Natural metric, 8.2
Negative element, 8.5
Jgative part, 8.5
J4;ilpotency class, 4.7
Nilpotent group, 4.7
Normal complex, 2.5
Normal series, 4.6
Normal subgroup, 2.6, 4.3, 8.4
Normal subset, 2.5
Normalizer, 4.2
Rank, 3.1, 5.3, 6.1
Reaches, 4.7
Real line, 8.3
Reducible representation, 7.3
Reflexive relation, 1.3
Regular conjugates, 2.5
Regular element, 2.5
Regular permutation, 3.3
Regular relation, 2.4
Regular semigroup, 2.5
Relation, 5.1
Representation, 7.1
Representation by left translations, 7.1
Representation of G by H, 7.2
Result, 2.1
Odd permutation, 3.3
One-to-one mapping, 1.2
Open homomorphism, 8.4
Open set, 8.3
Operation, 2.1
Operative, 2.1
Self-coincidence, 3.5
Semigroup, 2.5
Semigroup of transformations, 3.1
Semilattice, 7.1

240
Index
Separable metric space, 8.1
Serving subgroup, 6.4
Set, 1.1
Set of representatives, 4.1, 4.1
Solvable group, 4.6
Stabilized series, 4.6, 4.7
Stable relation, 2.4
Subdirect product, 4.9
Subgroup, 2.6, 8.4
Subsemigroup,2.5
Subspace,8.1
Sylow subgroup, 4.4
Symmetric group, 3.3
Symmetric relation, 1.3
System of elements, 1.1
Transitive closure, 1.3
Transitive group, 4.9, 7.2, 8.4
Transitive relation, 1.3
Transposition, 3.3
Type, 6.4
Type of group, 6.4
Type of semigroup, 2.5
Type poo, 6.4
Union, 1.1
Universal binary relation, 1.3
Universally minimal element, 1.3
Upper central series, 4.7
Value of a word, 2.5, 5.1
Topological group, 8.4
Topological isomorphism, 8.4
Topological space, 8.3
Torsion group, 6.1
Torsion free group, 6.1
Transcendental number, 1.2
Transformation, 3.1
Word, 2.5, 5.4
Word problem, 5.4
Zero element, 2.2, 8.5