Exercises in Group Theory

244 Pages·1972·10.24 MB·English

by  E. S. Lyapin

Exercises in
Group Theory
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
W olters-N oordhoff 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
ISBN-13:978-1-4613-4591-6 e-ISBN -13:978-1-4613-4589-3
DOl: 10.1007/978-1-4613-4589-3
© 1972 Plenum Press, N ew York
Softcover reprint of the hardcover 1st edition 1972
A Division of Plenum Publishing Corporation
227 West 17th Street, New York, N. Y. 10011
and Wolters-Noordhoff Publishing, Groningen
All rights reserved
No part of this publication may be reproduced in any form without
written permission from the publisher
TRANSLATOR'S PREFACE
The present book is a translation of E. S. Lyapin, A. Va. Aizenshtat, and
M. M. Lesokhin's Uprazhneniya po teorii grupp. I have departed somewhat
from the original text in the following respects.
I) 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 order to conform
with present English usage (e.g., "inverses" instead of "regular conjugates").
3) I have corrected a number of misprints 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 andcounterex::Imples
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
semmars.
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
I. 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
I. General Properties of the Composition of Transformations ... 51
2. hi.vertible 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
5. Commutators and the Commutator Subgroup ............. , 91
6. Solvable Groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 94
7. 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 Products of Groups ................................. 125
6. The Direct Product of Groups ............................ 127
Chapter 6
Abelian Groups
I. Elementary Properties of Abelian Groups .................. 133
2. Finite Abelian Groups ................................... 136
3. 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
1. Metric Spaces .......................................... 161
2. Groups of Continuous Transformations of a Metric Space .... 166
3. Topological Spaces ...................................... 170
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
Chapter 1 ................................................ 201
Chapter 2 ................................................ 204
Chapter 3 ................................................ 209
Chapter 4 ................................................ 216
Chapter 5 ................................................ 220
Chapter 6 ................................................ 222
Chapter 7 ................................................ 224
Chapter 8 ................................................ 227
Appendix. Multiplication Table for S4 ........................... 231
Bibliography ................................................ 233
Index ...................................................... 237