Exercises in Group Theory PDF
Preview Exercises in Group Theory
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
The list of books you might like
