Exercises in Group Theory

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.

Table of Contents:
1 Sets.- 1. Basic Concepts.- 2. Mappings of Sets.- 3. Binary Relations.- 4. Multiplication of Binary Relations.- 2 Algebraic Operations of a General Type.- 1. The Concept of an Algebraic Operation.- 2. Basic Properties of Operations.- 3. Multiplication of Subsets of a Multiplicative Set.- 4. Homomorphisms.- 5. Semigroups.- 6. Elementary Concepts of the Theory of Groups.- 3 Compositions of Transformations.- 1. General Properties of the Composition of Transformations.- 2. Invertible Transformations.- 3. Invertible Transformations of Finite Sets.- 4. Endomorphisms.- 5. Groups of Isometries.- 6. Partial Transformations.- 4 Groups and Their Subgroups.- 1. Decomposition of a Group by a Subgroup.- 2. Conjugate Classes.- 3. Normal Subgroups and Factor Groups.- 4. Subgroups of Finite Groups.- 5. Commutators and the Commutator Subgroup.- 6. Solvable Groups.- 7. Nilpotent Groups.- 8. Automorphisms of Groups.- 9. Transitive Groups of Transformations.- 5 Defining Sets of Relations.- 1. Defining Sets of Relations on Semigroups.- 2. Defining Sets of Relations on Groups.- 3. Free Groups.- 4. Groups Defined by Sets of Relations.- 5. Free Products of Groups.- 6. The Direct Product of Groups.- 6 Abelian Groups.- 1. Elementary Properties of Abelian Groups.- 2. Finite Abelian Groups.- 3. Finitely Generated Abelian Groups.- 4. Infinite Abelian Groups.- 7 Group Representations.- 1. Representations of a General Type.- 2. Representations of Groups by Transformations.- 3. Representations of Groups by Matrices.- 4. Groups of Homomorphisms of Abelian Groups.- 5. Characters of Groups.- 8 Topological and Ordered Groups.- 1. Metric Spaces.- 2. Groups of Continuous Transformations of a Metric Space.- 3. Topological Spaces.- 4. Topological Groups.- 5. Ordered Groups.- Hints.- 1.- 2.- 3.- 4.- 5.- 6.- 7.- 8.- Answers.- 1.- 2.- 3.- 4.- 5.- 6.- 7.- 8.