Group Theory is a vast subject and, in this Introduction (as well as in theearlier editions), I have tried to select important and representative theoremsand to organize them in a coherent way. Proofs must be clear, and examplesshould illustrate theorems and also explain the presence of restrictive hypo-theses. I also believe that some history should be given so that one canunderstand the origin of problems and the context in which the subject developed.

Preface to the Fourth Edition

From Preface to the Third Edition

To the Reader

CHAPTER 1 Groups and Homomorphisms

 Permutations

 Cycles

 Factorization into Disjoint Cycles

 Even and Odd Permutations

 Semigroups

 Groups

 Homomorphisms

CHAPTER 2 The Isomorphism Theorems

 Subgroups

 Lagrange's Theorem

 Cycic Groups

 Normal Subgroups

 Quotient Groups

 The Isomorphism Theorems

 Correspondence Theorem

 Direct Products

CHAPTER 3 Symmetric Groups and G-Sets

 Conjugates

 Symmetric Groups

 The Simplicity of A.

 Some Representation Theorems

 G-Sets

 Counting Orbits

 Some Geometry

CHAPTER 4 The Sylow Theorems

 p-Groups

 The Sylow Theorems

 Groups of Small Order

CHAPTER 5 Normal Series

 Some Galois Theory

 The Jordan-Ho1der Theorem

 Solvable Groups

 Two Theorems of P. Hall

 Central Series and Nilpotent Groups

 p-Groups

CHAPTER 6 Finite Direct Products

 The Basis Theorem

 The Fundamental Theorem of Finite Abelian Groups

 Canonical Forms; Existence

 Canonical Forms; Uniqueness

 The KrulI-Schmidt Theorem

 Operator Groups

CHAPTER 7 Extensions and Cohomology

 The Extension Problem

 Automorphism Groups

 Semidirect Products

 Wreath Products

 Factor Sets

 Theorems of Schur-Zassenhaus and GaschiJtz

 Transfer and Burnside's Theorem

 Projective Representations and the Schur Multiplier

 Derivations

CHAPTER 8 Some Simple Linear Groups

 Finite Fields

 The General Linear Group

 PSL(2, K)

 PSL(m, K)

 Classical Groups

CHAPTER 9 Permutations and the Mathieu Groups

 Multiple Transitivity

 Primitive G-Sets

 Simplicity Criteria

 Atline Geometry

 Projeetive Geometry

 Sharply 3-Transitive Groups

 Mathieu Groups

 Steiner Systems

CHAPTER 10 Abelian Groups

 Basics

 Free Abelian Groups

 Finitely Generated Abelian Groups

 Divisible and Reduced Groups

 Torsion Groups

 Subgroups of

 Character Groups

CHAPTER 11 Free Groups and Free Products

 Generators and Relations

 Semigroup Interlude

 Coset Enumeration

 Presentations and the Schur Multiplier

 Fundamental Groups of Complexes

 Tietze's Theorem

 Covering Complexes

 The Nielsen Schreier Theorem

 Free Products

 The Kurosh Theorem

 The van Kampen Theorem

 Amalgams

 HNN Extensions

CHAPTER 12 The Word Problem

 Introduction

 Turing Machines

 The Markov-Post Theorem

 The Novikov-Boone-Britton Theorem: Sufficiency of Boone's Lemma

 Cancellation Diagrams

 The Novikov-Boone-Britton Theorem: Necessity of Boone's Lemma

 The Higman Imbedding Theorem

 Some Applications

Epilogue

 APPENDIX Ⅰ Some Major Algebraic Systems

 APPENDIX Ⅱ Equivalence Relations and Equivalence Classes

 APPENDIX Ⅲ Functions

 APPENDIX Ⅳ Zorn's Lemma

 APPENDIX Ⅴ Countability

 APPENDIX Ⅵ Commutative Rings

Bibliography

Notation

Index