群论导论

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

Lagranges 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 Burnsides 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

Tietzes 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 Boones

Lemma

Cancellation Diagrams

The Novikov-Boone-Britton Theorem： Necessity of Boones

Lemma

The Higman Imbedding Theorem

Some Applications

Epilogue

APPENDIX I

Some Major Algebraic Systems

APPENDIX II

Equivalence Relations and Equivalence Classes

APPENDIX Ill

Functions

APPENDIX IV

Zorns Lemma

APPENDIX V

Countability

APPENDIX VI

Commutative Rings

Bibliography

Notation

Index

《群论导论(第4版)(英文版)》介绍了：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. ！ also believe that some history should be given so that one canunderstand the origin of problems and the context in which the subjectdeveloped. Just as each of the earlier editions differs from the previous one in a signifi-cant way, the present （fourth） edition is genuinely different from the third.Indeed, this is already apparent in the Table of Contents. The book nowbegins with the unique factorization of permutations into disjoint cycles andthe parity of permutations; only then is the idea of group introduced. This isconsistent with the history of Group Theory, for these first results on permu-tations can be found in an 1815 paper by Cauchy, whereas groups of permu-tations were not introduced until 1831 （by Galois）But even if history