內容大鋼
本版傳承了前幾版之表述清晰的風格，並在此基礎上對內容進行了全面修正與改編。開篇介紹了群與同態的基本性質，涵蓋的主題包括Lagrange定理、Noether同構定理、對稱群、G-集、Sylow定理，有限Abel群、Krull-Schmidt定理、可解及冪零群，Jordan-Holder定理。隨後中間幾章應用Jordan-Holder定理組織了關於擴張（同構群、半直積、Schur-Zassenhaus引理，Schur乘子）及單群（射影?模群的簡單性，以及回到G-集的討論后，構造零散Mathieu群）的討論。本書最後三章的內容有：無限Abel群（重點討論可數群）；自由群及群表示、陪集枚舉、自由積、共合及HNN擴張；以及有限表示群問題，和Higman嵌入定理與群同構問題的不可判定性的完全性證明。目次：群與同態；構定理；對稱群與G集；Sylow定理；正合列；有限直積；擴張與上同調；一些單線性群；置換與Mathieu群；Abel群；自由群與自由積；字問題。
目錄
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 An
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-Holder 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 Krull-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 Gaschutz
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
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