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內容簡介

《群論導論(第4版)(英文版)》介紹瞭：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

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
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

前言/序言

　　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 wereotherwise， I feel that it is usually good pedagogy to introduce a generalnotion only after becoming comfortable with an important special case. Ihave also added several new sections， and I have subtracted the chapter onHomologieal Algebra （although the section on Horn functors and charactergroups has been retained） and the section on Grothendieck groups.　The format of the book has been changed a bit： almost all exercises nowoccur at ends of sections， so as not to interrupt the exposition. There areseveral notational changes from earlier editions： I now write insteadof　to denote "H is a subgroup of G"; the dihedral group of order2n is now denoted by　instead of by ; the trivial group is denoted by ！instead of by {1}; in the discussion of simple linear groups， I now distinguishelementary traesvections from more general transvections;

群論導論（第4版）（英文版） [An Introduction to the Theory of Groups] 下載 mobi epub pdf txt 電子書

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