群論導論（第4版）（英文版） [An Introduction to the Theory of Groups] 下載 mobi epub pdf 電子書 2024

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《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 格式

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羅特曼是一個數學傢，依據如下的事實，他的批評是特彆中肯的：皮亞傑的基本模式是符閤邏輯的，他經常參照集體稱為“布爾巴基”的法國數學傢小組。羅特曼指齣，皮亞傑誤解瞭數學的本質，特彆是數學進步中證明的作用。數學的主體是一種連貫的結構，但是證明的技術不是該結構的組成部分。他說，“數學的確由證明關於結構的主張所組成。……隻有對語言尤其是數學語言待貧乏作用的觀點，纔能支持皮亞傑提供的分析。”

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我們可以把羅特曼的論證概括如下：個體數學傢通過語言的媒介交流思想。作為討論的結果，他們對數學概念的錶達形式取得一緻，而且這些錶達形式在時間過程中可以發生變化。皮亞傑低估瞭數學傢使用的語言，寜願乞求·認識主體（然而對這種主體來說，特彆的語言概念幾乎不適用）。羅特曼的觀點更多地是與數學有關。在所有領域的情況是，“他人的觀點是公共的實體――通過主體間的一緻和體現在語言中的慣例為個體主體取得意義”。忽視語言的相互交流，對於任何認知發展理論都是嚴重的限製。我們的認知發展大都通過語言的相互交流――多數是在兒童和他的老師之間――而發生。使教育實踐建立在把語言相互作用降低為微不足道的作用的理論上，就是把它建立在其影響必定是有害的理論的基礎上。皮亞傑的理論不關心個體兒童怎樣在智慧上發展――或者是單獨的，或者是與他人相互作用，而是關心假定的“認識主體”的發展――對這種主體來說，任何個體心靈的運算僅僅能提供一種說明。對於教育實踐來說，這是一種極端古怪的基礎，而教育實踐不可避免地要涉及到個體兒童的發展。

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