群論導(dǎo)論

前言

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;

內(nèi)容概要

　　《群論導(dǎo)論(第4版)(英文版)》介紹了：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

作者簡(jiǎn)介

作者：(美國(guó))羅曼(Joseph J.Rotman)

書籍目錄

Preface to the Fourth EditionFrom Preface to the Third EditionTo the ReaderCHAPTER 1　Groups and Homomorphisms　Permutations　Cycles　Factorization into Disjoint Cycles　Even and Odd Permutations　Semigroups　Groups　HomomorphismsCHAPTER 2　The Isomorphism Theorems　Subgroups　Lagrange's Theorem　Cyclic Groups　Normal Subgroups　Quotient Groups　The Isomorphism Theorems　Correspondence Theorem　Direct ProductsCHAPTER 3　Symmetric Groups and G-Sets　Conjugates　Symmetric Groups　The Simplicity of A.　Some Representation Theorems　G-Sets　Counting Orbits　Some GeometryCHAPTER 4　The Sylow Theorems　p-Groups　The Sylow Theorems　 Groups of Small OrderCHAPTER 5　Normal Series　 Some Galois Theory　 The Jordan-Ho1der Theorem　 Solvable Groups　 Two Theorems of P. Hall　 Central Series and Nilpotent Groups　 p-GroupsCHAPTER 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 GroupsCHAPTER 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　  DerivationsCHAPTER 8　Some Simple Linear Groups  ……CHAPTER 9　Permutations and the Mathieu GroupsCHAPTER 10　Abelian GroupsCHAPTER 11　Free Groups and Free ProductsCHAPTER 12　The Word ProblemEpilogueBibliographyNotationIndex

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