An Introduction to the Theory of Groups (Graduate Texts in Mathematics) (4 SUB)

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An Introduction to the Theory of Groups (Graduate Texts in Mathematics) (4 SUB)

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  • 製本 Hardcover:ハードカバー版
  • 言語 ENG
  • 商品コード 9780387942858
  • DDC分類 512.2

Full Description


Anyone who has studied abstract algebra and linear algebra as an undergraduate can understand this book. The first six chapters provide material for a first course, while the rest of the book covers more advanced topics. This revised edition retains the clarity of presentation that was the hallmark of the previous editions. From the reviews:"Rotman has given us a very readable and valuable text, and has shown us many beautiful vistas along his chosen route." --MATHEMATICAL REVIEWS

Table of Contents

Preface to the Fourth Edition                      vii
From Preface to the Third Edition ix
To the Reader xv
Groups and Homomorphisms 1 (19)
Permutations 2 (1)
Cycles 3 (3)
Factorization into Disjoint Cycles 6 (1)
Even and Odd Permutations 7 (3)
Semigroups 10 (2)
Groups 12 (4)
Homomorphisms 16 (4)
The Isomorphism Theorems 20 (23)
Subgroups 20 (4)
Lagrange's Theorem 24 (4)
Cyclic Groups 28 (1)
Normal Subgroups 29 (3)
Quotient Groups 32 (3)
The Isomorphism Theorems 35 (2)
Correspondence Theorem 37 (3)
Direct Products 40 (3)
Symmetric Groups and G-Sets 43 (30)
Conjugates 43 (3)
Symmetric Groups 46 (4)
The Simplicity of An 50 (1)
Some Representation Theorems 51 (4)
G-Sets 55 (3)
Counting Orbits 58 (5)
Some Geometry 63 (10)
The Sylow Theorems 73 (16)
p-Groups 73 (5)
The Sylow Theorems 78 (4)
Groups of Small Order 82 (7)
Normal Series 89 (36)
Some Galois Theory 91 (7)
The Jordan-Holder Theorem 98 (4)
Solvable Groups 102(6)
Two Theorems of P. Hall 108(4)
Central Series and Nilpotent Groups 112(7)
p-Groups 119(6)
Finite Direct Products 125(29)
The Basis Theorem 125(6)
The Fundamental Theorem of Finite Abelian 131(2)
Groups
Canonical Forms; Existence 133(8)
Canonical Forms; Uniqueness 141(3)
The Krull-Schmidt Theorem 144(7)
Operator Groups 151(3)
Extensions and Cohomology 154(63)
The Extension Problem 154(2)
Automorphism Groups 156(11)
Semidirect Products 167(5)
Wreath Products 172(6)
Factor Sets 178(10)
Theorems of Schur-Zassenhaus and Gaschutz 188(5)
Transfer and Burnside's Theorem 193(8)
Projective Representations and the Schur 201(10)
Multiplier
Derivations 211(6)
Some Simple Linear Groups 217(30)
Finite Fields 217(1)
The General Linear Groups 217(2)
The General Linear Group 219(5)
PSL(2, K) 224(3)
PSL(m, K) 227(7)
Classical Groups 234(13)
Permutations and the Mathieu Groups 247(60)
Multiple Transitivity 247(9)
Primitive G-Sets 256(3)
Simplicity Criteria 259(5)
Affine Geometry 264(8)
Projective Geometry 272(9)
Sharply 3-Transitive Groups 281(5)
Mathieu Groups 286(7)
Steiner Systems 293(14)
Abelian Groups 307(36)
Basics 307(5)
Free Abelian Groups 312(6)
Finitely Generated Abelian Groups 318(2)
Divisible and Reduced Groups 320(5)
Torsion Groups 325(6)
Subgroups of Q 331(4)
Character Groups 335(8)
Free Groups and Free Products 343(75)
Generators and Relations 343(6)
Semigroup Interlude 349(2)
Coset Enumeration 351(7)
Presentations and the Schur Multiplier 358(8)
Fundamental Groups of Complexes 366(8)
Tietze's Theorem 374(3)
Covering Complexes 377(6)
The Nielsen-Schreier Theorem 383(5)
Free Products 388(3)
The Kurosh Theorem 391(3)
The van Kampen Theorem 394(7)
Amalgams 401(6)
HNN Extensions 407(11)
The Word Problem 418(53)
Introduction 418(2)
Turing Machines 420(5)
The Markov-Post Theorem 425(5)
The Novikov-Boone-Britton Theorem: 430(3)
Sufficiency of Boone's Lemma
Cancellation Diagrams 433(5)
The Novikov-Boone-Britton Theorem: 438(12)
Necessity of Boone's Lemma
The Higman Imbedding Theorem 450(14)
Some Applications 464(7)
Epilogue 471(4)
Appendix I Some Major Algebraic Systems 475(2)
Appendix II Equivalence Relations and 477(2)
Equivalence Classes
Appendix III Functions 479(2)
Appendix IV Zorn's Lemma 481(2)
Appendix V Countability 483(2)
Appendix VI Commutative Rings 485(10)
Bibliography 495(3)
Notation 498(5)
Index 503