An introduction to the theory of groups

著者

• Rotman, Joseph J.

書誌事項

An introduction to the theory of groups

Joseph J. Rotman

（Graduate texts in mathematics, 148）

Springer-Verlag, c1995

4th ed

• : us
• : gw

注記

Includes bibliographical references and indexes

内容説明・目次

巻冊次

: us ISBN 9780387942858

内容説明

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

目次

• 1 Groups and Homomorphisms.- Permutations.- Cycles.- Factorization into Disjoint Cycles.- Even and Odd Permutations.- Semigroups.- Groups.- Homomorphisms.- 2 The Isomorphism Theorems.- Subgroups.- Lagrange's Theorem.- Cyclic Groups.- Normal Subgroups.- Quotient Groups.- The Isomorphism Theorems.- Correspondence Theorem.- Direct Products.- 3 Symmetric Groups and G-Sets.- Conjugates.- Symmetric Groups.- The Simplicity of An.- Some Representation Theorems.- G-Sets.- Counting Orbits.- Some Geometry.- 4 The Sylow Theorems.- p-Groups.- The Sylow Theorems.- Groups of Small Order.- 5 Normal Series.- Some Galois Theory.- The Jordan-Hoelder Theorem.- Solvable Groups.- Two Theorems of P. Hall.- Central Series and Nilpotent Groups.- p-Groups.- 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.- 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.- 8 Some Simple Linear Groups.- Finite Fields.- The General Linear Group.- PSL(2, K).- PSL(m, K).- Classical Groups.- 9 Permutations and the Mathieu Groups.- Multiple Transitivity.- Primitive G-Sets.- Simplicity Criteria.- Affine Geometry.- Projective Geometry.- Sharply 3-Transitivc Groups.- Mathieu Groups.- Steiner Systems.- 10 Abelian Groups.- Basics.- Free Abelian Groups.- Finitely Generated Abelian Groups.- Divisible and Reduced Groups.- Torsion Groups.- Subgroups of ?.- Character Groups.- 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 Nielscn-Schreier Theorem.- Free Products.- The Kurosh Theorem.- The van Kampen Theorem.- Amalgams.- HNN Extensions.- 12 The Word Problem.- 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.- Notation.

巻冊次

: gw ISBN 9783540942856

内容説明

Anyone who has studied abstract and linear algebra as an undergraduate will have the background to understand this book. The first six chapters provide ample material for a first course, beginning with the basic properties of groups and homomorphisms. The next section of text uses the Jordan-Holder Theorem to organize a discussion of extensions and simple groups. The book closes with three chapters on infinite Abelian groups, free groups and a complete proof of the unsolvability of the word problem for finitely presented groups.

目次

• Groups and homomorphisms
• the isomorphism theorems
• symmetric groups and G-sets
• the Sylow theorems
• normal series
• finite direct products
• extensions and cohomology
• some simple linear groups
• permutations and Mathieu groups
• Abelian groups
• free groups and free products
• the word problem. Appendices: some major algebraic systems
• equivalence relations and equivalence classes
• functions
• Zorn's lemma
• countability
• commutative rings.

「Nielsen BookData」 より

詳細情報

• NII書誌ID(NCID): BA23879214
• ISBN:
  • 0387942858
  • 3540942858
• LCCN: 94006507
• 出版国コード: us
• タイトル言語コード: eng
• 本文言語コード: eng
• 出版地: New York
• ページ数/冊数: xiv, 513 p.
• 大きさ: 25 cm
• 分類:
  • LCC : QA174.2
  • DC20 : 512/.2
  • NDC8 : 411.6
• 件名:
  • LCSH : Group theory
• 親書誌ID:
  • BA00010074