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An Introduction to the Theory of Groups

by Joseph J. Rotman

  • ISBN: 9780387942858
  • ISBN10: 0387942858

An Introduction to the Theory of Groups

by Joseph J. Rotman

  • List Price: $79.99
  • Binding: Hardcover
  • Edition: 4
  • Publisher: Springer Verlag
  • Publish date: 01/01/1995
  • ISBN: 9780387942858
  • ISBN10: 0387942858
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Description: 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-Hlder 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 Gaschtz.- 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.
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