An Introduction to the Theory of Groups

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-Hölder 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 KrullSchmidt Theorem. - Operator Groups. - 7 Extensions and Cohomology. - The Extension Problem. - Automorphism Groups. - Semidirect Products. - Wreath Products. - Factor Sets. - Theorems of Schur-Zassenhaus and Gaschütz. - 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. - SemigroupInterlude. - 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 MarkovPost Theorem. - The NovikovBooneBritton Theorem: Sufficiency of Boone's Lemma. - Cancellation Diagrams. - The NovikovBooneBritton 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. Language: English