1 Groups and Homomorphisms -- Permutations -- Cycles -- Factorization into Disjoint Cycles -- Even and Odd Permutations -- Semigroups -- Groups -- Homomorphisms -- 2 The Isomorphism Theorems -- Subgroups -- Lagrangeโs 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-Hรถlder 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 KrullโSchmidt Theorem -- Operator Groups -- 7 Extensions and Cohomology -- The Extension Problem -- Automorphism Groups -- Semidirect Products -- Wreath Products -- Factor Sets -- Theorems of Schur-Zassenhaus and Gaschรผtz -- Transfer and Burnsideโs 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 -- Tietzeโs 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 MarkovโPost Theorem -- The NovikovโBooneโBritton Theorem: Sufficiency of Booneโs Lemma -- Cancellation Diagrams -- The NovikovโBooneโBritton Theorem: Necessity of Booneโs 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 Zornโs Lemma -- APPENDIX V Countability -- APPENDIX VI Commutative Rings -- Notation

1. Mathematics
2. Group theory
3. Mathematics
4. Group Theory and Generalizations