Preface A Guide to this Guide 1 Algebra: Classical, Modern, and Ultramodern 1.1 The Beginnings of Modern Algebra 1.2 Modern Algebra 1.3 Ultramodern Algebra 1.4 What Next 2 Categories 2.1 Categories 2.2 Functors 2.3 Natural Transformations 2.4 Products, Coproducts, and Generalizations 3 Algebraic Structures 3.1 Structures with One Operation 3.2 Rings 3.3 Actions 3.4 Semirings 3.5 Algebras 3.6 Ordered Structures 4 Groups and their Representations 4.1 Definitions 4.1.1 Groups and homomorphisms 4.1.2 Subgroups 4.1.3 Actions 4.1.4 G acting on itself 4.2 Some Important Examples 4.2.1 Permutation groups 4.2.2 Symmetry groups 4.2.3 Other examples 4.2.4 Topological groups 4.2.5 Free groups 4.3 Reframing the Definitions 4.4 Orbits and Stabilizers 4.4.1 Stabilizers 4.4.2 Orbits 4.4.3 Acting by multiplication 4.4.4 Cosets 4.4.5 Counting cosets and elements 4.4.6 Double cosets 4.4.7 A nice example 4.5 Homomorphisms and Subgroups 4.5.1 Kernel, image, quotient 4.5.2 Homomorphism theorems 4.5.3 Exact sequences 4.5.4 H?lder's dream 4.6 Many Cheerful Subgroups 4.6.1 Generators,cyclic groups 4.6.2 Elements of finite order 4.6.3 Finitely generated groups and the Burnside problem 4.6.4 Other nice subgroups 4.6.5 Conjugation and the class equation 4.6.6 p-groups 4.6.7 Sylow's Theorem and Sylow subgroups 4.7 Sequences of Subgroups 4.7.1 Composition series 4.7.2 Central series, derived series, nilpotent, solvable 4.8 New Groups from Old 4.8.1 Direct products 4.8.2 Semidirect products 4.8.3 Isometries of R3 4.8.4 Free products 4.8.5 Direct sums of abelian groups 4.8.6 Inverse limits and direct limits 4.9 Generators and Relations 4.9.1 Definition and examples 4.9.2 Cayley graphs …… 5 Rings and Modules 6 Fields and Skew Fields Bibliography Index of Notations Index About the Author
