Algebra Math Notes • Study Guide Group Theory Table of Contents Groups..................................................................................................................................................................... 3 Binary Operations ............................................................................................................................................................. 3 Groups .............................................................................................................................................................................. 3 Examples of Groups ......................................................................................................................................................... 4 Cyclic Groups ................................................................................................................................................................... 5 Homomorphisms and Normal Subgroups ......................................................................................................................... 5 Cosets and Quotient Groups ............................................................................................................................................ 6 Isomorphism Theorems .................................................................................................................................................... 7 Product Groups ................................................................................................................................................................ 8 Finite Groups........................................................................................................................................................... 9 Permutations and G-sets .................................................................................................................................................. 9 Combinatorial Equations ................................................................................................................................................ 10 p-Sylow Subgroups ........................................................................................................................................................ 10 Automorphisms ..................................................................................................................................................... 12 Inner and Outer Automorphisms..................................................................................................................................... 12 Complete Groups ........................................................................................................................................................... 12 Holomorph ...................................................................................................................................................................... 13 Symmetry .............................................................................................................................................................. 14 Symmetry ....................................................................................................................................................................... 14 2-D Symmetry Groups .................................................................................................................................................... 14 3-D Symmetry Groups .................................................................................................................................................... 15 Permutation Groups .............................................................................................................................................. 16 Symmetric Group............................................................................................................................................................ 16 Transitive Groups ........................................................................................................................................................... 16 Primitive Groups ............................................................................................................................................................. 17 Steiner Systems; Affine and Projective Spaces .............................................................................................................. 18 Mathieu Groups .............................................................................................................................................................. 19 Pólya Enumeration ......................................................................................................................................................... 20 Normal Series ....................................................................................................................................................... 22 Composition Series ........................................................................................................................................................ 22 Commutators and Derived Groups ................................................................................................................................. 22 Solvable Groups ............................................................................................................................................................. 23 Hall Subgroups ............................................................................................................................................................... 23 Supersolvable and Nilpotent Groups .............................................................................................................................. 24 Frattini Subgroup ............................................................................................................................................................ 25 Extensions and Cohomology ................................................................................................................................ 26 The Extension Problem .................................................................................................................................................. 26 Operator Groups............................................................................................................................................................. 27 Semidirect Product ......................................................................................................................................................... 27 Cohomology: Background .............................................................................................................................................. 29 Factor Sets: The Second Cohomology Group ................................................................................................................ 29 Theorems ....................................................................................................................................................................... 31 Transfer .......................................................................................................................................................................... 31 Derivations: The First Cohomology Group ..................................................................................................................... 32 Projective Representations and the Schur Multiplier ...................................................................................................... 33 Abelian Groups ..................................................................................................................................................... 34 Finite Abelian Groups ..................................................................................................................................................... 34 Infinite Abelian Groups ................................................................................................................................................... 34 Divisible Groups ............................................................................................................................................................. 35 Pure Subgroups.............................................................................................................................................................. 36 Direct Sum Decompositions ........................................................................................................................................... 36 Subgroups of ℚ .............................................................................................................................................................. 37 1 Free Groups .......................................................................................................................................................... 38 Free Groups ................................................................................................................................................................... 38 Todd-Coxeter Algorithm ................................................................................................................................................. 39 Fundamental Group and Nielsen-Schreier Theorem .....................................................................................................