Group Theory

by J.S Milne



1 Basic Definitions and Results
Definitions and examples
Multiplication tables
Groups of small order
Normal subgroups
Kernels and quotients
Theorems concerning homomorphisms
Direct products
Commutative groups
The order of ab

2 Free Groups and Presentations; Coxeter Groups
Free monoids
Free groups
Generators and relations
Finitely presented groups
Coxeter groups

3 Automorphisms and Extensions
Automorphisms of groups
Characteristic subgroups
Semidirect products
Extensions of groups
The H¨older program

4 Groups Acting on Sets
Definition and examples
Permutation groups
The Todd-Coxeter algorithm
Primitive actions

5 The Sylow Theorems; Applications
The Sylow theorems
Alternative approach to the Sylow theorems

6 Subnormal Series; Solvable and Nilpotent Groups
Subnormal Series
Solvable groups
Nilpotent groups
Groups with operators
Krull-Schmidt theorem

7 Representations of Finite Groups
Matrix representations
Roots of 1 in fields
Linear representations
Maschke’s theorem
The group algebra; semisimplicity
Semisimple modules
Simple F -algebras and their modules
Semisimple F -algebras and their modules
The representations of G
The characters of G
The character table of a group

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