by J.S Milne

**Contents**

1 Basic Definitions and Results

Definitions and examples

Multiplication tables

Subgroups

Groups of small order

Homomorphisms

Cosets

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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