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Group Theory

  


by Filed under Mathematics

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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