Notes on Group Theory

✍ Scribed by Mark Reeder

• Year: 2015
• Tongue: English
• Leaves: 126
• Series: lecture notes
• Edition: version 27 Sep 2015
• Category: Library

✦ Table of Contents

Notation for sets and functions......Page 4
The definition of a group......Page 5
Group homomorphisms......Page 6
Subgroups......Page 7
Cosets and quotient spaces......Page 8
The first isomorphism theorem......Page 9
Exact sequences......Page 10
The second isomorphism theorem (correspondence theorem)......Page 11
Direct products......Page 12
Semidirect products (internal view)......Page 13
Conjugacy......Page 14
Cycle decomposition and conjugacy classes......Page 15
The group Sn......Page 17
Length......Page 18
Inversions......Page 19
Sign character and alternating group......Page 20
Group actions......Page 22
The left regular action......Page 24
Double cosets......Page 25
Conjugation......Page 27
Burnside's Lemma......Page 28
Application: Graph counting......Page 29
Linear groups......Page 31
Symmetric groups and Linear groups......Page 33
Conjugacy classes in 3942"613A``4547"603AGL2(F)......Page 34<br>Affine functions......Page 36<br>Affine Automorphisms......Page 37<br>Projective Groups......Page 38<br>The Bruhat decomposition......Page 39<br>Cyclic groups......Page 44<br>Finite abelian groups......Page 45<br>Unit Groups......Page 47<br>The dihedral groups Dn......Page 48<br>The quaternion and generalized quaternion groups Q4n......Page 50<br>p-groups, a first look......Page 51<br>Simple groups......Page 53<br>Simplicity of alternating groups......Page 54<br>Simplicity of3942"613A45`47`"603APSL2(F)......Page 55<br>Exceptional isomorphisms......Page 57<br>Applications to simple groups......Page 58<br>Finite linear groups......Page 59<br>Sylow p-subgroups......Page 60<br>Small examples.......Page 64<br>Groups of order pq......Page 65<br>Sylow subgroups in `39`42`"613A4547"603AGLn and flag varieties......Page 66
The Burnside Transfer Theorem......Page 68
Simple groups......Page 72
The simple group of order 60......Page 74
The simple group of order 168......Page 75
Simple groups of order 720......Page 79
Almost-simple groups of order 720......Page 82
Solvable and nilpotent groups......Page 86
Groups of order p3......Page 89
Automorphisms of the Heisenberg group......Page 92
Higher powers of p......Page 94
Projective limits and pro-p groups......Page 95
Toward the classification of p-groups......Page 97
Presentations of Groups......Page 98
Free Groups......Page 99
Generators and Relations......Page 101
A presentation of the symmetric group......Page 102
Coxeter groups and reflection groups......Page 103
Presentations of alternating groups......Page 105
A presentation of A5......Page 106
The Platonic Groups......Page 108
Automorphisms......Page 110
Automorphisms of Sn......Page 112
Semidirect Products (external view)......Page 114
Groups of order p2q......Page 118
Extensions......Page 120
Metacyclic groups and extensions......Page 122