Notes on Group Theory

Description

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 `39`42`"613A``45`47`"603AGL2(F)......Page 34 Affine functions......Page 36 Affine Automorphisms......Page 37 Projective Groups......Page 38 The Bruhat decomposition......Page 39 Cyclic groups......Page 44 Finite abelian groups......Page 45 Unit Groups......Page 47 The dihedral groups Dn......Page 48 The quaternion and generalized quaternion groups Q4n......Page 50 p-groups, a first look......Page 51 Simple groups......Page 53 Simplicity of alternating groups......Page 54 Simplicity of `39`42`"613A``45`47`"603APSL2(F)......Page 55 Exceptional isomorphisms......Page 57 Applications to simple groups......Page 58 Finite linear groups......Page 59 Sylow p-subgroups......Page 60 Small examples.......Page 64 Groups of order pq......Page 65 Sylow subgroups in `39`42`"613A``45`47`"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