Groups - Modular Mathematics Series

Front Cover......Page 1
Groups......Page 4
Copyright Page......Page 5
Table of Contents\n......Page 10
Series Preface......Page 6
Preface......Page 8
1.1 Symmetries of a square......Page 14
1.2 Symmetries of a circle......Page 20
1.3 Further exercises on Chapter 1......Page 22
2.1 The symmetric group S4\n......Page 25
2.2 Functions......Page 27
2.3 Permutations......Page 32
2.4 Basic properties of cycles......Page 34
2.5 Cycle decomposition......Page 35
2.6 Transpositions......Page 37
2.7 The 15-puzzle......Page 39
2.8 Further exercises on Chapter 2......Page 40
3.1 Matrix multiplication......Page 42
3.2 Linear transformations......Page 45
3.3 Orthogonal matrices......Page 47
3.4 Further exercises on Chapter 3......Page 49
4.1 Number systems......Page 51
4.2 Binary operations......Page 53
4.4 Examples of groups......Page 54
4.5 Consequences of the axioms......Page 56
4.6 Direct products......Page 59
4.7 Further exercises on Chapter 4......Page 60
5.1 Subgroups......Page 62
5.2 Examples of subgroups......Page 63
5.3 Groups of symmetries......Page 65
5.4 Further exercises on Chapter 5......Page 68
6.1 Cyclic groups......Page 70
6.2 Cyclic subgroups......Page 71
6.3 Order of elements......Page 72
6.4 Orders of products......Page 74
6.5 Orders of powers......Page 75
6.6 Subgroups of cyclic groups......Page 76
6.8 Further exercises on Chapter 6......Page 78
7.1 Groups acting on sets......Page 81
7.2 Orbits......Page 84
7.3 Stabilizers......Page 85
7.4 Permutations arising from group actions......Page 86
7.5 The alternating group......Page 87
7.6 Further exercises on Chapter 7......Page 91
8.1 Partitions......Page 94
8.2 Relations......Page 95
8.3 Equivalence classes......Page 97
8.4 Equivalence relations from group actions......Page 98
8.5 Modular arithmetic......Page 100
8.6 Further exercises on Chapter 8......Page 104
9.1 Comparing D3 and S3\n......Page 106
9.2 Properties of homomorphisms......Page 110
9.3 Homomorphisms arising from group actions......Page 111
9.4 Cayley\'s theorem......Page 115
9.5 Cyclic groups......Page 116
9.6 Further exercises on Chapter 9......Page 117
10.1 Left cosets......Page 119
10.2 Left cosets as equivalence classes......Page 121
10.3 Lagrange\'s theorem......Page 123
10.4 Consequences of Lagrange\'s theorem......Page 124
10.5 Applications to number theory......Page 126
10.7 Further exercises on Chapter 10......Page 127
11.1 The orbit-stabilizer theorem......Page 130
11.2 Fixed subsets......Page 132
11.3 Counting orbits......Page 134
11.4 Further exercises on Chapter 11......Page 135
12.1 Colouring problems......Page 137
12.2 Groups of symmetries in three dimensions......Page 140
12.3 Three-dimensional colouring problems......Page 142
12.4 Further exercises on Chapter 12......Page 143
13.2 Conjugacy classes......Page 146
13.3 Conjugacy classes in Sn......Page 148
13.5 Centres......Page 150
13.6 Conjugates and centralizers......Page 151
13.7 Further exercises on Chapter 13......Page 153
14.1 An action of S3\n on three-dimensional space......Page 155
14.2 Cauchy\'s theorem......Page 156
14.3 Direct products......Page 158
14.4 Further exercises on Chapter 14......Page 161
15.1 Kernels of homomorphisms......Page 163
15.2 Kernels of actions......Page 164
15.3 Conjugates of a subgroup......Page 165
15.4 Normal subgroups......Page 167
15.5 Normal subgroups and conjugacy classes......Page 170
15.6 Simple groups......Page 172
15.7 Further exercises on Chapter 15......Page 173
16.1 Cosets of the kernel of an action......Page 176
16.2 Factor groups......Page 179
16.3 Calculations in factor groups......Page 181
16.4 The first isomorphism theorem......Page 183
16.5 Groups of order p2 are Abelian......Page 185
16.6 Further exercises on Chapter 16......Page 187
17.2 Groups of order 2p......Page 189
17.3 Groups of order 8......Page 191
17.4 Groups of order 12......Page 194
17.5 Further exercises on Chapter 17......Page 197
18.1 History......Page 199
18.2 Topics for further study......Page 200
18.3 Projects......Page 203
Solutions......Page 206
Glossary......Page 216
Bibliography......Page 217
Index......Page 218