Discovering Group Theory: A Transition to Advanced Mathematics

Contents......Page 3
Preface......Page 7
The Need for Proof......Page 9
Proving by Contradiction......Page 11
If, and Only If......Page 12
Proving That Something Is False......Page 14
What You Should Know......Page 15
Examples of Sets: Notation......Page 17
Describing a Set......Page 18
Subsets......Page 19
Venn Diagrams......Page 20
Intersection and Union......Page 21
Proving That Two Sets Are Equal......Page 22
What You Should Know......Page 24
Binary Operations......Page 26
Examples of Binary Operations......Page 27
Tables......Page 28
Testing for Binary Operations......Page 29
What You Should Know......Page 30
The Division Algorithm......Page 32
Relatively Prime Pairs of Numbers......Page 33
Prime Numbers......Page 34
Residue Classes of Integers......Page 35
What You Should Know......Page 39
Two Examples of Groups......Page 41
Definition of a Group......Page 43
A Diversion on Notation......Page 45
Some Examples of Groups......Page 46
Some Useful Properties of Groups......Page 49
The Powers of an Element......Page 50
The Order of an Element......Page 52
What You Should Know......Page 55
Subgroups......Page 57
Examples of Subgroups......Page 58
Testing for a Subgroup......Page 59
The Subgroup Generated by an Element......Page 60
What You Should Know......Page 62
Introduction......Page 64
Cyclic Groups......Page 65
Some Definitions and Theorems about Cyclic Groups......Page 66
What You Should Know......Page 68
The Cartesian Product......Page 69
Direct Product Groups......Page 70
What You Should Know......Page 71
Functions: A Discussion......Page 73
Functions: Formalizing the Discussion......Page 74
Examples......Page 75
Injections and Surjections......Page 76
Injections and Surjections of Finite Sets......Page 79
What You Should Know......Page 81
Composite Functions......Page 84
Some Properties of Composite Functions......Page 85
Inverse Functions......Page 86
Inverse of a Composite Function......Page 89
The Bijections from a Set to Itself......Page 91
What You Should Know......Page 92
Introduction......Page 94
Isomorphism......Page 96
Proving Two Groups Are Isomorphic......Page 98
Proving Two Groups Are Not Isomorphic......Page 99
Finite Abelian Groups......Page 100
What You Should Know......Page 105
Introduction......Page 107
Another Look at Permutations......Page 109
Practice at Working with Permutations......Page 110
Even and Odd Permutations......Page 115
Cycles......Page 120
Transpositions......Page 123
The Alternating Group......Page 125
What You Should Know......Page 126
Introduction......Page 129
Towards a General Notation......Page 131
The General Dihedral Group......Page 133
Subgroups of Dihedral Groups......Page 134
What You Should Know......Page 136
Cosets......Page 138
Lagrange’s Theorem......Page 141
Deductions from Lagrange’s Theorem......Page 142
Two Number Theory Applications......Page 143
More Examples of Cosets......Page 144
What You Should Know......Page 145
Groups of Order 4......Page 147
Groups of Order 6......Page 148
Groups of Order 8......Page 149
Summary......Page 151
Equivalence Relations......Page 153
Partitions......Page 155
An Important Equivalence Relation......Page 157
What You Should Know......Page 159
Introduction......Page 160
Sets as Elements of Sets......Page 162
Cosets as Elements of a Group......Page 163
Normal Subgroups......Page 164
The Quotient Group......Page 166
What You Should Know......Page 168
Homomorphisms......Page 170
The Kernel of a Homomorphism......Page 173
What You Should Know......Page 174
More about the Kernel......Page 176
The Quotient Group of the Kernel......Page 177
The First Isomorphism Theorem......Page 178
What You Should Know......Page 181
Answers......Page 182
Chapter 2......Page 183
Chapter 4......Page 185
Chapter 5......Page 187
Chapter 6......Page 192
Chapter 7......Page 193
Chapter 8......Page 194
Chapter 9......Page 196
Chapter 10......Page 198
Chapter 11......Page 199
Chapter 12......Page 204
Chapter 13......Page 206
Chapter 14......Page 207
Chapter 15......Page 209
Chapter 16......Page 210
Chapter 17......Page 212
Chapter 18......Page 213
Chapter 19......Page 214
Index......Page 216