Discovering Group Theory

Description

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