CONTENTS xi 11 Homomorphisms 181 11.1 Group Homomorphisms..·····. 181 ll.2 The Isomorphism Theorems······ 183 11.3 Exercises 186 11.4 Additional Exercises:Automorphisms 187 11.5Sage...·..·····…···· 188 11.6 Sage Exercises 192 12 Matrix Groups and Symmetry 194 12.1 Matrix Groups 194 12.2 Symmetry 200 12.3 Exercises 206 References and Suggested Readings. 208 12.4Sage.············ 209 12.5 Sage Exercises 209 13 The Structure of Groups 210 13.1 Finite Abelian Groups 210 13.2 Solvable Groups 214 l3.3 Exercises··,······· 217 13.4 Programming Exercises 218 References and Suggested Readings.. 219 13.5Sage..············ 219 13.6 Sage Exercises ....... 221 14 Group Actions 222 14.1 Groups Acting on Sets 222 14.2 The Class Equation 225 14.3 Burnside's Counting Theorem.. 226 14.4 Exercises.·.·.·····.· 232 14.5 Programming Exercise .... 234 References and Suggested Reading 234 14.6Sage......·.. 235 14.7 Sage Exercises 238 15 The Sylow Theorems 240 15.1 The Sylow Theorems ... 240 15.2 Examples and Applications 243 15.3 Exercises 246 15.4 A Project 247 References and Suggested Readings.. 248 15.5Sage·..··············· 248 l5.6 Sage Exercises··.······ 254 16 Rings 256 16.1 Rings..·· 256 16.2 Integral Domains and Fields.. 259 16.3 Ring Homomorphisms and Ideals 261 l6.4 Maximal and Prime Ideals........·.......·.··. 264 16.5 An Application to Software Design 266 l6.6 Exercises...························· 269CONTENTS xi 11 Homomorphisms 181 11.1 Group Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 11.2 The Isomorphism Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 11.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 11.4 Additional Exercises: Automorphisms . . . . . . . . . . . . . . . . . . . . . 187 11.5 Sage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 11.6 Sage Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 12 Matrix Groups and Symmetry 194 12.1 Matrix Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 12.2 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 12.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 References and Suggested Readings . . . . . . . . . . . . . . . . . . . . . . . . . . 208 12.4 Sage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 12.5 Sage Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 13 The Structure of Groups 210 13.1 Finite Abelian Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 13.2 Solvable Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 13.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 13.4 Programming Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 References and Suggested Readings . . . . . . . . . . . . . . . . . . . . . . . . . . 219 13.5 Sage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 13.6 Sage Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 14 Group Actions 222 14.1 Groups Acting on Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 14.2 The Class Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 14.3 Burnside’s Counting Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 226 14.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 14.5 Programming Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 References and Suggested Reading . . . . . . . . . . . . . . . . . . . . . . . . . . 234 14.6 Sage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 14.7 Sage Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 15 The Sylow Theorems 240 15.1 The Sylow Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 15.2 Examples and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 15.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 15.4 A Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 References and Suggested Readings . . . . . . . . . . . . . . . . . . . . . . . . . . 248 15.5 Sage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 15.6 Sage Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 16 Rings 256 16.1 Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 16.2 Integral Domains and Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 16.3 Ring Homomorphisms and Ideals . . . . . . . . . . . . . . . . . . . . . . . . 261 16.4 Maximal and Prime Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 16.5 An Application to Software Design . . . . . . . . . . . . . . . . . . . . . . . 266 16.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269