Ranks of Groups (eBook)
John Wiley & Sons (Verlag)
978-1-119-08032-9 (ISBN)
A comprehensive guide to ranks and group theory
Ranks of Groups features a logical, straightforward presentation, beginning with a succinct discussion of the standard ranks before moving on to specific aspects of ranks of groups. Topics covered include section ranks, groups of finite 0-rank, minimax rank, special rank, groups of finite section p-rank, groups having finite section p-rank for all primes p, groups of finite bounded section rank, groups whose abelian subgroups have finite rank, groups whose abelian subgroups have bounded finite rank, finitely generated groups having finite rank, residual properties of groups of finite rank, groups covered by normal subgroups of bounded finite rank, and theorems of Schur and Baer.
This book presents fundamental concepts and notions related to the area of ranks in groups. Class-tested worldwide by highly qualified authors in the fields of abstract algebra and group theory, this book focuses on critical concepts with the most interesting, striking, and central results. In order to provide readers with the most useful techniques related to the various different ranks in a group, the authors have carefully examined hundreds of current research articles on group theory authored by researchers around the world, providing an up-to-date, comprehensive treatment of the subject.
• All material has been thoroughly vetted and class-tested by well-known researchers who have worked in the area of rank conditions in groups
• Topical coverage reflects the most modern, up-to-date research on ranks of groups
• Features a unified point-of-view on the most important results in ranks obtained using various methods so as to illustrate the role those ranks play within group theory
• Focuses on the tools and methods concerning ranks necessary to achieve significant progress in the study and clarification of the structure of groups
Ranks of Groups: The Tools, Characteristics, and Restrictions is an excellent textbook for graduate courses in mathematics, featuring numerous exercises, whose solutions are provided. This book will be an indispensable resource for mathematicians and researchers specializing in group theory and abstract algebra.
MARTYN R. DIXON, PhD, is Professor in the Department of Mathematics at the University of Alabama.
LEONID A. KURDACHENKO, PhD, DrS, is Distinguished Professor and Chair of the Department of Algebra at the University of Dnepropetrovsk, Ukraine.
IGOR YA SUBBOTIN, PhD, is Professor in the Department of Mathematics and Natural Sciences at National University in Los Angeles, California.
Martyn R. Dixon, PhD, is Professor in the Department of Mathematics at the University of Alabama.
Leonid A. Kurdachenko, PhD, DrS, is Distinguished Professor and Chair of the Department of Algebra at the University of Dnepropetrovsk, Ukraine.
Igor Ya Subbotin, PhD, is Professor in the Department of Mathematics and Natural Sciences at National University in Los Angeles, California.
A comprehensive guide to ranks and group theory Ranks of Groups features a logical, straightforward presentation, beginning with a succinct discussion of the standard ranks before moving on to specific aspects of ranks of groups. Topics covered include section ranks, groups of finite 0-rank, minimax rank, special rank, groups of finite section p-rank, groups having finite section p-rank for all primes p, groups of finite bounded section rank, groups whose abelian subgroups have finite rank, groups whose abelian subgroups have bounded finite rank, finitely generated groups having finite rank, residual properties of groups of finite rank, groups covered by normal subgroups of bounded finite rank, and theorems of Schur and Baer. This book presents fundamental concepts and notions related to the area of ranks in groups. Class-tested worldwide by highly qualified authors in the fields of abstract algebra and group theory, this book focuses on critical concepts with the most interesting, striking, and central results. In order to provide readers with the most useful techniques related to the various different ranks in a group, the authors have carefully examined hundreds of current research articles on group theory authored by researchers around the world, providing an up-to-date, comprehensive treatment of the subject. All material has been thoroughly vetted and class-tested by well-known researchers who have worked in the area of rank conditions in groups Topical coverage reflects the most modern, up-to-date research on ranks of groups Features a unified point-of-view on the most important results in ranks obtained using various methods so as to illustrate the role those ranks play within group theory Focuses on the tools and methods concerning ranks necessary to achieve significant progress in the study and clarification of the structure of groups Ranks of Groups: The Tools, Characteristics, and Restrictions is an excellent textbook for graduate courses in mathematics, featuring numerous exercises, whose solutions are provided. This book will be an indispensable resource for mathematicians and researchers specializing in group theory and abstract algebra. MARTYN R. DIXON, PhD, is Professor in the Department of Mathematics at the University of Alabama. LEONID A. KURDACHENKO, PhD, DrS, is Distinguished Professor and Chair of the Department of Algebra at the University of Dnepropetrovsk, Ukraine. IGOR YA SUBBOTIN, PhD, is Professor in the Department of Mathematics and Natural Sciences at National University in Los Angeles, California.
Martyn R. Dixon, PhD, is Professor in the Department of Mathematics at the University of Alabama. Leonid A. Kurdachenko, PhD, DrS, is Distinguished Professor and Chair of the Department of Algebra at the University of Dnepropetrovsk, Ukraine. Igor Ya Subbotin, PhD, is Professor in the Department of Mathematics and Natural Sciences at National University in Los Angeles, California.
Ranks of Groups 3
Contents 7
Preface 9
CHAPTER 1: Essential Toolbox 15
1.1. Ascending and Descending Series in Groups 15
1.1.1. Lemma 16
1.1.2. Corollary 16
1.1.3. Lemma 18
1.1.4. Corollary 19
1.1.5. Corollary 19
1.1.6. Proposition 19
1.1.7. Lemma 19
1.1.8. Corollary 20
1.2. Generalized Soluble Groups 21
1.2.1. Lemma 21
1.2.2. Lemma 21
1.2.3. Lemma 22
1.2.4. Proposition (Chernikov [39]) 22
1.2.5. Proposition 23
1.2.6. Corollary 23
1.2.7. Lemma 23
1.2.8. Corollary 24
1.2.9. Corollary 24
1.2.10. Proposition 24
1.2.11. Proposition 25
1.2.12. Lemma 27
1.2.13. Proposition 27
1.2.14. Corollary 28
1.2.15. Proposition 28
1.2.16. Lemma 29
1.2.17. Lemma 29
1.2.18. Proposition 30
1.2.19. Corollary 31
1.2.20. Proposition 31
1.2.21. Proposition (Kurdachenko [139]) 32
1.2.22. Theorem 33
1.3. Chernikov Groups and the Minimum Condition 33
1.3.1. Proposition 34
1.3.2. Proposition 34
1.3.3. Lemma 36
1.3.4. Lemma 37
1.3.5. Theorem 37
1.3.6. Corollary 38
1.4. Linear Groups 39
1.4.1. Lemma 40
1.4.2. Proposition 41
1.4.3. Theorem 41
1.4.4. Corollary 41
1.4.5. Corollary 41
1.4.6. Theorem 42
1.4.7. Theorem 43
1.4.8. Theorem 43
1.4.9. Corollary 44
1.4.10. Proposition 44
1.5. Some Relationships Between the Factors of the Upper and Lower Central Series 45
1.5.1. Lemma 46
1.5.2. Proposition 46
1.5.3. Corollary 47
1.5.4. Proposition 47
1.5.5. Lemma 48
1.5.6. Theorem 48
1.5.7. Corollary 49
1.5.8. Lemma 49
1.5.9. Corollary 49
1.5.10. Corollary 49
1.5.11. Corollary 49
1.5.12. Theorem 50
1.5.13. Theorem 50
1.5.14. Corollary 50
1.5.15. Theorem 50
1.5.16. Corollary 51
1.5.17. Corollary 51
1.5.18. Theorem 51
1.5.19. Theorem 52
1.5.20. Lemma 52
1.5.21. Theorem 52
1.5.22. Theorem 53
1.6. Some Direct Decompositions in Abelian Normal Subgroups 54
1.6.1. Lemma 55
1.6.2. Corollary 56
1.6.3. Lemma 56
1.6.4. Corollary 56
1.6.5. Corollary 57
1.6.6. Proposition 57
1.6.7. Proposition 58
1.6.8. Theorem 59
CHAPTER 2: Groups of Finite 0-Rank 60
2.1. The Z-Rank in Abelian Groups 61
2.1.1. Definition 62
2.1.2. Proposition 62
2.1.3. Proposition 62
2.1.4. Lemma 63
2.1.5. Definition 63
2.1.6. Proposition 63
2.1.7. Proposition 64
2.1.8. Corollary 64
2.2. The 0-Rank of a Group 65
2.2.1. Lemma 65
2.2.2. Definition 66
2.2.3. Lemma 66
2.2.4. Definition 67
2.2.5. Lemma 67
2.3. Locally Nilpotent Groups of Finite 0-Rank 67
2.3.1. Lemma 67
2.3.2. Lemma 68
2.3.3. Theorem 69
2.3.4. Corollary 70
2.3.5. Corollary 70
2.3.6. Corollary 71
2.4. Groups of Finite 0-Rank in General 71
2.4.1. Theorem 71
2.4.2. Corollary 71
2.4.3. Lemma 71
2.4.4. Lemma 72
2.4.5. Corollary 72
2.4.6. Lemma 73
2.4.7. Corollary 73
2.4.8. Corollary 74
2.4.9. Lemma 75
2.4.10. Lemma 75
2.4.11. Theorem 76
2.4.12. Corollary 77
2.4.13. Theorem 78
2.4.14. Corollary 78
2.5. Local Properties of Groups of Finite 0-Rank 78
2.5.1. Lemma 78
2.5.2. Proposition 79
2.5.3. Lemma 80
2.5.4. Proposition 81
2.5.5. Corollary 81
2.5.6. Proposition 82
2.5.7. Theorem 83
CHAPTER 3: Section p-Rank of Groups 85
3.1. p-Rank in Abelian Groups 85
3.1.1. Definition 85
3.1.2. Definition 85
3.1.3. Lemma 86
3.1.4. Proposition 87
3.1.5. Corollary 87
3.2. Finite Section p-Rank 87
3.2.1. Definition 88
3.2.2. Lemma 88
3.2.3. Proposition 89
3.2.4. Lemma 89
3.2.5. Lemma 90
3.2.6. Proposition 90
3.2.7. Corollary 90
3.2.8. Lemma 91
3.2.9. Lemma 91
3.2.10. Corollary 91
3.2.11. Proposition 91
3.2.12. Proposition 92
3.2.13. Lemma 92
3.2.14. Lemma 92
3.2.15. Corollary 92
3.2.16. Corollary 92
3.2.17. Theorem 93
3.2.18. Corollary 93
3.2.19. Proposition 93
3.2.20. Theorem 93
3.2.21. Corollary 94
3.2.22. Lemma 95
3.2.23. Corollary 96
3.2.24. Corollary 96
3.2.25. Corollary 97
3.2.26. Theorem 98
3.3. Locally Finite Groups with Finite Section p-Rank 99
3.3.1. Definition 100
3.3.2. Lemma 100
3.3.3. Theorem 102
3.3.4. Proposition 102
3.3.5. Theorem 103
3.3.6. Proposition 104
3.3.7. Corollary 104
3.3.8. Proposition 105
3.3.9. Lemma 105
3.3.10. Theorem 106
3.3.11. Corollary 108
3.3.12. Corollary 108
3.3.13. Corollary 109
3.4. Structure of Locally Generalized Radical Groups with Finite Section p-Rank 109
3.4.1. Lemma 109
3.4.2. Theorem 109
3.4.3. Theorem 110
3.4.4. Corollary 110
3.4.5. Corollary 111
CHAPTER 4: Groups of Finite Section Rank 112
4.1. Locally Finite Groups with Finite Section Rank 112
4.1.1. Definition 112
4.1.2. Lemma 113
4.1.3. Theorem 113
4.1.4. Theorem 114
4.1.5. Lemma 115
4.1.6. Theorem 115
4.1.7. Proposition 116
4.1.8. Lemma 116
4.1.9. Corollary 116
4.1.10. Corollary 116
4.1.11. Proposition 117
4.1.12. Corollary 118
4.1.13. Corollary 118
4.2. Structure of Locally Generalized Radical Groups with Finite Section Rank 119
4.2.1. Theorem 119
4.2.2. Corollary 119
4.2.3. Lemma 119
4.2.4. Corollary 120
4.2.5. Corollary 120
4.2.6. Proposition 120
4.2.7. Lemma 120
4.2.8. Proposition 120
4.2.9. Lemma 121
4.2.10. Theorem 122
4.2.11. Theorem 122
4.2.12. Theorem 123
4.3. Connections Between the Order of a Finite Group and Its Section Rank 124
4.3.1. Lemma 124
4.3.2. Corollary 124
4.3.3. Corollary 124
4.3.4. Proposition 124
4.3.5. Corollary 126
4.3.6. Theorem 126
4.3.7. Corollary 127
4.3.8. Corollary 127
4.3.9. Corollary 127
4.3.10. Corollary 128
4.3.11. Corollary 128
4.3.12. Theorem 128
4.3.13. Theorem 128
4.3.14. Theorem 129
4.3.15. Corollary 129
4.4. Groups of Finite Bounded Section Rank 129
4.4.1. Definition 129
4.4.2. Lemma 129
4.4.3. Theorem 130
4.4.4. Corollary 132
4.4.5. Corollary 133
4.4.6. Theorem 133
CHAPTER 5: Zaitsev Rank 135
5.1. The Zaitsev Rank of a Group 135
5.1.1. Definition 135
5.1.2. Definition 135
5.1.3. Lemma 136
5.1.4. Lemma 137
5.1.5. Lemma 139
5.1.6. Corollary 139
5.1.7. Corollary 139
5.1.8. Definition 139
5.1.9. Corollary 140
5.1.10. Corollary 140
5.1.11. Proposition 140
5.2. Zaitsev Rank and 0-Rank 141
5.2.1. Corollary 141
5.2.2. Corollary 141
5.2.3. Corollary 141
5.2.4. Corollary 141
5.2.5. Lemma 142
5.2.6. Proposition 144
5.2.7. Corollary 144
5.2.8. Theorem 144
5.3. Weak Minimal and Weak Maximal Conditions 145
5.3.1. Definition 145
5.3.2. Theorem 146
5.3.3. Theorem 146
5.3.4. Corollary 147
5.3.5. Corollary 147
CHAPTER 6: Special Rank 149
6.1. Elementary Properties of Special Rank 149
6.1.1. Definition 149
6.1.2. Lemma 150
6.1.3. Lemma 150
6.1.4. Lemma 151
6.1.5. Corollary 151
6.1.6. Proposition 152
6.1.7. Lemma 152
6.1.8. Lemma 152
6.1.9. Corollary 152
6.1.10. Theorem 152
6.1.11. Lemma 153
6.1.12. Lemma 153
6.1.13. Lemma 153
6.1.14. Lemma 154
6.2. The Structure of Groups Having Finite Special Rank 155
6.2.1. Lemma 155
6.2.2. Theorem 156
6.2.3. Corollary 157
6.2.4. Proposition 157
6.2.5. Example 158
6.2.6. Lemma 162
6.2.7. Corollary 162
6.2.8. Corollary 162
6.2.9. Corollary 162
6.2.10. Corollary 163
6.2.11. Theorem 163
6.2.12. Theorem 164
6.2.13. Corollary 164
6.2.14. Corollary 164
6.2.15. Corollary 164
6.2.16. Corollary 165
6.2.17. Corollary 165
6.2.18. Theorem 165
6.2.19. Corollary 165
6.2.20. Corollary 165
6.3. The Relationship Between the Special Rank and the Bounded Section Rank 166
6.3.1. Lemma 166
6.3.2. Lemma 166
6.3.3. Lemma 167
6.3.4. Lemma 167
6.3.5. Lemma 167
6.3.6. Lemma 168
6.3.7. Proposition 168
6.3.8. Lemma 169
6.3.9. Theorem 170
6.3.10. Corollary 170
6.3.11. Proposition 170
6.3.12. Proposition 171
6.3.13. Theorem 172
6.3.14. Theorem 172
6.3.15. Corollary 173
6.3.16. Corollary 173
6.3.17. Theorem 174
6.4. A Taste of the Exotic 174
6.4.1. Theorem 175
6.4.2. Proposition 175
6.4.3. Proposition 175
6.4.4. Proposition 175
6.4.5. Theorem 176
CHAPTER 7: The Relationship Between the Factors of the Upper Central Series and the Nilpotent Residual 178
7.1. Hypercentral Extensions by Groups of Finite 0-Rank 178
7.1.1. Lemma 178
7.1.2. Corollary 179
7.1.3. Lemma 179
7.1.4. Corollary 179
7.1.5. Corollary 180
7.1.6. Corollary 180
7.1.7. Corollary 181
7.1.8. Lemma 181
7.1.9. Corollary 181
7.1.10. Corollary 181
7.1.11. Lemma 182
7.1.12. Lemma 182
7.1.13. Proposition 183
7.1.14. Corollary 183
7.1.15. Corollary 183
7.1.16. Corollary 183
7.1.17. Corollary 184
7.1.18. Theorem 184
7.1.19. Theorem 185
7.1.20. Corollary 186
7.1.21. Theorem 186
7.1.22. Corollary 187
7.1.23. Lemma 187
7.1.24. Theorem 188
7.1.25. Theorem 188
7.1.26. Lemma 190
7.1.27. Corollary 190
7.1.28. Lemma 190
7.1.29. Corollary 191
7.1.30. Corollary 191
7.1.31. Corollary 191
7.1.32. Corollary 191
7.1.33. Theorem 192
7.2. Central Extensions by Groups of Finite Section Rank 192
7.2.1. Lemma 193
7.2.2. Lemma 193
7.2.3. Lemma 194
7.2.4. Lemma 195
7.2.5. Corollary 195
7.2.6. Proposition 196
7.2.7. Corollary 196
7.2.8. Corollary 196
7.2.9. Lemma 196
7.2.10. Lemma 197
7.2.11. Corollary 197
7.2.12. Proposition 197
7.2.13. Lemma 198
7.2.14. Proposition 198
7.2.15. Lemma 199
7.2.16. Proposition 199
7.2.17. Theorem 200
7.2.18. Corollary 201
7.2.19. Corollary 201
7.2.20. Corollary 201
7.2.21. Corollary 201
7.2.22. Corollary 201
7.2.23. Corollary 201
7.2.24. Corollary 201
7.2.25. Corollary 201
7.2.26. Corollary 202
7.2.27. Corollary 202
7.2.28. Lemma 202
7.2.29. Lemma 202
7.2.30. Theorem 203
7.2.31. Lemma 203
7.2.32. Theorem 203
7.2.33. Corollary 204
7.2.34. Corollary 205
7.2.35. Corollary 205
7.3. Hypercentral Extensions by Groups of Finite Section p-Rank 205
7.3.1. Lemma 205
7.3.2. Theorem 206
7.3.3. Corollary 206
7.3.4. Lemma 207
7.3.5. Corollary 207
7.3.6. Corollary 207
7.3.7. Corollary 207
7.3.8. Corollary 208
7.3.9. Corollary 208
7.3.10. Corollary 208
7.3.11. Corollary 208
7.3.12. Corollary 209
7.3.13. Corollary 209
7.3.14. Corollary 209
7.3.15. Corollary 209
7.3.16. Theorem 209
7.3.17. Lemma 210
7.3.18. Lemma 210
7.3.19. Corollary 210
7.3.20. Lemma 211
7.3.21. Lemma 211
7.3.22. Lemma 212
7.3.23. Corollary 213
7.3.24. Theorem 213
7.3.25. Corollary 214
7.3.26. Corollary 214
7.3.27. Example 214
7.3.28. Lemma 215
7.3.29. Corollary 216
7.3.30. Theorem 217
7.3.31. Corollary 217
7.3.32. Corollary 218
7.3.33. Corollary 218
7.3.34. Corollary 218
7.3.35. Corollary 218
7.3.36. Corollary 218
CHAPTER 8: Finitely Generated Groups of Finite Section Rank 219
8.1. The Z(G)-Decomposition in Some Abelian Normal Subgroups 219
8.1.1. Lemma 219
8.1.2. Lemma 220
8.1.3. Lemma 220
8.1.4. Corollary 221
8.1.5. Lemma 221
8.1.6. Corollary 221
8.1.7. Corollary 222
8.1.8. Lemma 222
8.1.9. Lemma 222
8.1.10. Lemma 223
8.1.11. Lemma 224
8.1.12. Proposition 224
8.1.13. Lemma 224
8.1.14. Proposition 224
8.1.15. Proposition 225
8.1.16. Corollary 225
8.1.17. Corollary 226
8.1.18. Lemma 226
8.1.19. Proposition 226
8.1.20. Proposition 226
8.1.21. Corollary 227
8.1.22. Corollary 228
8.2. Splittings over Some Normal Subgroups 228
8.2.1. Lemma 228
8.2.2. Lemma 229
8.2.3. Lemma 229
8.2.4. Lemma 229
8.2.5. Lemma 230
8.2.6. Proposition 230
8.2.7. Theorem 231
8.2.8. Corollary 232
8.2.9. Lemma 232
8.2.10. Lemma 233
8.2.11. Lemma 234
8.2.12. Proposition 234
8.2.13. Proposition 235
8.2.14. Theorem 236
8.2.15. Corollary 236
8.2.16. Corollary 236
8.3. Residually Finite Groups Having Finite 0-Rank 236
8.3.1. Proposition 237
8.3.2. Corollary 237
8.3.3. Proposition 237
8.3.4. Lemma 238
8.3.5. Lemma 238
8.3.6. Proposition 238
8.3.7. Lemma 238
8.3.8. Lemma 238
8.3.9. Lemma 238
8.3.10. Lemma 240
8.3.11. Corollary 240
8.3.12. Theorem 241
8.3.13. Corollary 241
8.3.14. Corollary 241
8.3.15. Corollary 241
8.3.16. Corollary 241
8.3.17. Corollary 242
8.4. Supplements to Divisible Abelian Normal Subgroups 242
8.4.1. Proposition 242
8.4.2. Lemma 244
8.4.3. Lemma 246
8.4.4. Theorem 247
8.4.5. Corollary 249
8.4.6. Corollary 249
8.4.7. Corollary 250
8.4.8. Corollary 251
8.4.9. Corollary 251
8.4.10. Corollary 251
8.4.11. Corollary 251
8.4.12. Corollary 251
8.4.13. Lemma 252
8.4.14. Theorem 252
8.4.15. Corollary 253
8.4.16. Corollary 253
CHAPTER 9: The Inuence of Important Families of Subgroups of Finite Rank 254
9.1. The Existence of Supplements to the Hirsch-Plotkin Radical 255
9.1.1. Lemma 255
9.1.2. Proposition 256
9.1.3. Lemma 257
9.1.4. Corollary 257
9.1.5. Proposition 257
9.1.6. Corollary 258
9.1.7. Lemma 258
9.1.8. Theorem 260
9.2. Groups Whose Locally Minimax Subgroups Have Finite Rank 261
9.2.1. Theorem 261
9.2.2. Lemma 263
9.2.3. Corollary 263
9.2.4. Theorem 263
9.2.5. Corollary 264
9.2.6. Corollary 264
9.2.7. Corollary 264
9.2.8. Lemma 264
9.2.9. Corollary 264
9.2.10. Corollary 265
9.2.11. Corollary 265
9.2.12. Lemma 265
9.2.13. Lemma 265
9.2.14. Theorem 266
9.2.15. Corollary 267
9.2.16. Corollary 267
9.2.17. Corollary 267
9.2.18. Theorem 267
9.2.19. Corollary 268
9.2.20. Corollary 268
9.2.21. Theorem 268
9.2.22. Corollary 268
9.2.23. Corollary 269
9.2.24. Theorem 269
9.2.25. Corollary 270
9.2.26. Corollary 270
9.3. Groups Whose Abelian Subgroups Have Finite Rank 270
9.3.1. Proposition 271
9.3.2. Proposition 271
9.3.3. Corollary 271
9.3.4. Corollary 272
9.3.5. Theorem 272
9.3.6. Corollary 272
9.3.7. Theorem 273
9.3.8. Corollary 273
9.3.9. Corollary 273
9.3.10. Corollary 274
9.3.11. Corollary 274
CHAPTER 10: A Brief Discussion of Other Interesting Results 275
10.1. Recent Work 275
10.1.1. Theorem 275
10.1.2. Corollary 275
10.1.3. Theorem 276
10.1.4. Theorem 276
10.1.5. Theorem 277
10.1.6. Theorem 277
10.1.7. Theorem 277
10.1.8. Theorem 277
10.1.9. Theorem 278
10.1.10. Theorem 278
10.1.11. Theorem 278
10.1.12. Theorem 278
10.1.13. Theorem 280
10.1.14. Theorem 280
10.1.15. Theorem 280
10.1.16. Theorem 281
10.1.17. Theorem 281
10.1.18. Theorem 281
10.1.19. Theorem 281
10.1.20. Theorem 283
10.1.21. Theorem 284
10.1.22. Theorem 285
10.1.23. Theorem 286
10.1.24. Theorem 286
10.1.25. Theorem 286
10.2. Questions 286
Bibliography 290
Author Index 309
Symbol Index 311
Subject Index 314
EULA 324
| Erscheint lt. Verlag | 15.6.2017 |
|---|---|
| Sprache | englisch |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Algebra |
| Schlagworte | abstract algebra • Algebra • Angewandte Mathematik • Applied mathematics • classes in locally finite groups • finite dimensional vector spaces • finitely generated groups having finite rank</p> • groups of finite 0-rank • groups of finite section rank • groups whose abelian subgroups have bounded finite ranks • groups whose abelian subgroups have finite ranks • group theory • group theory in the natural sciences • Gruppe (Math.) • Gruppenalgebra • <p>Groups and ranks • Mathematics • Mathematik • Modern/Abstract Algebra • Moderne u. abstrakte Algebra • principles of abstract algebra • ranks of groups for physicists • ranks of groups in cosmology • ranks of groups in particle theory • ranks of groups in physical chemistry • ranks of groups in quantum mechanics • residual properties of groups of finite rank • section p-rank of groups • zaitsev rank |
| ISBN-10 | 1-119-08032-0 / 1119080320 |
| ISBN-13 | 978-1-119-08032-9 / 9781119080329 |
| Informationen gemäß Produktsicherheitsverordnung (GPSR) | |
| Haben Sie eine Frage zum Produkt? |
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