Appendix 15. A criterion for a group to be nilpotent

Chapters in this book

1. Frontmatter i
2. Contents v
3. List of definitions and notations ix
4. Foreword xv
5. Preface xvii
6. Introduction 1
7. §1. Groups with a cyclic subgroup of index p. Frattini subgroup. Varia 22
8. §2. The class number, character degrees 58
9. §3. Minimal classes 69
10. §4. p-groups with cyclic Frattini subgroup 73
11. §5. Hall’s enumeration principle 81
12. §6. q'-automorphisms of q-groups 91
13. §7. Regular p-groups 98
14. §8. Pyramidal p-groups 109
15. §9. On p-groups of maximal class 114
16. §10. On abelian subgroups of p-groups 128
17. §11. On the power structure of a p-group 146
18. §12. Counting theorems for p-groups of maximal class 151
19. §13. Further counting theorems 161
20. §14. Thompson’s critical subgroup 185
21. §15. Generators of p-groups 189
22. §16. Classification of finite p-groups all of whose noncyclic subgroups are normal 192
23. §17. Counting theorems for regular p-groups 198
24. §18. Counting theorems for irregular p-groups 202
25. §19. Some additional counting theorems 215
26. §20. Groups with small abelian subgroups and partitions 219
27. §21. On the Schur multiplier and the commutator subgroup 222
28. §22. On characters of p-groups 229
29. §23. On subgroups of given exponent 242
30. §24. Hall’s theorem on normal subgroups of given exponent 246
31. §25. On the lattice of subgroups of a group 256
32. §26. Powerful p-groups 262
33. §27. p-groups with normal centralizers of all elements 275
34. §28. p-groups with a uniqueness condition for nonnormal subgroups 279
35. §29. On isoclinism 285
36. §30. On p-groups with few nonabelian subgroups of order pp and exponent p 289
37. §31. On p-groups with small p0-groups of operators 301
38. §32. W. Gaschütz’s and P. Schmid’s theorems on p-automorphisms of p-groups 309
39. §33. Groups of order pm with automorphisms of order pm-1, pm-2 or pm-3 314
40. §34. Nilpotent groups of automorphisms 318
41. §35. Maximal abelian subgroups of p-groups 326
42. §36. Short proofs of some basic characterization theorems of finite p-group theory 333
43. §37. MacWilliams’ theorem 345
44. §38. p-groups with exactly two conjugate classes of subgroups of small orders and exponentp > 2 348
45. §39. Alperin’s problem on abelian subgroups of small index 351
46. §40. On breadth and class number of p-groups 355
47. §41. Groups in which every two noncyclic subgroups of the same order have the same rank 358
48. §42. On intersections of some subgroups 362
49. §43. On 2-groups with few cyclic subgroups of given order 365
50. §44. Some characterizations of metacyclic p-groups 372
51. §45. A counting theorem for p-groups of odd order 377
52. Appendix 1. The Hall–Petrescu formula 379
53. Appendix 2. Mann’s proof of monomiality of p-groups 383
54. Appendix 3. Theorems of Isaacs on actions of groups 385
55. Appendix 4. Freiman’s number-theoretical theorems 393
56. Appendix 5. Another proof of Theorem 5.4 399
57. Appendix 6. On the order of p-groups of given derived length 401
58. Appendix 7. Relative indices of elements of p-groups 405
59. Appendix 8. p-groups withabsolutely regular Frattini subgroup 409
60. Appendix 9. On characteristic subgroups of metacyclic groups 412
61. Appendix 10. On minimal characters of p-groups 417
62. Appendix 11. On sums of degrees of irreducible characters 419
63. Appendix 12. 2-groups whose maximal cyclic subgroups of order > 2 are self-centralizing 422
64. Appendix 13. Normalizers of Sylow p-subgroups of symmetric groups 425
65. Appendix 14. 2-groups with an involution contained in only one subgroup of order 4 431
66. Appendix 15. A criterion for a group to be nilpotent 433
67. Research problems and themes I 437
68. Backmatter 480