§ 284 Nonabelian p-groups, p > 2, of exponent > p² all of whose minimal nonabelian subgroups are of order p³

Chapters in this book

1. Frontmatter I
2. Contents V
3. List of definitions and notations XIII
4. Preface XIX
5. § 257 Nonabelian p-groups with exactly one minimal nonabelian subgroup of exponent > p 1
6. § 258 2-groups with some prescribed minimal nonabelian subgroups 6
7. § 259 Nonabelian p-groups, p > 2, all of whose minimal nonabelian subgroups are isomorphic to M_p³ 13
8. § 260 p-groups with many modular subgroups M_pⁿ 15
9. § 261 Nonabelian p-groups of exponent > p with a small number of maximal abelian subgroups of exponent > p 18
10. § 262 Nonabelian p-groups all of whose subgroups are powerful 24
11. § 263 Nonabelian 2-groups G with C_G(x) ≤ H for all H ∈ Γ₁ and x ∈ H − Z(G) 25
12. § 264 Nonabelian 2-groups of exponent ≥ 16 all of whose minimal nonabelian subgroups, except one, have order 8 26
13. § 265 p-groups all of whose regular subgroups are either absolutely regular or of exponent p 29
14. § 266 Nonabelian p-groups in which any two distinct minimal nonabelian subgroups with a nontrivial intersection are non-isomorphic 33
15. § 267 Thompson’s A × B lemma 35
16. § 268 On automorphisms of some p-groups 36
17. § 269 On critical subgroups of p-groups 47
18. § 270 p-groups all of whose A_k-subgroups for a fixed k > 1 are metacyclic 50
19. § 271 Two theorems of Blackburn 55
20. § 272 Nonabelian p-groups all of whose maximal abelian subgroups, except one, are either cyclic or elementary abelian 57
21. § 273 Nonabelian p-groups all of whose noncyclic maximal abelian subgroups are elementary abelian 58
22. § 274 Non-Dedekindian p-groups in which any two nonnormal subgroups normalize each other 59
23. § 275 Nonabelian p-groups with exactly p normal closures of minimal nonabelian subgroups 61
24. § 276 2-groups all of whose maximal subgroups, except one, are Dedekindian 63
25. § 277 p-groups with exactly two conjugate classes of nonnormal maximal cyclic subgroups 67
26. § 278 Nonmetacyclic p-groups all of whose maximal metacyclic subgroups have index p 68
27. § 279 Subgroup characterization of some p-groups of maximal class and close to them 70
28. § 280 Nonabelian p-groups all of whose maximal subgroups, except one, are minimal nonmetacyclic 73
29. § 281 Nonabelian p-groups in which any two distinct minimal nonabelian subgroups have a cyclic intersection 80
30. § 282 p-groups with large normal closures of nonnormal subgroups 83
31. § 283 Nonabelian p-groups with many cyclic centralizers 86
32. § 284 Nonabelian p-groups, p > 2, of exponent > p² all of whose minimal nonabelian subgroups are of order p³ 87
33. § 285 A generalization of Lemma 57.1 88
34. § 286 Groups ofexponent p with many normal subgroups 90
35. § 287 p-groups in which the intersection of any two nonincident subgroups is normal 92
36. § 288 Nonabelian p-groups in which for every minimal nonabelian M < G and x ∈ G − M, we have C_M(x) = Z(M) 97
37. § 289 Non-Dedekindian p-groups all of whose maximal nonnormal subgroups are conjugate 98
38. § 290 Non-Dedekindian p-groups G with a noncyclic proper subgroup H such that each subgroup which is nonincident with H is normal in G 99
39. § 291 Nonabelian p-groups which are generated by a fixed maximal cyclic subgroup and any minimal nonabelian subgroup 100
40. § 292 Nonabelian p-groups generated by any two non-conjugate minimal nonabelian subgroups 101
41. § 293 Exercises 102
42. § 294 p-groups, p > 2, whose Frattini subgroup is nonabelian metacyclic 174
43. § 295 Any irregular p-group contains a non-isolated maximal regular subgroup 176
44. § 296 Non-Dedekindian p-groups all of whose nonnormal maximal cyclic subgroups are C-equivalent 178
45. § 298 Non-Dedekindian p-groups all of whose subgroups of order ≤ p^s (s ≥ 1 fixed) are normal 182
46. § 299 On p’-automorphisms of p-groups 184
47. § 300 On p-groups all of whose maximal subgroups of exponent p are normal and have order p^p 185
48. § 301 p-groups of exponent > p containing < p maximal abelian subgroups of exponent > p 188
49. § 302 Alternate proof of Theorem 109.1 190
50. § 303 Nonabelian p-groups of order > p⁴ all of whose subgroups of order p⁴ are isomorphic 192
51. § 304 Non-Dedekindian p-groups in which each nonnormal subgroup has a cyclic complement in its normalizer 196
52. § 305 Nonabelian p-groups G all of whose minimal nonabelian subgroups M satisfy Z(M) ≤ Z(G) 198
53. § 306 Nonabelian 2-groups all of whose maximal subgroups, except one, are quasi-Hamiltonian or abelian 200
54. § 307 Nonabelian p-groups, p > 2, all of whose maximal subgroups, except one, are quasi-Hamiltonian or abelian 205
55. § 308 Nonabelian p-groups with an elementary abelian intersection of any two distinct maximal abelian subgroups 210
56. § 309 Minimal non-p-central p-groups 211
57. § 310 Nonabelian p-groups in which each element in any minimal nonabelian subgroup is half-central 213
58. § 311 Nonabelian p-groups G of exponent p in which C_G(x) = <x> G for all noncentral x ∈ G 214
59. § 312 Nonabelian 2-groups all of whose minimal nonabelian subgroups, except one, are isomorphic to M₂(2, 2) = <a, b / a⁴ = b⁴ = 1, a^b = a⁻¹> 215
60. § 313 Non-Dedekindian 2-groups all of whose maximal Dedekindian subgroups have index 2 223
61. § 314 Theorem of Glauberman–Mazza on p-groups with a nonnormal maximal elementary abelian subgroup of order p² 224
62. § 315 p-groups with some non-p-central maximal subgroups 227
63. § 316 Nonabelian p-groups, p > 2, of exponent > p³ all of whose minimal nonabelian subgroups, except one, have order p³ 228
64. § 317 Nonabelian p-groups, p > 2, all of whose minimal nonabelian subgroups are isomorphic to M_p(2, 2) 231
65. § 318 Nonabelian p-groups, p > 2, of exponent > p² all of whose minimal nonabelian subgroups, except one, are isomorphic to M_p(2, 2) 233
66. § 319 A new characterization of p-central p-groups 236
67. § 320 Nonabelian p-groups with exactly one non-p-central minimal nonabelian subgroup 237
68. § 321 Nonabelian p-groups G in which each element in G − Φ(G) is half-central 239
69. § 322 Nonabelian p-groups G such that C_G(H) = Z(G) for any nonabelian H ≤ G 240
70. § 323 Nonabelian p-groups that are not generated by its noncyclic abelian subgroups 241
71. § 324 A separation of metacyclic and nonmetacyclic minimal nonabelian subgroups in nonabelian p-groups 242
72. § 325 p-groups which are not generated by their nonnormal subgroups, 2 243
73. § 326 Nonabelian p-groups all of whose maximal abelian subgroups are normal 244
74. Appendix 110 Non-absolutely regular p-groups all of whose maximal absolutely regular subgroups have index p 245
75. Appendix 111 Nonabelian p-groups of exponent > p all of whose maximal abelian subgroups of exponent > p are isolated 247
76. Appendix 112 Metacyclic p-groups with an abelian maximal subgroup 249
77. Appendix 113 Nonabelian p-groups with a cyclic intersection of any two distinct maximal abelian subgroups 251
78. Appendix 114 An analog of Thompson’s dihedral lemma 253
79. Appendix 115 Some results from Thompson’ papers and the Odd Order paper 255
80. Appendix 116 On normal subgroups of a p-group 257
81. Appendix 117 Theorem of Mann 263
82. Appendix 118 On p-groups with given isolated subgroups 264
83. Appendix 119 Two-generator normal subgroups of a p-group G that contained in Φ(G) are metacyclic 269
84. Appendix 120 Alternate proofs of some counting theorems 270
85. Appendix 121 On p-groups of maximal class 275
86. Appendix 122 Criteria of regularity 283
87. Appendix 123 Nonabelian p-groups in which any two nonincident subgroups have an abelian intersection 285
88. Appendix 124 Characterizations of the p-groups of maximal class and the primary ECF-groups 287
89. Appendix 125 Nonabelian p-groups all of whose proper nonabelian subgroups have exponent p 289
90. Appendix 126 On p-groups with abelian automorphism groups 291
91. Appendix 127 Alternate proof of Proposition 1.23 293
92. Appendix 128 Alternate proof of the theorem of Passman on p-groups all of whose subgroups of order ≤ p^s (s ≥ 1 is fixed) are normal 294
93. Appendix 129 Alternate proofs of Theorems 309.1 and 309.2 on minimal non-p-central p-groups 297
94. Appendix 130 Non-Dedekindian p-groups all of whose nonnormal maximal cyclic subgroups are conjugate 299
95. Appendix 131 A characterization of some 3-groups of maximal class 301
96. Appendix 132 Alternate approach to classification of minimal non-p-central p-groups 303
97. Appendix 133 Nonabelian p-groups all of whose minimal nonabelian subgroups are isomorphic to Mp(n, n) or M_p(n, n, 1) for a fixed natural n > 1 305
98. Appendix 134 On irregular p-groups G = Ω1(G) without subgroup of order p^p+1 and exponent p 308
99. Appendix 135 Nonabelian 2-groups of given order with maximal possible number of involutions 311
100. Appendix 136 On metacyclic p-groups 315
101. Appendix 137 Alternate proof of Lemma 207.1 317
102. Appendix 138 Subgroup characterization of a p-group of maximal class with an abelian subgroup of index p 319
103. Research problems and themes VI 321
104. Bibliography 365
105. Author index 377
106. Subject index 379