§ 207. Metacyclic groups of exponent p^e with a normal cyclic subgroup of order p^e

Chapters in this book

1. Frontmatter I
2. Contents V
3. List of definitions and notations XIII
4. Preface XIX
5. § 190. On p-groups containing a subgroup of maximal class and index p 1
6. § 191. p-groups G all of whose nonnormal subgroups contain G′ in its normal closure 4
7. § 192. p-groups with all subgroups isomorphic to quotient groups 7
8. § 193. Classification of p-groups all of whose proper subgroups are s-self-dual 15
9. § 194. p-groups all of whose maximal subgroups, except one, are s-self-dual 30
10. § 195. Nonabelian p-groups all of whose subgroups are q-self-dual 33
11. § 196. A p-group with absolutely regular normalizer of some subgroup 40
12. § 197. Minimal non-q-self-dual 2-groups 43
13. § 198. Nonmetacyclic p-groups with metacyclic centralizer of an element of order p 52
14. § 199. p-groups with minimal nonabelian closures of all nonnormal abelian subgroups 56
15. § 200. The nonexistence of p-groups G all of whose minimal nonabelian subgroups intersect Z(G) trivially 61
16. § 201. Subgroups of order p^p and exponent p in p-groups with an irregular subgroup of maximal class and index > p 64
17. § 202. p-groups all of whose A2-subgroups are metacyclic 67
18. § 203. Nonabelian p-groups G in which the center of each nonabelian subgroup is contained in Z(G) 71
19. § 204. Theorem of R. van der Waal on p-groups with cyclic derived subgroup, p > 2 73
20. § 205. Maximal subgroups of A₂-groups 75
21. § 206. p-groups all of whose minimal nonabelian subgroups are pairwise nonisomorphic 90
22. § 207. Metacyclic groups of exponent p^e with a normal cyclic subgroup of order p^e 94
23. § 208. Non-Dedekindian p-groups all of whose nonnormal maximal cyclic subgroups are maximal abelian 99
24. § 209. p-groups with many minimal nonabelian subgroups, 3 101
25. § 210. A generalization of Dedekindian groups 103
26. § 211. Nonabelian p-groups generated by the centers of their maximal subgroups 108
27. § 212. Nonabelian p-groups generated by any two nonconjugate maximal abelian subgroups 110
28. § 213. p-groups with A ∩ B being maximal in A or B for any two nonincident subgroups A and B 112
29. § 214. Nonabelian p-groups with a small number of normal subgroups 117
30. § 215. Every p-group of maximal class and order ≥ p^p, p > 3, has exactly p two-generator nonabelian subgroups of index p 120
31. § 216. On the theorem of Mann about p-groups all of whose nonnormal subgroups are elementary abelian 122
32. § 217. Nonabelian p-groups all of whose elements contained in any minimal nonabelian subgroup are of breadth < 2 129
33. § 218. A nonabelian two-generator p-group in which any nonabelian epimorphic image has the cyclic center 130
34. § 219. On “large” elementary abelian subgroups in p-groups of maximal class 132
35. § 220. On metacyclic p-groups and close to them 136
36. § 221. Non-Dedekindian p-groups in which normal closures of nonnormal abelian subgroups have cyclic centers 141
37. § 222. Characterization of Dedekindian p-groups, 2 143
38. § 223. Non-Dedekindian p-groups in which the normal closure of any nonnormal cyclic subgroup is nonabelian 147
39. § 224. p-groups in which the normal closure of any cyclic subgroup is abelian 154
40. § 225. Nonabelian p-groups in which any s (a fixed s ∈ {3, . . . , p + 1}) pairwise noncommuting elements generate a group of maximal class 156
41. § 226. Noncyclic p-groups containing only one proper normal subgroup of a given order 158
42. § 227. p-groups all of whose minimal nonabelian subgroups have cyclic centralizers 161
43. § 228. Properties of metahamiltonian p-groups 163
44. § 229. p-groups all of whose cyclic subgroups of order ≥ p³ are normal 170
45. § 230. Nonabelian p-groups of exponent p^e all of whose cyclic subgroups of order p^e are normal 179
46. § 231. p-groups which are not generated by their nonnormal subgroups 185
47. § 232. Nonabelian p-groups in which any nonabelian subgroup contains its centralizer 191
48. § 233. On monotone p-groups 194
49. § 234. p-groups all of whose maximal nonnormal abelian subgroups are conjugate 196
50. § 235. On normal subgroups of capable 2-groups 197
51. § 236. Non-Dedekindian p-groups in which the normal closure of any cyclic subgroup has a cyclic center 198
52. § 237. Noncyclic p-groups all of whose nonnormal maximal cyclic subgroups are self-centralizing 199
53. § 238. Nonabelian p-groups all of whose nonabelian subgroups have a cyclic center 200
54. § 239. p-groups G all of whose cyclic subgroups are either contained in Z(G) or avoid Z(G) 202
55. § 240. p-groups G all of whose nonnormal maximal cyclic subgroups are conjugate 203
56. § 241. Non-Dedekindian p-groups with a normal intersection of any two nonincident subgroups 205
57. § 242. Non-Dedekindian p-groups in which the normal closures of all nonnormal subgroups coincide 207
58. § 243. Nonabelian p-groups G with Φ(H) = H′ for all nonabelian H ≤ G 210
59. § 244. p-groups in which any two distinct maximal nonnormal subgroups intersect in a subgroup of order ≤ p 211
60. § 245. On 2-groups saturated by nonabelian Dedekindian subgroups 212
61. § 246. Non-Dedekindian p-groups with many normal subgroups 226
62. § 247. Nonabelian p-groups all of whose metacyclic sections are abelian 227
63. § 248. Non-Dedekindian p-groups G such that H^G = HZ(G) for all nonnormal H < G 228
64. § 249. Nonabelian p-groups G with A ∩ B = Z(G) for any two distinct maximal abelian subgroups A and B 229
65. § 250. On the number of minimal nonabelian subgroups in a nonabelian p-group 230
66. § 251. p-groups all of whose minimal nonabelian subgroups are isolated 236
67. § 252. Nonabelian p-groups all of whose maximal abelian subgroups are isolated 242
68. § 253. Maximal abelian subgroups of p-groups, 2 244
69. § 254. On p-groups with many isolated maximal abelian subgroups 246
70. § 255. Maximal abelian subgroups of p-groups, 3 248
71. § 256. A problem of D. R. Hughes for 3-groups 249
72. Appendix 58 – Appendix 109 251
73. Research problems and themes V 361
74. Bibliography 391
75. Author index 405
76. Subject index 407
77. Backmatter 413