§ 117 2-groups all of whose nonnormal subgroups are either cyclic or of maximal class

Chapters in this book

1. Frontmatter i
2. Contents v
3. List of definitions and notations ix
4. Preface xv
5. Prerequisites from Volumes 1 and 2 xvii
6. § 93 Nonabelian 2-groups all of whose minimal nonabelian subgroups are metacyclic and have exponent 4 1
7. § 94 Nonabelian 2-groups all of whose minimal nonabelian subgroups are nonmetacyclic and have exponent 4 8
8. § 95 Nonabelian 2-groups of exponent 2^e which have no minimal nonabelian subgroups of exponent 2^e 10
9. § 96 Groups with at most two conjugate classes of nonnormal subgroups 12
10. § 97 p-groups in which some subgroups are generated by elements of order p 24
11. § 98 Nonabelian 2-groups all of whose minimal nonabelian subgroups are isomorphic to M₂n+1 , n ≥ 3 fixed 31
12. § 99 2-groups with sectional rank at most 4 34
13. § 100 2-groups with exactly one maximal subgroup which is neither abelian nor minimal nonabelian 46
14. § 101 p-groups G with p > 2 and d(G)= 2 having exactly one maximal subgroup which is neither abelian nor minimal nonabelian 66
15. § 102 p-groups G with p > 2 and d(G) > 2 having exactly one maximal subgroup which is neither abelian nor minimal nonabelian 77
16. § 103 Some results of Jonah and Konvisser 93
17. § 104 Degrees of irreducible characters of p-groups associated with finite algebras 97
18. § 105 On some special p-groups 102
19. § 106 On maximal subgroups of two-generator 2-groups 110
20. § 107 Ranks of maximal subgroups of nonmetacyclic two-generator 2-groups 113
21. § 108 p-groups with few conjugate classes of minimal nonabelian subgroups 120
22. § 109 On p-groups with metacyclic maximal subgroup without cyclic subgroup of index p 122
23. § 110 Equilibrated p-groups 125
24. § 111 Characterization of abelian and minimal nonabelian groups 134
25. § 112 Non-Dedekindian p-groups all of whose nonnormal subgroups have the same order 140
26. § 113 The class of 2-groups in § 70 is not bounded 148
27. § 114 Further counting theorems 152
28. § 115 Finite p-groups all of whose maximal subgroups except one are extraspecial 157
29. § 116 Groups covered by few proper subgroups 162
30. § 117 2-groups all of whose nonnormal subgroups are either cyclic or of maximal class 176
31. § 118 Review of characterizations of p-groups with various minimal nonabelian subgroups 179
32. § 119 Review of characterizations of p-groups of maximal class 185
33. § 120 Nonabelian 2-groups such that any two distinct minimal nonabelian subgroups have cyclic intersection 192
34. § 121 p-groups of breadth 2 197
35. § 122 p-groups all of whose subgroups have normalizers of index at most p 204
36. § 123 Subgroups of finite groups generated by all elements in two shortest conjugacy classes 237
37. § 124 The number of subgroups of given order in a metacyclic p-group 239
38. § 125 p-groups G containing a maximal subgroup H all of whose subgroups are G-invariant 269
39. § 126 The existence of p-groups G1 < G such that Aut(G1) ≈ Aut(G) 272
40. § 127 On 2-groups containing a maximal elementary abelian subgroup of order 4 275
41. § 128 The commutator subgroup of p-groups with the subgroup breadth 1 277
42. § 129 On two-generator 2-groups with exactly one maximal subgroup which is not two-generator 285
43. § 130 Soft subgroups of p-groups 287
44. § 131 p-groups with a 2-uniserial subgroup of order p 292
45. § 132 On centralizers of elements in p-groups 295
46. § 133 Class and breadth of a p-group 300
47. § 134 On p-groups with maximal elementary abelian subgroup of order p² 304
48. § 135 Finite p-groups generated by certain minimal nonabelian subgroups 315
49. § 136 p-groups in which certain proper nonabelian subgroups are two-generator 328
50. § 137 p-groups all of whose proper subgroups have its derived subgroup of order at most p 338
51. § 138 p-groups all of whose nonnormal subgroups have the smallest possible normalizer 343
52. § 139 p-groups with a noncyclic commutator group all of whose proper subgroups have a cyclic commutator group 355
53. § 140 Power automorphisms and the norm of a p-group 363
54. § 141 Nonabelian p-groups having exactly one maximal subgroup with a noncyclic center 368
55. § 142 Nonabelian p-groups all of whose nonabelian maximal subgroups are either metacyclic or minimal nonabelian 370
56. § 143 Alternate proof of the Reinhold Baer theorem on 2-groups with nonabelian norm 373
57. § 144 p-groups with small normal closures of all cyclic subgroups 376
58. A.27 Wreathed 2-groups 384
59. A.28 Nilpotent subgroups 393
60. A.29 Intersections of subgroups 405
61. A.30 Thompson’s lemmas 416
62. A.31 Nilpotent p'-subgroups of class 2 in GL(n, p) 428
63. A.32 On abelian subgroups of given exponent and small index 434
64. A.33 On Hadamard 2-groups 437
65. A.34 Isaacs–Passman’s theorem on character degrees 440
66. A.35 Groups of Frattini class 2 446
67. A.36 Hurwitz’ theorem on the composition of quadratic forms 449
68. A.37 On generalized Dedekindian groups 452
69. A.38 Some results of Blackburn and Macdonald 457
70. A.39 Some consequences of Frobenius’ normal p-complement theorem 460
71. A.40 Varia 472
72. A.41 Nonabelian 2-groups all of whose minimal nonabelian subgroups have cyclic centralizers 514
73. A.42 On lattice isomorphisms of p-groups of maximal class 516
74. A.43 Alternate proofs of two classical theorems on solvable groups and some related results 519
75. A.44 Some of Freiman’s results on finite subsets of groups with small doubling 527
76. Research problems and themes III 536
77. Author index 630
78. Subject index 632