Groups of Prime Power Order 3 (eBook)

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Yakov Berkovich, University of Haifa, Israel; Zvonimir Janko,Heidelberg University, Germany.

Contents 6
List of definitions and notations 10
Preface 16
Prerequisites from Volumes 1 and 2 18
§93 Nonabelian 2-groups all of whose minimal nonabelian subgroups are metacyclic and have exponent 4 28
§94 Nonabelian 2-groups all of whose minimal nonabelian subgroups are nonmetacyclic and have exponent 4 35
§95 Nonabelian 2-groups of exponent 2e which have no minimal nonabelian subgroups of exponent 2e 37
§96 Groups with at most two conjugate classes of nonnormal subgroups 39
§97 p-groups in which some subgroups are generated by elements of order p 51
§98 Nonabelian 2-groups all of whose minimal nonabelian subgroups are isomorphic to M2n+1, n> 3 fixed
§99 2-groups with sectional rank at most 4 61
§100 2-groups with exactly one maximal subgroup which is neither abelian nor minimal nonabelian 73
§101 p-groups G with p > 2 and d(G) = 2 having exactly one maximal subgroup which is neither abelian nor minimal nonabelian
§102 p-groups G with p > 2 and d(G) >
§103 Some results of Jonah and Konvisser 120
§104 Degrees of irreducible characters of p-groups associated with finite algebras 124
§105 On some special p-groups 129
§106 On maximal subgroups of two-generator 2-groups 137
§107 Ranks of maximal subgroups of nonmetacyclic two-generator 2-groups 140
§108 p-groups with few conjugate classes of minimal nonabelian subgroups 147
§109 On p-groups with metacyclic maximal subgroup without cyclic subgroup of index p 149
§110 Equilibrated p-groups 152
§111 Characterization of abelian and minimal nonabelian groups 161
§112 Non-Dedekindian p-groups all of whose nonnormal subgroups have the same order 167
§113 The class of 2-groups in §70 is not bounded 175
§114 Further counting theorems 179
§115 Finite p-groups all of whose maximal subgroups except one are extraspecial 184
§116 Groups covered by few proper subgroups 189
§117 2-groups all of whose nonnormal subgroups are either cyclic or of maximal class 203
§118 Review of characterizations of p-groups with various minimal nonabelian subgroups 206
§119 Review of characterizations of p-groups of maximal class 212
§120 Nonabelian 2-groups such that any two distinct minimal nonabelian subgroups have cyclic intersection 219
§121 p-groups of breadth 2 224
§122 p-groups all of whose subgroups have normalizers of index at most p 231
§123 Subgroups of finite groups generated by all elements in two shortest conjugacy classes 264
§124 The number of subgroups of given order in a metacyclic p-group 266
§125 p-groups G containing a maximal subgroup H all of whose subgroups are G-invariant 296
§126 The existence of p-groups G1 < G such that Aut(G1) Aut(G)
§127 On 2-groups containing a maximal elementary abelian subgroup of order 4 302
§128 The commutator subgroup of p-groups with the subgroup breadth 1 304
§129 On two-generator 2-groups with exactly one maximal subgroup which is not two-generator 312
§130 Soft subgroups of p-groups 314
§131 p-groups with a 2-uniserial subgroup of order p 319
§132 On centralizers of elements in p-groups 322
§133 Class and breadth of a p-group 327
§134 On p-groups with maximal elementary abelian subgroup of order p2 331
§135 Finite p-groups generated by certain minimal nonabelian subgroups 342
§136 p-groups in which certain proper nonabelian subgroups are two-generator 355
§137 p-groups all of whose proper subgroups have its derived subgroup of order at most p 365
§138 p-groups all of whose nonnormal subgroups have the smallest possible normalizer 370
§139 p-groups with a noncyclic commutator group all of whose proper subgroups have a cyclic commutator group 382
§140 Power automorphisms and the norm of a p-group 390
§141 Nonabelian p-groups having exactly one maximal subgroup with a noncyclic center 395
§142 Nonabelian p-groups all of whose nonabelian maximal subgroups are either metacyclic or minimal nonabelian 397
§143 Alternate proof of the Reinhold Baer theorem on 2-groups with nonabelian norm 400
§144 p-groups with small normal closures of all cyclic subgroups 403
Appendix 27 Wreathed 2-groups 411
Appendix 28 Nilpotent subgroups 420
Appendix 29 Intersections of subgroups 432
Appendix 30 Thompson’s lemmas 443
Appendix 31 Nilpotent p’-subgroups of class 2 in GL(n, p) 455
Appendix 32 On abelian subgroups of given exponent and small index 461
Appendix 33 On Hadamard 2-groups 464
Appendix 34 Isaacs–Passman’s theorem on character degrees 467
Appendix 35 Groups of Frattini class 2 473
Appendix 36 Hurwitz’ theorem on the composition of quadratic forms 476
Appendix 37 On generalized Dedekindian groups 479
Appendix 38 Some results of Blackburn and Macdonald 484
Appendix 39 Some consequences of Frobenius’ normal p-complement theorem 487
Appendix 40 Varia 499
Appendix 41 Nonabelian 2-groups all of whose minimal nonabelian subgroups have cyclic centralizers 541
Appendix 42 On lattice isomorphisms of p-groups of maximal class 543
Appendix 43 Alternate proofs of two classical theorems on solvable groups and some related results 546
Appendix 44 Some of Freiman’s results on finite subsets of groups with small doubling 554
Research problems and themes III 563
Author index 657
Subject index 659