: Yakov Berkovich, Zvonimir Janko
: Groups of Prime Power Order 3
: Walter de Gruyter GmbH& Co.KG
: 9783110254488
: De Gruyter Expositions in MathematicsISSN
: 1
: CHF 176,60
:
: Arithmetik, Algebra
: English
: 664
: Wasserzeichen
: PC/MAC/eReader/Tablet
: PDF
< PAN lang=EN>

This is the third volume of a comprehensive and elementary treatment of finitep-group theory. Topics covered in this volume: (a) impact of minimal nonabelian subgroups on the structure ofp-groups, (b) classification of groups all of whose nonnormal subgroups have the same order, (c) degrees of irreducible characters ofp-groups associated with finite algebras, (d) groups covered by few proper subgroups, (e)p-groups of element breadth 2 and subgroup breadth 1, (f) exact number of subgroups of given order in a metacyclicp-group, (g) soft subgroups, (h)p-groups with a maximal elementary abelian subgroup of orderp2, (i)p-groups generated by certain minimal nonabelian subgroups, (j)p-groups in which certain nonabelian subgroups are 2-generator.

The book contains many dozens of original exercises (with difficult exercises being solved) and a list of about 900 research problems and themes.



< >Yakov Berkovich, University of Haifa, Israel;Zvonimir Janko,Heidelberg University, Germany.
Contents6
List of definitions and notations10
Preface16
Prerequisites from Volumes 1 and 218
§93 Nonabelian 2-groups all of whose minimal nonabelian subgroups are metacyclic and have exponent 428
§94 Nonabelian 2-groups all of whose minimal nonabelian subgroups are nonmetacyclic and have exponent 435
§95 Nonabelian 2-groups of exponent 2e which have no minimal nonabelian subgroups of exponent 2e37
§96 Groups with at most two conjugate classes of nonnormal subgroups39
§97 p-groups in which some subgroups are generated by elements of order p51
§98 Nonabelian 2-groups all of whose minimal nonabelian subgroups are isomorphic to M2n+1, n51
5851
§99 2-groups with sectional rank at most 461
§100 2-groups with exactly one maximal subgroup which is neither abelian nor minimal nonabelian73
§101 p-groups G with p73
9373
§102 p-groups G with p73
§102 p-groups G with p73
10473
§103 Some results of Jonah and Konvisser120
§104 Degrees of irreducible characters of p-groups associated with finite algebras124
§105 On some special p-groups129
§106 On maximal subgroups of two-generator 2-groups137
§107 Ranks of maximal subgroups of nonmetacyclic two-generator 2-groups140
§108 p-groups with few conjugate classes of minimal nonabelian subgroups147
§109 On p-groups with metacyclic maximal subgroup without cyclic subgroup of index p149
§110 Equilibrated p-groups152
§111 Characterization of abelian and minimal nonabelian groups161
§112 Non-Dedekindian p-groups all of whose nonnormal subgroups have the same order167
§113 The class of 2-groups in §70 is not bounded175
§114 Further counting theorems179
§115 Finite p-groups all of whose maximal subgroups except one are extraspecial184
§116 Groups covered by few proper subgroups189
§117 2-groups all of whose nonnormal subgroups are either cyclic or of maximal class203
§118 Review of characterizations of p-groups with various minimal nonabelian subgroups206
§119 Review of characterizations of p-groups of maximal class212
§120 Nonabelian 2-groups such that any two distinct minimal nonabelian subgroups have cyclic intersection219
§121 p-groups of breadth 2224
§122 p-groups all of whose subgroups have normalizers of index at most p231
§123 Subgroups of finite groups generated by all elements in two shortest conjugacy classes264
§124 The number of subgroups of given order in a metacyclic p-group266
§125 p-groups G containing a maximal subgroup H all of whose subgroups are G-invariant296
§126 The existence of p-groups G1296
299296
§127 On 2-groups containing a maximal elementary abelian subgroup of order 4302
§128 The commutator subgroup of p-groups with the subgroup breadth 1304
§129 On two-generator 2-groups with exactly one maximal subgroup which is not two-generator312
§130 Soft subgroups of p-groups314
§131 p-groups with a 2-uniserial subgroup of order p319
§132 On centralizers of elements in p-groups322
§133 Class and breadth of a p-group327
§134 On p-groups with maximal elementary abelian subgroup of order p2331
§135 Finite p-groups generated by certain minimal nonabelian subgroups342
§136 p-groups in which certain proper nonabelian subgroups are two-generator355
§137 p-groups all of whose proper subgroups have its derived subgroup of order at most p365
§138 p-groups all of whose nonnormal subgroups have the smallest possible normalizer370
§139 p-groups with a noncyclic commutator group all of whose proper subgroups have a cyclic commutator group382
§140 Power automorphisms and the norm of a p-group390
§141 Nonabelian p-groups having exactly one maximal subgroup with a noncyclic center395
§142 Nonabelian p-groups all of whose nonabelian maximal subgroups are either metacyclic or minimal nonabelian397
§143 Alternate proof of the Reinhold Baer theorem on 2-groups with nonabelian norm400
§144 p-groups with small normal closures of all cyclic subgroups403
Appendix 27 Wreathed 2-groups411
Appendix 28 Nilpotent subgroups420
Appendix 29 Intersections of subgroups432
Appendix 30 Thompson’s lemmas443
Appendix 31 Nilpotent p’-subgroups of class 2 in GL(n, p)455
Appendix 32 On abelian subgroups of given exponent and small index461
Appendix 33 On Hadamard 2-groups464
Appendix 34 Isaacs–Passman’s theorem on character degrees467
Appendix 35 Groups of Frattini class 2473
Appendix 36 Hurwitz’ theorem on the composition of quadratic forms476
Appendix 37 On generalized Dedekindian groups479
Appendix 38 Some results of Blackburn and Macdonald484
Appendix 39 Some consequences of Frobenius’ normal p-complement theorem487
Appendix 40 Varia499
Appendix 41 Nonabelian 2-groups all of whose minimal nonabelian subgroups have cyclic centralizers541
Appendix 42 On lattice isomorphisms of p-groups of maximal class543
Appendix 43 Alternate proofs of two classical theorems on solvable groups and some related results546
Appendix 44 Some of Freiman’s results on finite subsets of groups with small doubling554
Research problems and themes III563
Author index657
Subject index659