Published by De Gruyter 2011

Volume 3

Yakov Berkovich and Zvonimir Janko

Part of the multi-volume work Groups of Prime Power Order

Volume 56 in the series De Gruyter Expositions in Mathematics

About this book

This is the third volume of a comprehensive and elementary treatment of finite p-group theory. Topics covered in this volume:

impact of minimal nonabelian subgroups on the structure of p-groups,
classification of groups all of whose nonnormal subgroups have the same order,
degrees of irreducible characters of p-groups associated with finite algebras,
groups covered by few proper subgroups,
p-groups of element breadth 2 and subgroup breadth 1,
exact number of subgroups of given order in a metacyclic p-group,
soft subgroups,
p-groups with a maximal elementary abelian subgroup of order p²,
p-groups generated by certain minimal nonabelian subgroups,
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.

Author / Editor information

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

Topics

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

(Deutsch)

Language: English
Publisher: De Gruyter
Audience: Researchers, Graduate Students of Mathematics; Academic Libraries
Pages
- Front matter: 25
- Main content: 639
Keywords: Group Theory; Order; Primes

eBook

Published: June 30, 2011
ISBN: 9783110254488

Hardcover

Published: June 16, 2011
ISBN: 9783110207170