Volume 5

Yakov G. Berkovich; Zvonimir Janko

Book

Volume 5

Yakov G. Berkovich and Zvonimir Janko

Part of the multi-volume work

Groups of Prime Power Order

Language: English

Published/Copyright: 2016

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This book is in the series

Volume 62 | De Gruyter Expositions in Mathematics

About this book

This is the fifth volume of a comprehensive and elementary treatment of finite p-group theory. Topics covered in this volume include theory of linear algebras and Lie algebras.

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.

Reviews

"Seeing over all five volumes now, one is able to conclude that the authors have enriched the mathematical literature with an important topic." Zentralblatt für Mathematik

Topics

Mathematics Algebra and Number Theory

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

Publishing information

Pages and Images/Illustrations in book

eBook published on:

January 15, 2016

eBook ISBN:

9783110295351

Hardcover published on:

January 29, 2016

Hardcover ISBN:

9783110295344

Pages and Images/Illustrations in book

Front matter:

20

Main content:

413

https://doi.org/10.1515/9783110295351

eBook ISBN: 9783110295351

Hardcover ISBN: 9783110295344

Keywords for this book

Cyclic Order; Group theory; Prime

Audience(s) for this book

Researchers, Graduate Students of Mathematics; Academic Libraries

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Manufacturer information:
Walter de Gruyter GmbH
Genthiner Straße 13
10785 Berlin
productsafety@degruyterbrill.com

Volume 5

Overview

About this book

Author / Editor information

Reviews

Topics

Table of contents

Bibliographic data