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

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Frontmatter
I

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Contents
V

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List of definitions and notations
XIII
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Preface
XIX
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§ 190. On p-groups containing a subgroup of maximal class and index p
1
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§ 191. p-groups G all of whose nonnormal subgroups contain G′ in its normal closure
4
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§ 192. p-groups with all subgroups isomorphic to quotient groups
7
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§ 193. Classification of p-groups all of whose proper subgroups are s-self-dual
15
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§ 194. p-groups all of whose maximal subgroups, except one, are s-self-dual
30
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§ 195. Nonabelian p-groups all of whose subgroups are q-self-dual
33
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§ 196. A p-group with absolutely regular normalizer of some subgroup
40
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§ 197. Minimal non-q-self-dual 2-groups
43
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§ 198. Nonmetacyclic p-groups with metacyclic centralizer of an element of order p
52
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§ 199. p-groups with minimal nonabelian closures of all nonnormal abelian subgroups
56
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§ 200. The nonexistence of p-groups G all of whose minimal nonabelian subgroups intersect Z(G) trivially
61
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§ 201. Subgroups of order p^p and exponent p in p-groups with an irregular subgroup of maximal class and index > p
64
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§ 202. p-groups all of whose A2-subgroups are metacyclic
67
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§ 203. Nonabelian p-groups G in which the center of each nonabelian subgroup is contained in Z(G)
71
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§ 204. Theorem of R. van der Waal on p-groups with cyclic derived subgroup, p > 2
73
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§ 205. Maximal subgroups of A₂-groups
75
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§ 206. p-groups all of whose minimal nonabelian subgroups are pairwise nonisomorphic
90
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§ 207. Metacyclic groups of exponent p^e with a normal cyclic subgroup of order p^e
94
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§ 208. Non-Dedekindian p-groups all of whose nonnormal maximal cyclic subgroups are maximal abelian
99
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§ 209. p-groups with many minimal nonabelian subgroups, 3
101
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§ 210. A generalization of Dedekindian groups
103
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§ 211. Nonabelian p-groups generated by the centers of their maximal subgroups
108
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§ 212. Nonabelian p-groups generated by any two nonconjugate maximal abelian subgroups
110
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§ 213. p-groups with A ∩ B being maximal in A or B for any two nonincident subgroups A and B
112
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§ 214. Nonabelian p-groups with a small number of normal subgroups
117
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§ 215. Every p-group of maximal class and order ≥ p^p, p > 3, has exactly p two-generator nonabelian subgroups of index p
120
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§ 216. On the theorem of Mann about p-groups all of whose nonnormal subgroups are elementary abelian
122
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§ 217. Nonabelian p-groups all of whose elements contained in any minimal nonabelian subgroup are of breadth < 2
129
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§ 218. A nonabelian two-generator p-group in which any nonabelian epimorphic image has the cyclic center
130
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§ 219. On “large” elementary abelian subgroups in p-groups of maximal class
132
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§ 220. On metacyclic p-groups and close to them
136
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§ 221. Non-Dedekindian p-groups in which normal closures of nonnormal abelian subgroups have cyclic centers
141
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§ 222. Characterization of Dedekindian p-groups, 2
143
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§ 223. Non-Dedekindian p-groups in which the normal closure of any nonnormal cyclic subgroup is nonabelian
147
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§ 224. p-groups in which the normal closure of any cyclic subgroup is abelian
154
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§ 225. Nonabelian p-groups in which any s (a fixed s ∈ {3, . . . , p + 1}) pairwise noncommuting elements generate a group of maximal class
156
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§ 226. Noncyclic p-groups containing only one proper normal subgroup of a given order
158
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§ 227. p-groups all of whose minimal nonabelian subgroups have cyclic centralizers
161
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§ 228. Properties of metahamiltonian p-groups
163
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§ 229. p-groups all of whose cyclic subgroups of order ≥ p³ are normal
170
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§ 230. Nonabelian p-groups of exponent p^e all of whose cyclic subgroups of order p^e are normal
179
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§ 231. p-groups which are not generated by their nonnormal subgroups
185
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§ 232. Nonabelian p-groups in which any nonabelian subgroup contains its centralizer
191
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§ 233. On monotone p-groups
194
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§ 234. p-groups all of whose maximal nonnormal abelian subgroups are conjugate
196
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§ 235. On normal subgroups of capable 2-groups
197
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§ 236. Non-Dedekindian p-groups in which the normal closure of any cyclic subgroup has a cyclic center
198
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§ 237. Noncyclic p-groups all of whose nonnormal maximal cyclic subgroups are self-centralizing
199
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§ 238. Nonabelian p-groups all of whose nonabelian subgroups have a cyclic center
200
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§ 239. p-groups G all of whose cyclic subgroups are either contained in Z(G) or avoid Z(G)
202
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§ 240. p-groups G all of whose nonnormal maximal cyclic subgroups are conjugate
203
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§ 241. Non-Dedekindian p-groups with a normal intersection of any two nonincident subgroups
205
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§ 242. Non-Dedekindian p-groups in which the normal closures of all nonnormal subgroups coincide
207
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§ 243. Nonabelian p-groups G with Φ(H) = H′ for all nonabelian H ≤ G
210
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§ 244. p-groups in which any two distinct maximal nonnormal subgroups intersect in a subgroup of order ≤ p
211
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§ 245. On 2-groups saturated by nonabelian Dedekindian subgroups
212
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§ 246. Non-Dedekindian p-groups with many normal subgroups
226
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§ 247. Nonabelian p-groups all of whose metacyclic sections are abelian
227
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§ 248. Non-Dedekindian p-groups G such that H^G = HZ(G) for all nonnormal H < G
228
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§ 249. Nonabelian p-groups G with A ∩ B = Z(G) for any two distinct maximal abelian subgroups A and B
229
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§ 250. On the number of minimal nonabelian subgroups in a nonabelian p-group
230
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§ 251. p-groups all of whose minimal nonabelian subgroups are isolated
236
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§ 252. Nonabelian p-groups all of whose maximal abelian subgroups are isolated
242
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§ 253. Maximal abelian subgroups of p-groups, 2
244
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§ 254. On p-groups with many isolated maximal abelian subgroups
246
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§ 255. Maximal abelian subgroups of p-groups, 3
248
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§ 256. A problem of D. R. Hughes for 3-groups
249
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Appendix 58 – Appendix 109
251
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Research problems and themes V
361
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Bibliography
391
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Author index
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Subject index
407
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Backmatter
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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Walter de Gruyter GmbH
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Volume 5

Overview

About this book

Author / Editor information

Reviews

Topics

Table of contents

Frontmatter

Contents

List of definitions and notations

Preface

§ 190. On p-groups containing a subgroup of maximal class and index p

§ 191. p-groups G all of whose nonnormal subgroups contain G′ in its normal closure

§ 192. p-groups with all subgroups isomorphic to quotient groups

§ 193. Classification of p-groups all of whose proper subgroups are s-self-dual

§ 194. p-groups all of whose maximal subgroups, except one, are s-self-dual

§ 195. Nonabelian p-groups all of whose subgroups are q-self-dual

§ 196. A p-group with absolutely regular normalizer of some subgroup

§ 197. Minimal non-q-self-dual 2-groups

§ 198. Nonmetacyclic p-groups with metacyclic centralizer of an element of order p

§ 199. p-groups with minimal nonabelian closures of all nonnormal abelian subgroups

§ 200. The nonexistence of p-groups G all of whose minimal nonabelian subgroups intersect Z(G) trivially

§ 201. Subgroups of order pp and exponent p in p-groups with an irregular subgroup of maximal class and index > p

§ 202. p-groups all of whose A2-subgroups are metacyclic

§ 203. Nonabelian p-groups G in which the center of each nonabelian subgroup is contained in Z(G)

§ 204. Theorem of R. van der Waal on p-groups with cyclic derived subgroup, p > 2

§ 205. Maximal subgroups of A2-groups

§ 206. p-groups all of whose minimal nonabelian subgroups are pairwise nonisomorphic

§ 207. Metacyclic groups of exponent pe with a normal cyclic subgroup of order pe

§ 208. Non-Dedekindian p-groups all of whose nonnormal maximal cyclic subgroups are maximal abelian

§ 209. p-groups with many minimal nonabelian subgroups, 3

§ 210. A generalization of Dedekindian groups

§ 211. Nonabelian p-groups generated by the centers of their maximal subgroups

§ 212. Nonabelian p-groups generated by any two nonconjugate maximal abelian subgroups

§ 213. p-groups with A ∩ B being maximal in A or B for any two nonincident subgroups A and B

§ 214. Nonabelian p-groups with a small number of normal subgroups

§ 215. Every p-group of maximal class and order ≥ pp, p > 3, has exactly p two-generator nonabelian subgroups of index p

§ 216. On the theorem of Mann about p-groups all of whose nonnormal subgroups are elementary abelian

§ 217. Nonabelian p-groups all of whose elements contained in any minimal nonabelian subgroup are of breadth < 2

§ 218. A nonabelian two-generator p-group in which any nonabelian epimorphic image has the cyclic center

§ 219. On “large” elementary abelian subgroups in p-groups of maximal class

§ 220. On metacyclic p-groups and close to them

§ 221. Non-Dedekindian p-groups in which normal closures of nonnormal abelian subgroups have cyclic centers

§ 222. Characterization of Dedekindian p-groups, 2

§ 223. Non-Dedekindian p-groups in which the normal closure of any nonnormal cyclic subgroup is nonabelian

§ 224. p-groups in which the normal closure of any cyclic subgroup is abelian

§ 225. Nonabelian p-groups in which any s (a fixed s ∈ {3, . . . , p + 1}) pairwise noncommuting elements generate a group of maximal class

§ 226. Noncyclic p-groups containing only one proper normal subgroup of a given order

§ 227. p-groups all of whose minimal nonabelian subgroups have cyclic centralizers

§ 228. Properties of metahamiltonian p-groups

§ 229. p-groups all of whose cyclic subgroups of order ≥ p3 are normal

§ 230. Nonabelian p-groups of exponent pe all of whose cyclic subgroups of order pe are normal

§ 231. p-groups which are not generated by their nonnormal subgroups

§ 232. Nonabelian p-groups in which any nonabelian subgroup contains its centralizer

§ 233. On monotone p-groups

§ 234. p-groups all of whose maximal nonnormal abelian subgroups are conjugate

§ 235. On normal subgroups of capable 2-groups

§ 236. Non-Dedekindian p-groups in which the normal closure of any cyclic subgroup has a cyclic center

§ 237. Noncyclic p-groups all of whose nonnormal maximal cyclic subgroups are self-centralizing

§ 238. Nonabelian p-groups all of whose nonabelian subgroups have a cyclic center

§ 239. p-groups G all of whose cyclic subgroups are either contained in Z(G) or avoid Z(G)

§ 240. p-groups G all of whose nonnormal maximal cyclic subgroups are conjugate

§ 241. Non-Dedekindian p-groups with a normal intersection of any two nonincident subgroups

§ 242. Non-Dedekindian p-groups in which the normal closures of all nonnormal subgroups coincide

§ 243. Nonabelian p-groups G with Φ(H) = H′ for all nonabelian H ≤ G

§ 244. p-groups in which any two distinct maximal nonnormal subgroups intersect in a subgroup of order ≤ p

§ 245. On 2-groups saturated by nonabelian Dedekindian subgroups

§ 246. Non-Dedekindian p-groups with many normal subgroups

§ 247. Nonabelian p-groups all of whose metacyclic sections are abelian

§ 248. Non-Dedekindian p-groups G such that HG = HZ(G) for all nonnormal H < G

§ 249. Nonabelian p-groups G with A ∩ B = Z(G) for any two distinct maximal abelian subgroups A and B

§ 250. On the number of minimal nonabelian subgroups in a nonabelian p-group

§ 251. p-groups all of whose minimal nonabelian subgroups are isolated

§ 252. Nonabelian p-groups all of whose maximal abelian subgroups are isolated

§ 253. Maximal abelian subgroups of p-groups, 2

§ 254. On p-groups with many isolated maximal abelian subgroups

§ 255. Maximal abelian subgroups of p-groups, 3

§ 256. A problem of D. R. Hughes for 3-groups

Appendix 58 – Appendix 109

Research problems and themes V

§ 201. Subgroups of order p^p and exponent p in p-groups with an irregular subgroup of maximal class and index > p

§ 205. Maximal subgroups of A₂-groups

§ 207. Metacyclic groups of exponent p^e with a normal cyclic subgroup of order p^e

§ 215. Every p-group of maximal class and order ≥ p^p, p > 3, has exactly p two-generator nonabelian subgroups of index p

§ 229. p-groups all of whose cyclic subgroups of order ≥ p³ are normal

§ 230. Nonabelian p-groups of exponent p^e all of whose cyclic subgroups of order p^e are normal

§ 248. Non-Dedekindian p-groups G such that H^G = HZ(G) for all nonnormal H < G