Groups of prime power order. Vol. 5.
type: Item
instance of: scholarly article
title: Groups of prime power order. Vol. 5.
zbMATH Open document ID: 1344.20002
DOI: 10.1515/9783110295351
published in: De Gruyter Expositions in Mathematics
publication date: 2016-01-18T00:00:00Z
review text: A few years ago the reviewer was pleasantly surprized with the announcement that after the successfull three volumes regarding explicit structure of \(p\)-groups by \textit{Y. G. Berkovich} [Groups of prime power order. Vol. 1. de Gruyter Expositions in Mathematics 46. Berlin: Walter de Gruyter (2008; Zbl 1168.20001)] and \textit{Y. G. Berkovich} and \textit{Z. Janko} [Groups of prime power order. Vol. 2. de Gruyter Expositions in Mathematics 47. Berlin: Walter de Gruyter (2008; Zbl 1168.20002); Groups of prime power order. Vol. 3. de Gruyter Expositions in Mathematics 56. Berlin: Walter de Gruyter (2011; Zbl 1229.20001)], another two volumes were in progress to appear. Seeing over all five volumes now, one is able to conclude that the authors have enriched the mathematical literature with an important topic. The two books here under review do contain a wealth of new results. To give a ``numerical'' impression firstly: In volume 4 that are at least 24 new characaterizations of certain classes of \(p\)-groups together with 46 sections filled with new structure theorems; 13 so-called Appendices, hundreds of exercises and 508 (unsolved mostly) problems and research themes. In volume 5 these figures are respectively (in short) 34; 66; 52; hundreds; 572. All these things constitute a remarkable treasure cove for researchers and other curious mathematiciens in the years to come. Here are some snapshorts to be found in volume 5: (partly quoted from backover) \(\bullet\) \(p\)-groups containing a subgroup of maximal class of index \(p\); \(\bullet\) normal closures of nonnormal subgroups; \(\bullet\) \(p\)-groups all of whose subgroups are isomorphic to quotient groups and \(p\)-groups all of whose quotient groups are isomorphic to subgroups; \(\bullet\) \(p\)-groups with many minimal nonabelian subgroups; \(\bullet\) generalizations of Dedekindian groups; \(\bullet\) \(p\)-groups all of whose nonnormal subgroups have exponent \(p\); \(\bullet\) structure of \(p\)-groups wi…
Mathematics Subject Classification ID: 20-02
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20D25
Mathematics Subject Classification ID: 20D35
Mathematics Subject Classification ID: 20D45
Mathematics Subject Classification ID: 00A07
zbMATH DE Number: 6532320
zbMATH Keywords: finite \(p\)-groups
zbMATH Keywords: particular structures of \(p\)-groups
zbMATH Keywords: cyclic derived subgroups
zbMATH Keywords: non-Dedekindian groups
zbMATH Keywords: metacyclic groups
zbMATH Keywords: minimal nonabelian groups
zbMATH Keywords: Hughes subgroup
author: Yakov Berkovich
author: Zvonimir Janko
reviewed by: Robert W. van der Waall
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1515/9783110295351
OpenAlex ID: W4299544198
Recommended article: Groups of prime power order. Volume 4.
Recommended article: Groups of prime power order. Vol. 2.
Recommended article: Groups of prime power order. Vol. 6.
Recommended article: Groups of prime power order. Vol. 3.
Recommended article: Finite non-abelian 2-groups such that any two distinct minimal non-abelian subgroups have cyclic intersection
Recommended article: The classification of prime-power groups.
Recommended article: Short proofs of some basic characterization theorems of finite p-group theory
Recommended article: Finite nonabelian p-groups of exponent >p with a small number of maximal abelian subgroups of exponent >p
Recommended article: Finite \(p\)-groups with many minimal nonabelian subgroups.
Recommended article: Finite \(p\)-groups with complemented maximal cyclic subgroups
description: scientific article; zbMATH DE number 6532320