Subgroups of powerful groups.
type: Item
instance of: scholarly article
title: Subgroups of powerful groups.
zbMATH Open document ID: 1053.20019
DOI: 10.1007/BF02783416
author: Avinoam Mann
author: Fania Posnick-Fradkin
published in: Israel Journal of Mathematics
publication date: 2004-03-17T00:00:00Z
review text: A finite \(p\)-group \(G\) is said to be powerful if \(p>2\) and \(G'\leq G^p\) or \(p=2\) and \(G'\leq G^4\), where \(G^n\) denotes the subgroup generated by all \(n\)-th powers. In this paper the authors give a new proof of Lubotzky's theorem [\textit{J. D. Dixon, M. P. F. du Sautoy, A. Mann} and \textit{D. Segal}, Analytic pro-\(p\)-groups (1999; Zbl 0934.20001)]. However, the question of which \(p\)-groups appear as subgroups of powerful groups is still open. The authors prove some results about the structure of 2-generator groups which occur as normal subgroups of powerful groups. Also they prove the following results: A finite \(p\)-group \(H\) can be embedded in a powerful group \(G\) such that \(G=H\cdot Z(G)\) if and only if \(H/Z(H)\) is powerful. If \(H\) is a \(p\)-group of class two then \(H\) can be embedded in a powerful group. If \(G\) is a \(p\)-group, \(p>2\), with a cyclic commutator subgroup then \(G\) can be embedded as a maximal subgroup of a powerful \(p\)-group.
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20F12
Mathematics Subject Classification ID: 20E07
zbMATH DE Number: 2060050
zbMATH Keywords: finite \(p\)-groups
zbMATH Keywords: subgroups of powerful \(p\)-groups
zbMATH Keywords: 2-generator \(p\)-groups
zbMATH Keywords: normal subgroups
zbMATH Keywords: cyclic commutator subgroup
reviewed by: Francisco Pérez-Monasor
MaRDI profile type: Publication
cites work: Generalizations of Certain Elementary Theorems on p -Groups
cites work: On finite \(p\)-groups with cyclic commutator subgroup
cites work: Q4264394
cites work: Omega subgroups of powerful \(p\)-groups.
cites work: On the structure of normal subgroups of potent \(p\)-groups.
cites work: Endliche Gruppen I
cites work: Powerful \(p\)-groups. I: Finite groups.
cites work: Regular p-groups. III
cites work: On the power structure of powerful p-groups
description: scientific article; zbMATH DE number 2060050