On power margins
type: Item
instance of: scholarly article
title: On power margins
zbMATH Open document ID: 0672.20016
DOI: 10.1016/0021-8693(89)90220-2
author: Luise-Charlotte Kappe
published in: Journal of Algebra
publication date: 1989-01-01T00:00:00Z
review text: Let \(n\) be an integer and let \(G\) be a group. Then the \(n\)-power margin (in the sense of \textit{P. Hall} [J. Reine Angew. Math. 182, 156-157 (1940; Zbl 0023.29902)]) of \(G\) is the characteristic subgroup \(M_ n(G)=\{a\in G\mid (ag)^ n=g^ n\) for all \(g\in G\}\). Moreover, let \(R_ n[m]=\{a\in G\mid a^ m=1\) and \([a,_ nx]=1\) for all \(x\in G\}\) be the set of all right \(n\)-Engel elements in \(G\) of order dividing \(m\). It is shown that \(M_ 3(G)=R_ 2[3]\) and \(M_ 3(G)\leq Z_ 3(G)\) and if \(G\) is metabelian and \(p\) is a prime, then \(M_ p(G)=R_{p-1}[p]\) and \(M_ p(G)\leq Z_ p(G)\); the bounds being best possible. Also, for finite groups, the \(p^ m\)-power margin is contained in the hypercentre.
Mathematics Subject Classification ID: 20F12
Mathematics Subject Classification ID: 20F45
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20E10
Mathematics Subject Classification ID: 20F14
zbMATH DE Number: 4099553
zbMATH Keywords: marginal subgroups
zbMATH Keywords: Engel conditions
zbMATH Keywords: metabelian groups
zbMATH Keywords: \(n\)-power margin
zbMATH Keywords: characteristic subgroup
zbMATH Keywords: right \(n\)-Engel elements
zbMATH Keywords: hypercentre
reviewed by: Rolf Brandl
MaRDI profile type: Publication
full work available at URL: 0021-8693(89)90220-2
OpenAlex ID: W2019097242
cites work: Verbal and marginal subgroups.
cites work: On the H p -Problem for Finite p-Groups
cites work: Engel margins in metabelian groups
cites work: Die \(A\)-Norm einer Gruppe
cites work: Über eine besondere Klasse von Gruppen
cites work: Q5613235
cites work: Metabelsche Gruppen
cites work: Q4177758
cites work: On the Engel margin
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description: scientific article; zbMATH DE number 4099553