On a theorem of Rhemtulla on polycyclic groups.
type: Item
instance of: scholarly article
title: On a theorem of Rhemtulla on polycyclic groups.
zbMATH Open document ID: 1211.20029
published in: Journal of Pure and Applied Algebra
publication date: 2010-04-22T00:00:00Z
review text: Let \(G\) be a polycyclic-by-finite group. If \(\Gamma\) is a subgroup of \(\Aut\,G\) and \(H\) is a subgroup of \(G\), denote by \(L(\Gamma,H)\) the poset of all intersections of subgroups \(H^\gamma\) (\(\gamma\in\Gamma\)). In particular, put \(L(G,H)=L(\text{Inn\,}G,G)\). In 1967 A. H. Rhemtulla proved that in a polycyclic-by-finite group any intersection of conjugates of a subgroup is the intersection of finitely many of the conjugates. Moreover, he showed that for every subgroup \(H\) of a polycyclic-by-finite group \(G\) the poset \(L(G,H)\) satisfies the minimal condition. In the article under review the author generalizes the latter result. The main theorem is the following: Theorem. Let \(H\) be a subgroup of a polycyclic-by-finite group \(G\) and set \(\Gamma=\Aut\,G\). Then there is a faithful \(\Gamma G\)-module \(V\) (\(\Gamma G\) is the holomorph of \(G\)) which is additively free Abelian of finite rank, a \(G\)-submodule \(U\) of \(V\) with \(V/U\) additively free Abelian and an element \(x\) of \(V\) such that \(H=C_G(x\text{\,mod\,}U)=\{g\in G\mid x(g-1)\in U\}\). Corollary. Let \(H\) be a subgroup of a polycyclic-by-finite group \(G\) and set \(\Gamma=\Aut\,G\). Then there is an integer \(d\) such that the chains in the poset \(L(\Gamma,H)\) have length at most \(d\). In particular \(L(\Gamma,H)\) satisfies the minimal condition and the characteristic core \(H_\Gamma\) of \(H\) in \(G\) is the intersection of at most \(d+1\) subgroups \(H^\gamma\) (\(\gamma\in\Gamma\)).
Mathematics Subject Classification ID: 20F16
Mathematics Subject Classification ID: 20F28
Mathematics Subject Classification ID: 20E15
Mathematics Subject Classification ID: 20E07
zbMATH DE Number: 5697070
zbMATH Keywords: polycyclic-by-finite groups
zbMATH Keywords: minimal condition
zbMATH Keywords: intersections of conjugates of subgroups
zbMATH Keywords: chains of subgroups
author: B. A. F. Wehrfritz
reviewed by: Alessio Russo
MaRDI profile type: Publication
full work available at URL: j.jpaa.2010.02.004
OpenAlex ID: W2021517651
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DOI: 10.1016/J.JPAA.2010.02.004
description: scientific article; zbMATH DE number 5697070