Residually nilpotent groups whose closed subgroups are subnormal.
type: Item
instance of: scholarly article
title: Residually nilpotent groups whose closed subgroups are subnormal.
zbMATH Open document ID: 1235.20027
author: D. Kharzeev
published in: Journal of Algebra
publication date: 2011-09-22T00:00:00Z
review text: A result of \textit{H. Smith} [J. Algebra 244, No. 2, 845-850 (2001; Zbl 0988.20018)] states that a periodic residually nilpotent group \(G\) in which all subgroups are subnormal is nilpotent. In this article the authors extend this result further. They take a particular residual nilpotency system for \(G\). By this is meant a family \(\mathcal N\) of normal subgroups of \(G\) with the properties: (1) \(G/N\) is nilpotent, for all \(N\in\mathcal N\); (2) for all \(N_1,N_2\in\mathcal N\) there exists \(N\in\mathcal N\) such that \(N\leq N_1\cap N_2\) and (3) \(\bigcap_{N\in\mathcal N}N=1\). A subgroup \(H\) of \(G\) is then said to be a closed subgroup of \(G\) if \(H=\bigcap_{N\in\mathcal N}HN\). The authors prove that if \(G\) is a periodic group admitting a residual nilpotency system with the property that all closed subgroups are subnormal then \(G\) is nilpotent. Interestingly, despite the natural topological idea involved, the proofs are for the most part group theoretic.
Mathematics Subject Classification ID: 20E15
Mathematics Subject Classification ID: 20F19
Mathematics Subject Classification ID: 20F50
Mathematics Subject Classification ID: 20E26
zbMATH DE Number: 5949287
zbMATH Keywords: closed subgroups
zbMATH Keywords: subnormal subgroups
zbMATH Keywords: subnormal intersection property
zbMATH Keywords: residually nilpotent groups
zbMATH Keywords: periodic groups
reviewed by: Martyn Dixon
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2010.10.022
OpenAlex ID: W2027228968
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DOI: 10.1016/J.JALGEBRA.2010.10.022
description: scientific article; zbMATH DE number 5949287