Groups with dense subnormal subgroups
type: Item
instance of: scholarly article
title: Groups with dense subnormal subgroups
zbMATH Open document ID: 0936.20023
author: F. de Giovanni
author: Alessio Russo
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 1999-12-06T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108485
review text: Let \(G\) be a group, \(S\) a family of subgroups. The set \(S\) is said to be dense in \(G\) if for any two subgroups \(H,K\) such that \(H<K\) and \(H\) is not maximal in \(K\) the set \(S\) contains a subgroup \(X\) such that \(H<X<K\). The authors consider groups, in which a family of subnormal subgroups is dense. The main results are the following. Theorem 2.4. Let \(G\) be an infinite group with dense family of subnormal subgroups. Then every subgroup of \(G\) is subnormal. Theorem 3.1. Let \(G\) be an infinite group with dense family of subnormal subgroups of bounded defect. Then \(G\) is nilpotent. Theorem 3.6. Let \(G\) be an infinite group for which the set of subnormal subgroups with defect at most 2 is dense. Then \(G\) is soluble with derived length at most 4.
Mathematics Subject Classification ID: 20E15
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20F16
Mathematics Subject Classification ID: 20F18
zbMATH DE Number: 1319589
zbMATH Keywords: subnormal subgroups
zbMATH Keywords: dense families of subgroups
reviewed by: L. A. Kurdachenko
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1319589