Groups with dense pronormal subgroups
type: Item
instance of: scholarly article
title: Groups with dense pronormal subgroups
zbMATH Open document ID: 0754.20011
author: Giovanni Vincenzi
published in: Ricerche di Matematica
publication date: 1992-09-27T00:00:00Z
review text: Let \(\chi\) be a property pertaining to subgroups. A group \(G\) is said to have dense (respectively: weakly dense) \(\chi\)-subgroups if for each pair \((H,K)\) of subgroups of \(G\) such that \(H<K\) and \(H\) is not maximal in \(K\), there exists a \(\chi\)-subgroup \(X\) of \(G\) with \(H<X<K\) (respectively: \(H\leq X\leq K)\). In this paper groups with dense pronormal subgroups are considered. Here a subgroup \(P\) of a group \(G\) is called pronormal if for each element \(x\) of \(G\) the subgroups \(P\) and \(P^ x\) are conjugate in \(\langle P,P^ x\rangle\). Improving a result of \textit{A. Mann} [Isr. J. Math. 6, 13-25 (1968; Zbl 0155.050)] the author shows that a locally soluble non-periodic group with weakly dense pronormal subgroups is abelian. Moreover, it is proved that a locally finite group with dense pronormal subgroups is hypercyclic and its commutator subgroup is nilpotent. This last result does not hold for groups with weakly dense pronormal subgroups, as the alternating group of degree 5 has this property.
Mathematics Subject Classification ID: 20F16
Mathematics Subject Classification ID: 20F50
Mathematics Subject Classification ID: 20E25
Mathematics Subject Classification ID: 20E07
zbMATH DE Number: 60236
zbMATH Keywords: dense pronormal subgroups
zbMATH Keywords: locally soluble non-periodic group
zbMATH Keywords: weakly dense pronormal subgroups
zbMATH Keywords: locally finite group
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 60236