On groups with many almost normal subgroups
type: Item
instance of: scholarly article
title: On groups with many almost normal subgroups
zbMATH Open document ID: 0853.20024
DOI: 10.1007/BF01759348
author: F. de Giovanni
author: Silvana Franciosi
author: L. A. Kurdachenko
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 1996-07-08T00:00:00Z
review text: In an FC group every finitely generated subgroup has finitely many conjugates. The authors define an anti-FC-group to be a group in which all non-finitely generated subgroups have only finitely many conjugates. Clearly centre-by-finite groups and groups satisfying the maximal condition are anti-FC. The authors describe the soluble anti-FC-groups not in these two classes. A soluble anti-FC-group is first shown to be an \(S_1\)-group and then different cases are considered. If \(G\) is a soluble anti-FC-group containing a \(p^\infty\)-subgroup \(P\) then different characterisations are given depending on whether \(P\) is central or not. In both cases, \(\text{Spec }G=\{p\}\). If \(G\) is an \(S_1\)-group without \(p^\infty\)-subgroups then its maximal normal periodic subgroup \(T\) is finite. Then \(G\) is anti-FC if and only if \(G/T\) is anti-FC. The description is then completed by considering soluble anti-FC-groups with no periodic normal subgroup.
Mathematics Subject Classification ID: 20F24
Mathematics Subject Classification ID: 20F16
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20E15
zbMATH DE Number: 878424
zbMATH Keywords: almost normal subgroups
zbMATH Keywords: non-finitely generated subgroups
zbMATH Keywords: finitely many conjugates
zbMATH Keywords: centre-by-finite groups
zbMATH Keywords: groups satisfying the maximal condition
zbMATH Keywords: soluble anti-FC-groups
zbMATH Keywords: \(S_ 1\)-groups
zbMATH Keywords: maximal normal periodic subgroup
reviewed by: M. J. Tomkinson
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01759348
OpenAlex ID: W2063554915
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description: scientific article; zbMATH DE number 878424