Locally (soluble-by-finite) groups with all proper insoluble subgroups of finite rank
type: Item
instance of: scholarly article
title: Locally (soluble-by-finite) groups with all proper insoluble subgroups of finite rank
zbMATH Open document ID: 0871.20031
author: Martyn Dixon
author: Martin J. Evans
author: Howard Smith
published in: Archiv der Mathematik
publication date: 1997-10-05T00:00:00Z
review text: A group \(G\) has finite rank \(r\) if every finitely generated subgroup of \(G\) is at most \(r\)-generator. If \(\mathfrak C\) is a class of groups then we let \(\mathfrak C^*\) denote the class of groups \(G\) in which every proper subgroup of \(G\) is either of finite rank or in \(\mathfrak C\). We let \(\mathfrak S\) denote the class of soluble groups and \({\mathfrak S}_d\) the class of soluble groups of derived length at most \(d\), where \(d\) is a positive integer. We let \(\Lambda\) denote the set of closure operations \(\{{\mathbf L},{\mathbf R},\text{\textbfṔ},\text{\textbf{\`P}}\}\) and let \(\mathfrak X\) denote the \(\Lambda\)-closure of the class of periodic locally graded groups. Amongst other results we prove that a soluble \({\mathfrak S}^*_d\)-group is either of finite rank or of derived length at most \(d\) and also that a group in the class \({\mathfrak X}\cap({\mathbf L}{\mathfrak S})^*\) is either locally soluble, or has finite rank, or is isomorphic to one of \(\text{SL}(2,\mathbb{F})\), \(\text{PSL}(2,\mathbb{F})\) or \(Sz(\mathbb{F})\) for suitable locally finite fields \(\mathbb{F}\).
Mathematics Subject Classification ID: 20F19
Mathematics Subject Classification ID: 20E25
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20F14
Mathematics Subject Classification ID: 20F50
zbMATH DE Number: 1026264
zbMATH Keywords: finitely generated subgroups
zbMATH Keywords: soluble groups
zbMATH Keywords: derived lengths
zbMATH Keywords: periodic locally graded groups
zbMATH Keywords: groups of finite rank
zbMATH Keywords: locally finite fields
zbMATH Keywords: Heineken-Mohamed groups
MaRDI profile type: Publication
full work available at URL: s000130050037
OpenAlex ID: W2034736275
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DOI: 10.1007/S000130050037
description: scientific article; zbMATH DE number 1026264