Some results concerning simple locally finite groups of 1-type.
type: Item
instance of: scholarly article
title: Some results concerning simple locally finite groups of 1-type.
zbMATH Open document ID: 1072.20042
author: Felix Leinen
author: Orazio Puglisi
published in: Journal of Algebra
publication date: 2005-06-01T00:00:00Z
review text: Let \(G\) be an infinite simple, locally finite group. Then \(G\) has a Kegel cover; that is, a collection of finite subgroups \(H\) with a specified maximal normal subgroup \(M\) such that for every finite subgroup \(F\) of \(G\) there is some pair \((H,M)\) in the Kegel cover such that \(F\leq H\) and \(F\cap M=\langle 1\rangle\). The group is said to be Type 1 if every Kegel cover of \(G\) contains a pair \((H,M)\) with \(H/M\) an alternating group. This is a highly technical paper and I cannot do better to summarize its contents than to quote the authors' abstract: Several aspects of infinite simple locally finite groups of 1-type are considered. In the first part, the classes of diagonal limits of finite alternating groups, of diagonal limits of finite direct products of alternating groups, and of absolutely simple groups of 1-type are distinguished from each other. In the second part, inductive systems of representations over fields of characteristic zero (which are known to correspond to ideals in the group algebra) are studied in general for groups of 1-type. The roles of primitive respectively imprimitive representations in inductive systems are investigated. Moreover it is shown that in any proper inductive system the depths of the representations of certain alternating subgroups are bounded.
Mathematics Subject Classification ID: 20F50
Mathematics Subject Classification ID: 20E32
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 20C07
zbMATH DE Number: 2173128
zbMATH Keywords: group rings
zbMATH Keywords: simple groups
zbMATH Keywords: locally finite groups
zbMATH Keywords: inductive systems
zbMATH Keywords: groups of 1-type
zbMATH Keywords: Kegel covers
reviewed by: B. A. F. Wehrfritz
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2004.12.021
OpenAlex ID: W2162424639
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DOI: 10.1016/J.JALGEBRA.2004.12.021
description: scientific article; zbMATH DE number 2173128