Locally finite simple groups of 1-type
type: Item
instance of: scholarly article
title: Locally finite simple groups of 1-type
zbMATH Open document ID: 0997.20041
author: Stefaan Delcroix
author: U. Meierfrankenfeld
published in: Journal of Algebra
publication date: 2002-11-17T00:00:00Z
review text: Basically, this paper is about large locally finite simple groups \(G\), where by large we mean \(G\) has no non-trivial finitary representations over any field, that is, where \(G\) is non-finitary. It is difficult to describe the contents of this paper adequately without giving a large number of technical definitions. We will try to give some idea of its flavour by selecting a few titbits. Let \(G\) be a non-finitary locally finite simple group. Suppose that for every finite subgroup \(F\) of \(G\) there is a finite subgroup \(H\) of \(G\) containing \(F\) and a normal subgroup \(K\) of \(H\) with \(F\cap K=\langle 1\rangle\) such that \(H/K\) is isomorphic to an alternating group. If \(F\) is a finite subgroup of \(G\) for which there is a finite subgroup \(F^*\) of \(G\) containing \(F\) such whenever we have a finite subgroup \(H\) of \(G\) containing \(F^*\) and \(K\) normal in \(H\) with \(H/K\) an alternativing group on \(\Omega\) of degree at least 7, then \(F\) has no regular orbit on \(\Omega\), then the authors describe the normal closure of \(F\) in \(L\) for all large enough finite subgroups \(L\) of \(G\) containing \(F\). They call such an \(F\) non-regular. If \(F\) is any finite subgroup of \(G\), the authors give a necessary and sufficient condition for \(F\) to be non-regular in terms of Kegel covers. Basically it hangs on the existence or otherwise of a bound on the number of regular orbits of \(F\) on any \(\Omega\), where \(H\), \(K\) and \(\Omega\) are as above and \((H,K)\) is in the Kegel cover. Finally, there is a group \(G\) as above, with a non-regular finite subgroup that is not absolutely simple, meaning that \(G\) has a proper serial subgroup.
Mathematics Subject Classification ID: 20F50
Mathematics Subject Classification ID: 20E32
Mathematics Subject Classification ID: 20E07
zbMATH DE Number: 1740524
zbMATH Keywords: locally finite simple groups
zbMATH Keywords: normal closures
zbMATH Keywords: finite subgroups
zbMATH Keywords: Kegel covers
zbMATH Keywords: regular orbits
zbMATH Keywords: serial subgroups
reviewed by: B. A. F. Wehrfritz
MaRDI profile type: Publication
full work available at URL: jabr.2001.8997
OpenAlex ID: W1978673902
cites work: On the centers of maximal subgroups of locally finite simple groups of alternating type
cites work: Infinite alternating groups as finitary linear transformation groups
cites work: Q4856481
cites work: Q4856476
cites work: Q4856482
DOI: 10.1006/JABR.2001.8997
description: scientific article; zbMATH DE number 1740524