Block-diagonal characterization of locally finite simple groups of \(p\)-type.
type: Item
instance of: scholarly article
zbMATH Open document ID: 1291.20036
author: Stefaan Delcroix
published in: Journal of Group Theory
publication date: 2014-01-07T00:00:00Z
Mathematics Subject Classification ID: 20F50
Mathematics Subject Classification ID: 20E32
zbMATH DE Number: 6243115
zbMATH Keywords: locally finite groups
zbMATH Keywords: locally finite simple groups
zbMATH Keywords: Kegel covers
MaRDI profile type: Publication
full work available at URL: jgt-2013-0004
OpenAlex ID: W1990663780
DOI: 10.1515/JGT-2013-0004
title: Block-diagonal characterization of locally finite simple groups of \(p\)-type.
review text: Let \(G\) be a simple, locally finite group. Then \(G\) always has a Kegel cover, that is, a set of pairs \((H,M)\), where \(H\) is a finite subgroup of \(G\) and \(M\) is a maximal normal subgroup of \(H\), such that for every finite subgroup \(X\) of \(G\) there exists at least one pair \((H,M)\) in the Kegel cover such that \(X\) is contained in \(H\) and intersects \(M\) in \(\langle 1\rangle\). This group \(G\) is said to be of \(p\)-type, for \(p\) some prime, if \(G\) is not isomorphic to any finitary linear group and if every Kegel cover of \(G\) contains at least one pair \((H,M)\) with \(H/M\) isomorphic to a classical group defined over a field of characteristic \(p\).NEWLINENEWLINE In earlier work [J. Algebra 315, No. 1, 419-453 (2007; Zbl 1128.20023)] the author proved that if our group \(G\) is of \(p\)-type, then it has at least one Kegel cover of a particularly complex and restricted type.NEWLINENEWLINE In this present paper the author proves the converse of the theorem and uses it to construct a general family of simple, locally finite groups of \(p\)-type.
reviewed by: B. A. F. Wehrfritz
description: scientific article; zbMATH DE number 6243115