Ideals in group algebras of simple locally finite groups of 1-type.
type: Item
instance of: scholarly article
title: Ideals in group algebras of simple locally finite groups of 1-type.
zbMATH Open document ID: 1065.16023
author: Felix Leinen
author: Orazio Puglisi
published in: Pacific Journal of Mathematics
publication date: 2004-09-16T00:00:00Z
review text: This paper continues the work of \textit{A.~E.~Zalesskij} and many others on the ideal lattice of characteristic 0 group algebras \(K[G]\) of infinite locally finite simple groups \(G\) [see Finite and locally finite groups, NATO ASI Ser., Ser. C, Math. Phys. Sci. 471, 219-246 (1995; Zbl 0839.16021)]. It is known that many of these group algebras have only three 2-sided ideals, namely 0, \(K[G]\) itself and the augmentation ideal \(\omega(K[G])\). Indeed, even when this is not the case, it appears that \(K[G]\) has surprisingly few ideals in general. In the PhD thesis of \textit{S.~Delcroix} [Non-finitary locally finite simple groups (Univ. Gent, Belgium) (2000)] the infinite locally finite simple groups were divided into four categories, namely finitary linear groups, groups of 1-type, groups of \(p\)-type, and groups of \(\infty\)-type. The main results here concern groups of 1-type. Specifically, it is shown that if \(G\) is of 1-type, then \(K[G]\) has at least four ideals. Furthermore, if \(G\) is a direct limit of finite direct products of alternating groups, then the ideal lattice of \(K[G]\) consists of a strictly descending chain \(K[G]=I_0\supset I_1=\omega(K[G])\supset I_2\supset\cdots\) with intersection 0. A key ingredient in this work is a result of \textit{U.~Meierfrankenfeld} [in Finite and locally finite groups, NATO ASI Ser., Ser. C, Math. Phys. Sci. 471, 189-212 (1995; Zbl 0858.20026)].
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 20C07
Mathematics Subject Classification ID: 20F50
Mathematics Subject Classification ID: 20E32
Mathematics Subject Classification ID: 16D25
Mathematics Subject Classification ID: 20E25
zbMATH DE Number: 2101030
zbMATH Keywords: group algebras
zbMATH Keywords: locally finite groups
zbMATH Keywords: groups of 1-type
zbMATH Keywords: ideal lattices
zbMATH Keywords: lattices of two-sided ideals
zbMATH Keywords: groups of \(p\)-type
zbMATH Keywords: groups of alternating type
zbMATH Keywords: Kegel covers
zbMATH Keywords: finite subgroups
zbMATH Keywords: finitary locally finite simple groups
zbMATH Keywords: non-finitary locally finite simple groups
zbMATH Keywords: projective special linear groups
zbMATH Keywords: augmentation ideals
reviewed by: D. S. Passman
MaRDI profile type: Publication
full work available at URL: pjm.2002.207.433
OpenAlex ID: W2060850172
DOI: 10.2140/PJM.2002.207.433
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description: scientific article; zbMATH DE number 2101030