Maximal subgroups of \(\text{GL}_n(D)\)
type: Item
instance of: scholarly article
title: Maximal subgroups of \(\text{GL}_n(D)\)
zbMATH Open document ID: 1016.20031
DOI: 10.1016/S0021-8693(02)00549-5
author: R. Ebrahimian
author: H. Momenaee Kermani
author: S. Akbari
author: Alireza Salehi Golsefidy
published in: Journal of Algebra
publication date: 2003-04-02T00:00:00Z
review text: In this interesting paper the authors continue their study of maximal subgroups of skew linear groups. They start by exhibiting a metabelian maximal subgroup of the multiplicative group of the quaternions. Suppose \(M\) is a maximal subgroup of the multiplicative group \(D^*\) of the division ring \(D\) with centre \(F\). If \(M\) is nilpotent they show that \(M\) is metabelian, if \(M\) is nilpotent and \(M\setminus F\) contains an element algebraic over \(F\) they prove that \(M\) is Abelian and if \(M\) is non-Abelian, locally soluble and algebraic over \(F\), they describe \(M\) in some detail. Then they focus their attention on the maximal subgroups \(M\) of the general linear group \(\text{GL}(n,D)\) for \(n\) a positive integer. If \(M\) is locally finite, they show that \(D=F\) is a locally finite field. If \(D\) is infinite and locally finite-dimensional over \(F\) and if \(M\) is locally nilpotent, they prove that \(M\) is Abelian. We refer the reader to the paper where there are further interesting results.
Mathematics Subject Classification ID: 20H25
Mathematics Subject Classification ID: 20E28
Mathematics Subject Classification ID: 16U60
Mathematics Subject Classification ID: 16K40
zbMATH DE Number: 1891319
zbMATH Keywords: division rings
zbMATH Keywords: nilpotent maximal subgroups
zbMATH Keywords: skew linear groups
zbMATH Keywords: metabelian maximal subgroups
zbMATH Keywords: multiplicative groups
zbMATH Keywords: general linear groups
zbMATH Keywords: locally finite groups
zbMATH Keywords: locally finite fields
reviewed by: B. A. F. Wehrfritz
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1891319