Maximal subgroups of \(\text{GL}_n(D)\) with finite conjugacy classes.
type: Item
instance of: scholarly article
title: Maximal subgroups of \(\text{GL}_n(D)\) with finite conjugacy classes.
zbMATH Open document ID: 1184.20043
author: Dariush Kiani
author: Mojtaba Ramezan-Nassab
published in: Manuscripta Mathematica
publication date: 2009-11-06T00:00:00Z
review text: Let \(D\) be an infinite division ring and \(M\) a maximal subgroup of the matrix group \(\text{GL}_n(D)\), for some positive integer \(n\). Suppose that \(M\) is an FC-group, i.e. each \(g\in M\) has a finite number of conjugates in \(M\). The paper under review shows that then \(M\) is the multiplicative group of some maximal subfield \(M'\) of the matrix ring \(M_n(D)\); in addition, it proves that if \(M'\) is an algebraic extension of the centre \(Z(D)\) of \(D\), then \(D\) is a finite-dimensional algebra over \(Z(D)\). The crucial step towards the authors' proof of this result is to show that the group \(M\) is Abelian.
reviewed by: Ivan D. Chipchakov
Mathematics Subject Classification ID: 20H25
Mathematics Subject Classification ID: 20E28
Mathematics Subject Classification ID: 20F24
Mathematics Subject Classification ID: 16U60
Mathematics Subject Classification ID: 16K40
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 12E15
Mathematics Subject Classification ID: 15B33
zbMATH DE Number: 5627042
zbMATH Keywords: FC-groups
zbMATH Keywords: division rings
zbMATH Keywords: absolutely irreducible subgroups
zbMATH Keywords: PI-rings
zbMATH Keywords: multiplicative groups
zbMATH Keywords: maximal subfields
MaRDI profile type: Publication
full work available at URL: s00229-009-0290-3
OpenAlex ID: W2042657864
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DOI: 10.1007/S00229-009-0290-3
description: scientific article; zbMATH DE number 5627042