Maximal subgroups of \(\text{GL}_1(D)\)
type: Item
instance of: scholarly article
title: Maximal subgroups of \(\text{GL}_1(D)\)
zbMATH Open document ID: 0933.16033
author: M. Mahdavi-Hezavehi
author: M. G. Mahmoudi
author: S. Akbari
published in: Journal of Algebra
publication date: 2000-01-25T00:00:00Z
review text: Let \(D\) be finite dimensional division algebra. By a well-known result of I.~N.~Herstein, every finite normal subgroup of the multiplicative group \(D^*\) of \(D\) is central. The authors extend this by proving that every finitely generated normal subgroup of \(D^*\) is central. The authors then study maximal subgroups of \(D^*\) for \(D\) a not necessarily finite dimensional division algebra. They give a number of results, some involving interaction (or rather a lack of it) between the degree and the characteristic of \(D\) and some rather technical. As a snippet, to give something of the flavour of this second part of the paper, we quote the following. Let \(D\) be a division ring that is algebraic over its centre. If \(D^*\) has a soluble maximal subgroup, then \(D\) is locally finite-dimensional. If \(D^*\) has an Abelian maximal subgroup, then \(D\) is finite dimensional.
Mathematics Subject Classification ID: 16U60
Mathematics Subject Classification ID: 16K40
Mathematics Subject Classification ID: 20E28
Mathematics Subject Classification ID: 16K20
Mathematics Subject Classification ID: 16P10
zbMATH DE Number: 1340554
zbMATH Keywords: finite dimensional division algebras
zbMATH Keywords: multiplicative groups
zbMATH Keywords: finitely generated normal subgroups
zbMATH Keywords: maximal subgroups
zbMATH Keywords: soluble maximal subgroups
zbMATH Keywords: locally finite-dimensional algebras
zbMATH Keywords: Abelian maximal subgroups
reviewed by: B. A. F. Wehrfritz
MaRDI profile type: Publication
full work available at URL: jabr.1998.7792
OpenAlex ID: W2592875436
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DOI: 10.1006/JABR.1998.7792
description: scientific article; zbMATH DE number 1340554