Free subgroups in linear groups over some skew fields
type: Item
instance of: scholarly article
title: Free subgroups in linear groups over some skew fields
zbMATH Open document ID: 0604.20052
DOI: 10.1016/0021-8693(87)90177-3
author: A. I. Lichtman
published in: Journal of Algebra
publication date: 1987-01-01T00:00:00Z
review text: Let D be the division ring of quotients of the universal enveloping algebra of a finite dimensional Lie algebra over a field of characteristic zero. In Theorem A the author analyses the structure of a subgroup of GL(n,D) with no non-cyclic free subgroups. There are several corollaries. For example Corollary 2 gives a bound to the derived length of a soluble subgroup of GL(n,D) in terms of n and the dimension of L. The proof uses a translation of the problem to one in positive characteristic and the application of an interesting theorem (Theorem B) on the structure of subgroups of GL(n,E), where E is a division ring of positive characteristic p and finite dimension \(q^ m\) over its centre, q a prime with \(n<q-1\). Finally the author considers similar questions over the division ring of quotients of the group algebra (over a field) of a free metabelian group and over the universal field of fractions of the group algebra of a free group.
Mathematics Subject Classification ID: 20H25
Mathematics Subject Classification ID: 16Kxx
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 17B35
Mathematics Subject Classification ID: 16S34
zbMATH DE Number: 3975225
zbMATH Keywords: skew linear groups
zbMATH Keywords: division ring of quotients
zbMATH Keywords: universal enveloping algebra
zbMATH Keywords: finite dimensional Lie algebra
zbMATH Keywords: subgroup of GL(n,D)
zbMATH Keywords: derived length
zbMATH Keywords: soluble subgroup
zbMATH Keywords: group algebra
zbMATH Keywords: free metabelian group
MaRDI profile type: Publication
full work available at URL: 0021-8693(87)90177-3
OpenAlex ID: W2077473401
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description: scientific article; zbMATH DE number 3975225