Sharply transitive linear algebraic groups
type: Item
instance of: scholarly article
title: Sharply transitive linear algebraic groups
zbMATH Open document ID: 0769.20022
DOI: 10.1007/BF01264521
author: Peter Müller
published in: Geometriae Dedicata
publication date: 1993-05-16T00:00:00Z
review text: The author studies Zariski-closed linear groups \(G \leq GL_ n k\) which act sharply transitively on the non-zero vectors of \(k^ n\), where \(k\) is a field of characteristic zero. If \(n\) is square-free or if the cohomological dimension of \(k\) is at most 1, then he proves that \(G\) is the multiplicative group of a field extension of degree \(n\) over \(k\). If \(n \leq 15\), then \(G\) is contained in a semi-linear group \(\Gamma L_ 1D\) for some skew field \(D\) of dimension \(n\) over \(k\). These results are proved by passing to the Lie algebra \(L\) of \(G\) and by investigating the representation of \(L\) in \(k^ n\). The author's arguments simplify parts of a related paper by the reviewer [Forum Math. 1, 81-101 (1989; Zbl 0649.20045)]. The results on \(G\) (or \(L\)) can be restated in terms of nearfields (or of left-symmetric division algebras).
reviewed by: Theo Grundhöfer
Mathematics Subject Classification ID: 20G30
Mathematics Subject Classification ID: 12K05
Mathematics Subject Classification ID: 17B45
Mathematics Subject Classification ID: 20G25
Mathematics Subject Classification ID: 12F99
Mathematics Subject Classification ID: 16K20
Mathematics Subject Classification ID: 16U60
zbMATH DE Number: 166879
zbMATH Keywords: sharply transitive
zbMATH Keywords: Zariski-closed linear groups
zbMATH Keywords: multiplicative group
zbMATH Keywords: field extension
zbMATH Keywords: semi-linear group
zbMATH Keywords: skew field
zbMATH Keywords: Lie algebra
zbMATH Keywords: nearfields
zbMATH Keywords: left-symmetric division algebras
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01264521
OpenAlex ID: W2020434307
description: scientific article; zbMATH DE number 166879