Subgroups of classical groups generated by transvections or Siegel transvections. I: Embeddings in linear groups
type: Item
instance of: scholarly article
title: Subgroups of classical groups generated by transvections or Siegel transvections. I: Embeddings in linear groups
zbMATH Open document ID: 0891.20034
DOI: 10.1023/A:1004996924764
author: Anja Steinbach
published in: Geometriae Dedicata
publication date: 1998-07-19T00:00:00Z
review text: Transvections and Siegel transvections play an important role in the theory of classical groups. Let \(Y\) be a finite-dimensional classical group and \(G\) a subgroup of \(Y\) generated by (Siegel) transvections. Assume \(G\) is defined over a field and has rank at least 3. Under some weak restrictions in characteristic 2, the author proves that \(G/Z(G)\) is isomorphic to a finite-dimensional projective classical group \(S/Z(S)\). In this situation the classical group \(S\) is said to embed in the classical group \(Y\) such that (Siegel) transvections act as (Siegel) transvections. The author gives all the possible such embeddings in linear groups by using methods of geometric algebra and the fundamental theorem of projective geometry.
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 51A50
Mathematics Subject Classification ID: 51A45
Mathematics Subject Classification ID: 20E07
zbMATH DE Number: 1121162
zbMATH Keywords: long root elements
zbMATH Keywords: polar spaces
zbMATH Keywords: embeddings of projective spaces
zbMATH Keywords: Siegel transvections
zbMATH Keywords: classical groups
zbMATH Keywords: finite-dimensional projective classical groups
zbMATH Keywords: embeddings in linear groups
zbMATH Keywords: fundamental theorem of projective geometry
MaRDI profile type: Publication
full work available at URL: a:1004996924764
OpenAlex ID: W1559001421
description: scientific article; zbMATH DE number 1121162