Groups satisfying Scherk's length theorem
type: Item
instance of: scholarly article
title: Groups satisfying Scherk's length theorem
zbMATH Open document ID: 0907.20044
DOI: 10.1007/BF01237492
author: Erich W. Ellers
author: Wolfgang Nolte
published in: Journal of Geometry
publication date: 1999-03-07T00:00:00Z
review text: Let \(V\) be a vector space over a field \(K\) of characteristic \(\neq 2\), and let \(G\) be a subgroup of the general linear group \(\text{GL}(n,K)\). Assume \(G\) is generated by its simple involutions such that \(-1_V\in G\) and Scherk's length theorem holds for \(G\) (i.e., \(G\) is a reflection group). The authors show that in any such irreducible group \(G\) there is a sequence of subspaces, each having an inner point, such that its sum is the entire space. They construct successively a quadratic form on sums of such spaces. Thus any group \(G\) having the above-mentioned properties is a subgroup of an orthogonal group.
reviewed by: Huberta Lausch
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 15A23
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20H15
zbMATH DE Number: 1146246
zbMATH Keywords: general linear groups
zbMATH Keywords: simple involutions
zbMATH Keywords: Scherk's length theorem
zbMATH Keywords: reflection groups
zbMATH Keywords: quadratic forms
zbMATH Keywords: orthogonal groups
MaRDI profile type: Publication
cites work: Q3973539
cites work: On the Decomposition of Orthogonalities into Symmetries
full work available at URL: https://doi.org/10.1007/bf01237492
OpenAlex ID: W2080009098
Recommended article: The Herzog–Schönheim conjecture for finitely generated groups
Recommended article: On groups of hyperbolic length
Recommended article: Lengths of conjugacy classes of finite solvable groups
Recommended article: GROUPS OF FINITE NORMAL LENGTH
Recommended article: ON LENGTHS ON SEMISIMPLE GROUPS
Recommended article: Q3705544
Recommended article: Groups of Wielandt length two
Recommended article: Q3032473
Recommended article: Q3645797
description: scientific article; zbMATH DE number 1146246