Three-dimensional classical groups acting on polytopes
type: Item
instance of: scholarly article
title: Three-dimensional classical groups acting on polytopes
zbMATH Open document ID: 1206.52012
author: Peter A. Brooksbank
author: Deborah A. Vicinsky
published in: Discrete \& Computational Geometry
publication date: 2010-11-08T00:00:00Z
review text: The authors show that any irreducible subgroup \(G\) of \(\mathrm{GL}(V)\), where \(V\) is a three-dimensional vector space over a finite field \(K\), that arises as automorphism group of an abstract regular polytope preserves a non-degenerate symmetric bilinear form \(f\) of \(V\). In any such case \(K\) is of characteristic \(\neq 2\). Moreover, let \(I(V,f)\leq\mathrm{GL}(V)\) be the group of \(f\)-isometries of \(V\). Then \(I(V,f)'\leq G\leq I(V,f)\), where \(I(V,f)'\) is the commutator subgroup of \(I(V,f)\).
reviewed by: Dimitris P. Vartziotis
Mathematics Subject Classification ID: 52B11
Mathematics Subject Classification ID: 20G40
zbMATH DE Number: 5813739
zbMATH Keywords: abstract regular polytope
zbMATH Keywords: string C-group
zbMATH Keywords: classical group
zbMATH Keywords: symmetric bilinear form
MaRDI profile type: Publication
full work available at URL: s00454-009-9212-0
OpenAlex ID: W2054436148
cites work: Q3996618
cites work: Groups of type L 2(q) acting on polytopes
cites work: Abstract Regular Polytopes
cites work: Reflection groups and polytopes over finite fields. I.
cites work: Q4311821
DOI: 10.1007/S00454-009-9212-0
description: scientific article; zbMATH DE number 5813739