On generalized quadrangles with some concurrent axes of symmetry
type: Item
instance of: scholarly article
title: On generalized quadrangles with some concurrent axes of symmetry
zbMATH Open document ID: 1042.51005
author: Koen Thas
published in: Bulletin of the Belgian Mathematical Society - Simon Stevin
publication date: 2004-01-22T00:00:00Z
review text: The author basically considers the following question: Given a generalized quadrangle \({\mathcal S}\), a point \(x\) in it and \(k\) axes of symmetry passing through \(x\) (an axis of symmetry \(L\) is a line such that the group of automorphisms fixing all lines concurrent with \(L\) has order \(s\), with \((s, t)\) the order of \({\mathcal S}\), i.e., every line of \(S\) carries exactly \(1+s\) points, and every point is incident with precisely \(1+t\) lines), under which additional conditions can we conclude that (a) all lines through \(x\) are axes of symmetry, (b) the related symmetry groups generate the symmetry groups corresponding to all lines through \(x\). The author offers a wealth of answers, going from using additional combinatorial properties, to using only a numerical bound on \(k\). It is also explained how his investigation relates with flock generalized quadrangles, and new characterizations emerge. In fact, there are so many results mentioned and proved in this paper, which relate to so many different aspects, that it is not easy to have an overview of them after a first reading. Personally, I most like the last main result stating that, with the above notation, if \(t\geq s\neq 1\) and \(k> t- s+2\), then every line through \(x\) is an axis of symmetry, and every symmetry group is generated by \(t- s+ 2\) arbitrary symmetry groups (provided \(s\neq t\); if \(s= t\), then one needs 3 symmetry groups).
Mathematics Subject Classification ID: 51E12
zbMATH DE Number: 2031081
zbMATH Keywords: translation generalized quadrangle
zbMATH Keywords: flock quadrangle
zbMATH Keywords: elation generalized quadrangle
reviewed by: Hendrik Van Maldeghem
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2031081