Characterizations of generalized quadrangles by generalized homologies
type: Item
instance of: scholarly article
title: Characterizations of generalized quadrangles by generalized homologies
zbMATH Open document ID: 0584.51005
DOI: 10.1016/0097-3165(85)90094-9
published in: Journal of Combinatorial Theory. Series A
publication date: 1985-01-01T00:00:00Z
review text: Let \(S=(P,B,I)\) be a finite generalized quadrangle of order (s,t) and \({\mathcal H}(x,y)\) the group of all collineations of S fixing the noncollinear points x and y linewise. S is called (x,y)-transitive if \({\mathcal H}(x,y)\) is transitive on each set \(^{\sim}-\{x,z\}\) and \(^{\sim}-\{y,z\},\) where z is any point collinear with both x and y and \(^{\sim}\) denotes the set of all points incident with the line xz. The author proves the following theorem: If \(s>1\), \(t>1\) and S is (x,y)- transitive for all noncollinear x,y\(\in P\), then one of the following cases occurs: S arises from (i) a symplectic polarity, (ii) a nonsingular hermitian variety in PG(4,s), (iii)-(iv) a nonsingular quadric in PG(4,s) or PG(5,s), respectively, (v) \(s=t^ 2\) and all points are regular.
Mathematics Subject Classification ID: 51E20
Mathematics Subject Classification ID: 05B25
zbMATH DE Number: 3935866
zbMATH Keywords: finite geometries
zbMATH Keywords: finite generalized quadrangle
author: Joseph A. Thas
MaRDI profile type: Publication
full work available at URL: 0097-3165(85)90094-9
OpenAlex ID: W2065860700
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description: scientific article; zbMATH DE number 3935866