Generalized quadrangles of order \((q,q^ 2)\), \(q\) even, containing \(W(q)\) as a subquadrangle
type: Item
instance of: scholarly article
title: Generalized quadrangles of order \((q,q^ 2)\), \(q\) even, containing \(W(q)\) as a subquadrangle
zbMATH Open document ID: 0834.51002
DOI: 10.1007/BF01263571
published in: Geometriae Dedicata
publication date: 1995-10-17T00:00:00Z
review text: Let \({\mathcal S}\) be a generalized quadrangle of order \((q, q^2)\), with \(q\) even. Suppose that \({\mathcal S}\) contains a subquadrangle \({\mathcal S}'\) of order \(q\) isomorphic to the symplectic quadrangle \(W(q)\). It is well- known that every point \(x\) outside \({\mathcal S}'\) is collinear to the points of an ovoid of \({\mathcal S}'\). If by varying \(x\), all such ovoids are classical ovoids, i.e., ovoids with the 2-transitive group \(L_2 (q^2)\) acting on it, then \textit{J. A. Thas} and \textit{S. E. Payne} [Geom. Dedicata 52, No. 3, 227-253 (1994; Zbl 0804.51007)] show that \({\mathcal S}\) is isomorphic to \(Q (5,q)\) (a quadric of projective Witt-index 1 in 5- dimensional projective space over the field with \(q\) elements). The main result of the paper under review is to weaken the previous assumption as follows: suppose every ovoid obtained by a point not in \({\mathcal S}'\) is either classical, or semi-classical (i.e., the Suzuki group \(Sz (q)\) acts on it as a 2-transitive automorphism group, and the ovoid is called a Suzuki-Tits ovoid), then the conclusion above is still true, i.e., \({\mathcal S}\) is isomorphic to \(Q (5,q)\).
Mathematics Subject Classification ID: 51E12
zbMATH DE Number: 806267
zbMATH Keywords: subquadrangle
zbMATH Keywords: symplectic quadrangle
zbMATH Keywords: generalized quadrangle
zbMATH Keywords: Suzuki-Tits ovoid
author: Matthew Brown
reviewed by: Hendrik Van Maldeghem
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 806267