Beiträge zur Algebra und GeometrieContributions to Algebra and Geometry
Vol. 48, No. 1, pp. 49-58 (2007)
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Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Vol. 48, No. 1, pp. 49-58 (2007) |
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Some notes on generalized quadrangles of order $s$ with a span of regular pointsBart De Bruyn and Stanley E. PayneDepartment of Pure Mathematics and Computer Algebra, Ghent University, Galglaan 2, 9000 Gent, Belgium, e-mail: bdb@cage.ugent.be; Department of Mathematics, University of Colorado at Denver, Campus Box 170, P.O. box 173364, Denver, Colorado 80217-3364, United States, e-mail: stanpayne@mac.comAbstract: Let $Q$ be a generalized quadrangle of order $s \geq 2$ with a regular point $x$. The set $x^\perp$ together with all spans which are contained in $x^\perp$ define a projective plane $\pi_x$ of order $s$. We introduce a property ($P_y$) for every point $y$ of $Q$ noncollinear with $x$ and prove that this property is equivalent with the regularity of the point $y$. We will use this to give an elementary proof for the following result: every generalized quadrangle of order $q$ which has a center of symmetry $x$ such that $\pi_x \cong {\mathrm{PG}}(2,q)$ and a regular point $y$ noncollinear with $x$ is isomorphic to $W(q)$. Keywords: generalized quadrangle, center of symmetry Classification (MSC2000): 51E12 Full text of the article: Electronic version published on: 14 May 2007. This page was last modified: 27 Jan 2010.
© 2007 Heldermann Verlag
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