Beiträge zur Algebra und Geometrie
Contributions to Algebra and Geometry
Vol. 51, No. 1, pp. 191-207 (2010)
Generalized quadrangles and projective axes of symmetry
Günter F. Steinke and Hendrik Van MaldeghemDepartment of Mathematics and Statistics, University of Canterbury, Private Bag 4800, Christchurch, New Zealand, e-mail: G.Steinke@math.canterbury.ac.nz; Department of Pure Mathematics and Computer Algebra, Ghent University, Galglaan 2, B-9000 Gent, Belgium, e-mail: hvm@cage.UGent.be
Abstract: We investigate generalized quadrangles $\Gamma$ that admit at least two projective axes of symmetry. We show that if there are three such axes incident with a common point $x$, then $x$ is a translation point of $\Gamma$. In case that $\Gamma$ is moreover a compact connected quadrangle with topological parameters $(p,p)$, $p\in\N$, then $\Gamma$ is a topological translation generalized quadrangle. We further investigate the case of two opposite projective axes of symmetry and obtain a characterization of the dual of the symplectic quadrangle over $\R$ or $\C$ among compact connected quadrangles with equal topological parameters.
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Electronic version published on: 27 Jan 2010. This page was last modified: 28 Jan 2013.