Slanted symplectic quadrangles
type: Item
instance of: scholarly article
title: Slanted symplectic quadrangles
zbMATH Open document ID: 0801.51006
DOI: 10.1007/BF01610617
author: Michael Joswig
author: Theo Grundhöfer
author: Markus Stroppel
published in: Geometriae Dedicata
publication date: 1994-05-18T00:00:00Z
review text: The paper investigates slanted symplectic quadrangles. A slanted symplectic quadrangle over a field \(F\) is constructed from the symplectic quadrangle over \(F\) as follows: The points of the slanted quadrangle are all points of the symplectic quadrangle not collinear with a given point \(s\) of the symplectic quadrangle, the lines are all lines of the symplectic quadrangle not incident with \(s\) plus all lines of the underlying projective space that are incident with \(s\) but that are not lines of the symplectic quadrangle. Lines of the first kind are called absolute lines, lines of the second kind are called special lines. The construction does not depend on the choice of \(s\). For finite fields this construction is well-known. The slanted symplectic quadrangle over \(F\) is represented by the group \(\Gamma= (\text{PSp}_ 4 F)_ s\), i.e., the group \(\Gamma\) acts transitively on the point set and the stabilizer \(\Gamma_ p\) of a point \(p\) fixes the special line through \(p\) and acts transitively on the absolute lines through \(p\). If the field has at least five elements, then the group \(\Delta\) of automorphisms of the slanted symplectic quadrangle over \(F\) is the semidirect product of the group \(\Gamma\) with the group of automorphisms of \(F\). Moreover, the group \(\Delta\) acts transitively on the point set and has precisely two orbits in the set of lines, namely the set of special lines and the set of absolute lines. In particular, this implies that the slanted symplectic quadrangle over a field with at least five elements is not a Moufang quadrangle. For slanted symplectic quadrangles as well as for the dual quadrangles of slanted symplectic quadrangles the groups of (even) projectivities are computed. It turns out that if \(F\) is an infinite field, then the group of even projectivities of the slanted symplectic quadrangle over \(F\) contains the finitary symmetric group on \(F\), i.e., the group of all permutations moving only finitely many e…
reviewed by: Andreas E. Schroth
Mathematics Subject Classification ID: 51E12
Mathematics Subject Classification ID: 51H10
zbMATH DE Number: 569278
zbMATH Keywords: generalized quadrangle
zbMATH Keywords: group of projectivities
zbMATH Keywords: slanted quadrangle
zbMATH Keywords: symplectic quadrangle
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 569278