Translation generalized quadrangles
type: Item
instance of: scholarly article
title: Translation generalized quadrangles
zbMATH Open document ID: 0856.51004
DOI: 10.1007/BF01195241
author: Michael Joswig
published in: Archiv der Mathematik
publication date: 1997-02-03T00:00:00Z
review text: A translation generalized quadrangle (TGQ) is a generalized quadrangle admitting a (uniquely determined) abelian group \(T\) of collineations fixing each line passing through some point \(\infty\) and acting transitively (and therefore regularly) on the set of points opposite to \(\infty\). Finite TGQ have been introduced by \textit{J. A. Thas} in Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. fis. mat. nat. 56, 303-314 (1974; Zbl 0327.05028)]. In this paper arbitrary TGQ are investigated. It is shown that there is an entirely algebraic description for TGQ which plays the same rôle as spreads do for translation (projective) planes. In particular, it is proved that the kernel of a TGQ, a certain subring of \(\text{End} (T)\) which is an invariant of the geometry, is in fact a skew field. Furthermore, TGQ with multiple translation centers and axes are examined.
reviewed by: Michael Joswig
Mathematics Subject Classification ID: 51E12
Mathematics Subject Classification ID: 05B30
zbMATH DE Number: 938601
zbMATH Keywords: translation generalized quadrangle
Wikidata QID: Q117245128
MaRDI profile type: Publication
cites work: Über nicht-Desarguessche Ebenen mit transitiver Translationsgruppe
cites work: Q5545191
cites work: Algebraic properties of quadratic quaternary rings
cites work: Q5683869
cites work: Isotopy of polygonal domains for generalized polygons
cites work: Generalized quadrangles associated with \(G_ 2(\)q)
cites work: Some generalized quadrangles with parameters \(q^ 2,q\)
cites work: Translation planes. Foundations and construction principles
cites work: Some Constructions of Infinite Mobius Planes
cites work: Finite translation planes
cites work: Semi-regular graph automorphisms and generalized quadrangles
cites work: Q4692803
cites work: Half-regular and regular points in compact polygons
cites work: Projektive Ebenen und Kegelschnitte
cites work: Q4091961
cites work: Moufang polygons. I: Root data
cites work: Q3814301
description: scientific article; zbMATH DE number 938601