Generalized quadrangles and flocks of cones
type: Item
instance of: scholarly article
title: Generalized quadrangles and flocks of cones
zbMATH Open document ID: 0646.51019
DOI: 10.1016/S0195-6698(87)80051-3
published in: European Journal of Combinatorics
publication date: 1987-01-01T00:00:00Z
review text: A flock of the quadratic cone K of PG(3,q) is a partition of K but its vertex into disjoint conics. It is called linear if the planes of the q conics of such a flock all contain a common line. A flock is linear if and only if there corresponds a Desarguesian translation plane to it. W. M. Kantor has proved that with a set of q upper triangular \(2\times 2\) matrices over GF(q) of a certain type, there corresponds a generalized quadrangle of order \((q^ 2,q)\). In this paper, the author proves that with such a set of q matrices, there corresponds a flock of the quadratic cone of PG(3,q), and conversely with each flock of the quadratic cone there corresponds such a set of matrices. Using this relationship, new flocks and new generalized quadrangles are obtained. Some interesting open problems are also noted.
Mathematics Subject Classification ID: 51E20
zbMATH DE Number: 4054454
zbMATH Keywords: translation plane
zbMATH Keywords: quadratic cone
zbMATH Keywords: flocks
zbMATH Keywords: generalized quadrangles
author: Joseph A. Thas
reviewed by: T. Thrivikraman
MaRDI profile type: Publication
cites work: Q4755274
cites work: 4-dimensionale Translationsebenen mit 8-dimensionaler Kollineationsgruppe
cites work: Q5184989
cites work: A Theorem on Permutations in a Finite Field
cites work: Q3684747
cites work: Q5545191
cites work: Geometry According to Euclid
cites work: Flocks in \(\mathrm{PG}(3,q)\)
cites work: Q3676729
cites work: Q3851217
cites work: Q3721310
cites work: Generalized quadrangles associated with \(G_ 2(\)q)
cites work: Ovoids and Translation Planes
cites work: Q3713056
cites work: Q3769506
cites work: Q3344217
cites work: Q4163985
cites work: Q4129398
cites work: Construction of maximal arcs and partial geometries
cites work: Q4091961
cites work: Q4131245
cites work: Ovoids and spreads of finite classical polar spaces
cites work: Sur la trialité et certains groupes qui s'en déduisent
cites work: A class of translation planes
cites work: Flocks of a quadratic cone in PG(3,q), q\(\leq 8\)
cites work: Flocks of quadratic cones, generalized quadrangles and translation planes
cites work: Q3352045
description: scientific article; zbMATH DE number 4054454