Extending polar spaces of rank at least 3
type: Item
instance of: scholarly article
title: Extending polar spaces of rank at least 3
zbMATH Open document ID: 0845.51004
DOI: 10.1016/0097-3165(95)90062-4
published in: Journal of Combinatorial Theory. Series A
publication date: 1996-09-02T00:00:00Z
review text: An incidence system is a pair \(\Gamma = \Gamma ({\mathcal P}, {\mathcal B})\), where \({\mathcal P}\) is a set of points and \({\mathcal B}\) is a family of subsets of \({\mathcal P}\) called blocks such that each block of \({\mathcal B}\) has at least two distinct points, and the incidence between points and blocks is defined by inclusion. Two points \(p,q\) are said to be adjacent if they lie in a common block. The point graph of \(\Gamma\) is a graph with vertex set \({\mathcal P}\) and with adjacency same as in \(\Gamma\). The residue \(\Gamma_p\) of a point \(p\) is the incidence system of points adjacent to \(p\) excluding \(p\) itself together with blocks on \(p\) with the point \(p\) removed. We say that \(\Gamma\) is connected if its point graph is connected. And finally, connected incidence system \(\Gamma = \Gamma ({\mathcal P}, {\mathcal B})\) is an extended polar space or extended (projective) plane if its point residues are finite, thick, nondegenerate polar spaces, or finite nondegenerate projective planes, respectively. In this paper under review, which is a continuation of an earlier author's work [J. Comb. Theory, Ser. A 68, No. 1, 100-114 (1994; Zbl 0808.05060)], the following main theorem is proved: If \(\Gamma\) is an extended polar space admitting extended planes, then either (i) \(\Gamma\) is a standard quotient of an affine polar space over \(GF(2)\), or (ii) \(\Gamma\) is the extended polar space for the sporadic Fischer group \(Fi_{22}\), or (iii) there are exactly two extended planes on each block and \(\Gamma\) is an extended \(Q^+_5 (4)\)-polar space. From this theorem it immediately follows that the extended polar space for the sporadic Fischer group \(Fi_{22}\) is the only extended polar space which has more than two extended planes on a block and is not isomorphic to a quotient of an affine polar space over \(GF(2)\). New examples of \(EGQ(4,1)\) and \(EGQ(4,2)\) are presented, as well.
reviewed by: J.Libicher
Mathematics Subject Classification ID: 51A50
Mathematics Subject Classification ID: 51E12
Mathematics Subject Classification ID: 05B30
zbMATH DE Number: 823494
zbMATH Keywords: extended polar space
Wikidata QID: Q56872298
author: Dmitrii V. Pasechnik
describes a project that uses: GAP
describes a project that uses: GRAPE
MaRDI profile type: Publication
full work available at URL: 0097-3165(95)90062-4
OpenAlex ID: W2071197262
cites work: Near polygons and Fischer spaces
cites work: Q3992965
cites work: Extensions of polar spaces and the doubly transitive symplectic groups
cites work: Diagrams for geometries and groups
cites work: Locally polar spaces and related rank 3 groups
cites work: Extended generalized quadrangles
cites work: Affine polar spaces
cites work: Locally polar geometries with affine planes
cites work: Q5545191
cites work: Flag-transitive extensions of \(C_ n\) geometries
cites work: Locally cotriangular graphs
cites work: Extensions of designs and groups: projective, symplectic and certain affine groups
cites work: On geometries of the Fischer groups
cites work: On vector enumeration
cites work: Some polar towers
cites work: Q4276299
cites work: Geometric characterization of the sporadic groups \(Fi_{22}\), \(Fi_{23}\), and \(Fi_{24}\)
cites work: Q4273617
cites work: Buildings of spherical type and finite BN-pairs
cites work: A characterization of the groups Fi 22, Fi 23 and Fi 24
description: scientific article; zbMATH DE number 823494