A certain uniqueness result for the tilde geometry for \(F_4(2)\)
type: Item
instance of: scholarly article
title: A certain uniqueness result for the tilde geometry for \(F_4(2)\)
zbMATH Open document ID: 0918.51017
author: Corinna Wiedorn
published in: Journal of Algebra
publication date: 1999-08-04T00:00:00Z
review text: In her PhD thesis [Some semiclassical parabolic systems of rank 4, Dissertation, Martin-Luther-Universität Halle-Wittenberg (1997); see also J. Algebra 211, No. 2, 472-513 (1999)] the author has investigated semiclassical flag-transitive geometries \(\Gamma\) of rank 4 with connected diagram, such that \(\Gamma\) has a rank 3 residue of type \((\widetilde C_3)\) and every such residue has an induced automorphism group isomorphic to \(3^7 Sp_6(2)\) (here a geometry of type \((\widetilde C_3)\) is one of the geometries classified by the reviewer in [J. Algebra 142, No. 1, 188-200 (1991; Zbl 0735.20007)]. One case of particular interest is the classification of geometries of type \((\widetilde F_4)\). (These are geometries similar to the \(F_4(2)\)-geometry, where instead of the rank 2 residues isomorphic to the generalized \(Sp_4(2)\)-quadrangle there are tilde geometries for \(3 S_6\).) The author has shown in her thesis, that the automorphism group \(G\) of such a geometry admits a homomorphism \(\varphi\) onto \(F_4(2)\). In the paper under review it is shown that under the hypothesis that the kernel of \(\varphi\) is abelian, it follows that \(G\) is a non-split extension \(3^{833} F_4(2)\). It has also been shown by the author that indeed there exists a geometry of type \((\widetilde F_4)\) with automorphism group isomorphic to \(3^{833} F_4(2)\) [Beitr. Algebra Geom. 38, No. 2, 337-342 (1997; Zbl 0890.51010)]. Meanwhile, a classification of semiclassical parabolic systems has been obtained by A. Fukshansky, G. Stroth and the author.
Mathematics Subject Classification ID: 51E24
Mathematics Subject Classification ID: 20D06
Mathematics Subject Classification ID: 20E06
Mathematics Subject Classification ID: 20B25
zbMATH DE Number: 1088605
zbMATH Keywords: flag-transitive geometries
zbMATH Keywords: semiclassical parabolic systems
zbMATH Keywords: rank 4 amalgams
reviewed by: Stefan Heiss
MaRDI profile type: Publication
full work available at URL: jabr.1997.7056
OpenAlex ID: W2075849946
cites work: Q3684278
cites work: On a parabolic system of type \(M_{24}\)
cites work: An infinite family of simply connected flag-transitive tilde geometries
cites work: A geometric characterization of Fischer's Baby Monster
cites work: Minimal parabolic geometries for the sporadic groups
cites work: Natural representations of some tilde and Petersen type geometries
cites work: Some semiclassical parabolic systems of rank \(4\)
DOI: 10.1006/JABR.1997.7056
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description: scientific article; zbMATH DE number 1088605