A certain uniqueness result for the tilde geometry for \(F_4(2)\)
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.
scientific article; zbMATH DE number 1088605