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A geometric characterization of the groups Suz and HS. (English) Zbl 0704.20015
A rank 3-geometry \(\Gamma =({\mathcal P},{\mathcal L},{\mathcal C};*)\) is called a c.C\({}_ 2\)-geometry if (1) the residues of elements of \({\mathcal P}\) (points) are generalized quadrangles (2) the residues of elements of \({\mathcal L}\) (lines) are generalized 2-gons, and (3) the residues of elements of \({\mathcal C}\) (circles) are isomorphic to the geometry of vertices and edges of a complete graph.
In [Geom. Dedicata (to appear)] A. Del Fra et al. studied \(c.C_ 2\)-geometries whose point residues are classical thick quadrangles, admitting a flag transitive group G. They succeeded in classifying these geometries, except in the case where point residues are isomorphic to the \(U_ 4(3)\)-quadrangle \(H_ 3(3^ 2)\). This paper settles this open question, i.e. it is shown:
Theorem. Let \(\Gamma\) be a connected \(c.C_ 2\)-geometry, whose point residues are isomorphic to the \(U_ 4(3)\)-quadrangle \(H_ 3(3^ 2)\). Suppose \(\Gamma\) admits a flag transitive group G. Then one of the following occurs. (1) The group G is isomorphic to either the sporadic Suzuki group Suz or Aut(Suz) and \(\Gamma\) is uniquely determined. (2) The group G is isomorphic to Aut(HS) \((HS=Higman\)-Sims group), and \(\Gamma\) is uniquely determined.
The construction of these geometries can be found in [M. Ronan, Lond. Math. Soc. Lect. Note Ser. 49, 316-331 (1981; Zbl 0467.51015)] and [the second author, Eur. J. Comb. 11, 81-93 (1990; Zbl 0697.51008)].
Reviewer: U.Dempwolff

MSC:
20D08 Simple groups: sporadic groups
20F65 Geometric group theory
51E25 Other finite nonlinear geometries
51D20 Combinatorial geometries and geometric closure systems
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