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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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