A geometric characterization of the groups Suz and HS

Abstract

Buekenhout and Hubaut [BH] studied finite connected c.C,-geometries (more generally, c.C,,-geometries) r whose point-residues are classical quadrangles, admitting a flag-transitive group G whose point stabilizer induces an automorphism group containing the corresponding classical simple group. In their Lemma 7.3, however, they overlooked the group M,, as a transitive extension of M,, acting on the 10 points of a line in a U,(3)quadrangle H,(3’). There is, in fact, an interesting c.C,-geometry associated with the sporadic Suzuki group Suz with such point residues (see [BF], [Bu]). This is a 3-local geometry; it was described in [St]; [Ro, 7.2, pp,326-3271. The set 9 is a certain conjugacy class of subgroups of order 3 in SUZ, and the sets 9 and 9 consist of pairs and maximal sets of mutually commuting subgroups in 9”, respectively. Recently, A. Del Fra, D. Ghinelli, T. Meixner, and A. Pasini [FGMP] studied c.C,-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 again isomorphic to the U,(3)-quadrangle H,(3’).