Minimal parabolic geometries for the sporadic groups
Most of the finite simple groups are of Lie type, and these act on geometries called Tits buildings [\textit{J. Tits}, Buildings of spherical type, Lect. Notes Math. 386 (1974; Zbl 0295.20047)]. Analogous geometries for sporadic simple groups were first studied by \textit{F. Buekenhout} [J. Comb. Theory, Ser. A 27, 121-151 (1979; Zbl 0419.51003)] with further examples given by \textit{W. Kantor} [Eur. J. Comb. 2, 239-247 (1981; Zbl 0514.51006)]. The idea of concentrating on geometries defined by p-local subgroups was introduced by \textit{M. Ronan} and the reviewer [Proc. Symp. Pure Math. 37, 283-289 (1980; Zbl 0478.20015)]. Whereas that paper considered ''maximal parabolics'' as basic, the present paper begins with ''minimal parabolics'', subgroups minimal with respect to containing some fixed subgroup B (typically the normalizer of a Sylow p-group). A set of such subgroups defines a chamber system in the sense of \textit{J. Tits} [in The Geometric Vein, 519-547 (1982; Zbl 0496.51001)]. The authors determine, for the various sporadic groups and their prime divisors, such minimal parabolic geometries. This work has been very influential in the modern study of groups and geometries. It contains much useful information on the subgroup structure of sporadic groups.
scientific article; zbMATH DE number 3916544
scientific article