The residually weakly primitive geometries of S5×2

Abstract

We classify all firm and residually connected coset geometries satisfying the intersection property (IP) 2, and on which the group S 5×2 acts flag-transitively and residually weakly primitively. This work was motivated by a study of the Ivanov–Shpectorov coset geometry for the O'Nan sporadic simple group (see Buekenhout, Leemens, J. Combin. Theory Ser. A 85(2) (1999) 148). The importance of having a list of residually weakly primitive coset geometries for S 5×2 is also shown by Buekenhout et al. (in: A. Pasini et al. (Eds.), Groups and Geometries, Birkhäuser, Basel 1998, pp. 39–54) and Buekenhout and Dony (Bull. Soc. Math. Belg. XLII (1990) 471). We extend the concept of direct sum introduced by Valette in (Simon Stevin 56(3) (1982) 167) to coset geometries and show that all coset geometries are either direct sums of coset geometries of S 5 and 2 satisfying the same properties or are extensions of lower rank coset geometries given by a theorem of Leemans’ (see Leemons, Master's Thesis, Vol. 1, Université Libre de Bruxelles, 1994.).