Highly transitive imprimitivities

Some new observations are made about imprimitive permutation groups associated with subfactors of von Neuman algebras. Of particular interest are examples of a group G containing two maximal subgroups H and K such that G≠HK, and such that the action of G on the space of cosets of H∩K has small rank (few suborbits). The rank 6 case turns out to correspond to the action of the collineation group on flags of a Desarguesian projective plane, and a special case of interest for rank 7 corresponds to the action of a 4-transitive group on ordered pairs of distinct points. Some other new (and unexpected) fundamental properties of groups are described along the way.