On the Orbits of Singer Groups and Their Subgroups

We study the action of Singer groups of projective geometries (and their subgroups) on (d − 1)-flats for arbitrary d. The possibilities which can occur are determined, and a formula for the number of orbits of each possible size is given. Motivated by an old problem of J.R. Isbell on the existence of certain permutation groups we pose the problem of determining, for given q and h, the maximum co-dimension fq(n, h) of a flat of PG(n − 1, q) whose orbit under a subgroup of index h of some Singer group covers all points of PG(n − 1, q). It is clear that fq(n, h) < n− logq(h); on the other hand we show that fq(n, h) ≥ n− 1− 2 logq(h).