Orbit-homogeneity in permutation groups

This paper introduces the concept of orbit-homogeneity of permutation groups: a group G is orbit-t-homogeneous if two sets of cardinality t lie in the same orbit of G whenever their intersections with each G-orbit have the same cardinality. For transitive groups, this coincides with the usual notion of t-homogeneity. This concept is also compatible with the idea of partition transitivity introduced by Martin and Sagan. Further, this paper shows that any group generated by orbit-t-homogeneous subgroups is orbit-t-homogeneous, and that the condition becomes stronger as t increases up to [n/2], where n is the degree. So any group G has a unique maximal orbit-t-homogeneous subgroup Ω_t(G), and Ω_t(G) ≤ Ω_{t t-1}(G). Some structural results for orbit-t-homogeneous groups, and a number of examples, are also given.