Alex Dent
Further results on the transitivity of intransitive permutation groups.
We return to the concept of a orbit-homogeneous permutation group. An orbit-t-homogeneous permutation group is one in which any two sets of t points which contain equal numbers of points in each orbit can be mapped onto each other. This generalises the notion of homogeneity into intransitive groups. Following the approach of Martin and Sagan, we examine the action of group on partitions of a point set and, in particular, groups that acts transitively on the different partitions of a set of points. Again, we examine the case where the group acts intransitively on the point set and define a new notion, which we term orbit-transitivity, to cover this problem. This generalises the idea of orbit-homoegenity. (This extends results of us presented in this seminar years ago.)