Two generalizations of homogeneity in groups with applications to regular semigroups

Let X be a finite set such that |X| = n and let n i ≤ j ≤ n. A group G ≤ S_n is said to be (i, j)-homogeneous if for every I, J ⊆ X, such that |I| = i and |J| = j, there exists g ∈ G such that I_g ⊆ J. (Clearly (i, i)-homogeneity is i-homogeneity in the usual sense.)

A group G ≤ S_n is said to have the k-universal transversal property if given any set I ⊆ X (with |I| = k) and any partition P ofX into k blocks, there exists g ∈ G such that I_g is a section for P. (That is, the orbit of each k-subset of X contains a section for each k-partition of X.
In this paper we classify the groups with the k-universal transversal property (with the exception of two classes of 2-homogeneous groups) and the (k-1,k)-homogeneous groups (for 2 <k ≤  (n+1)/2). As a corollary of the classification we prove that a (k-1,k)-homogeneous group is also (k-2,k-1)-homogeneous, with two exceptions; and similarly, but with no exceptions, groups having the k-universal transversal property have the (k-1)-universal transversal property.
A corollary of all the previous results is a classification of the groupsthat together with any rank k transformation on X generate aregular semigroup (for 1 ≤ k ≤ (n+1)/2).
The paper ends with a number of challenges for experts in number theory,group and/or semigroup theory, linear algebra and matrix theory.