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Let X be a finite set such that |X| = n and let i 6 j 6 n. A group G 6 Sn 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 Ig ⊆ J. (Clearly (i,i)-homogeneity is i-homogeneity in the usual sense.) A group G 6 Sn is said to have the k-universal transversal property if given any set I ⊆ X (with |I| = k) and any partition P of X into k blocks, there exists g ∈ G such that Ig 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 prop- erty (with the exception of two classes of 2-homogeneous groups) and the (k−1,k)-homogeneous groups (for 2 < k 6 ⌊ n+1 ⌋). As a corollary of the clas- sification 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 groups that together with any rank k transformation on X generate a regular semigroup (for 1 6 k 6 ⌊ n+1 ⌋). The paper ends with a number of challenges for experts in number theory, group and/or semigroup theory, linear algebra and matrix theory.

Two generalizations of homogeneity in groups with applications to regular semigroups