Two Generalizations of Homogeneity in Groups with Applications to Regular Semigroups

Let $X$ be a finite set such that $\mathrm{\mid }X\mathrm{\mid }=n$ and let $i\le j\le n$. A group $G\le \text{sym}$ is said to be $(i,j)$-homogeneous if for every $I,J\subseteq X$, such that $\mathrm{\mid }I\mathrm{\mid }=i$ and $\mathrm{\mid }J\mathrm{\mid }=j$, there exists $g\in G$ such that $Ig\subseteq J$. (Clearly $(i,i)$-homogeneity is $i$-homogeneity in the usual sense.) A group $G\le \text{sym}$ is said to have the $k$-universal transversal property if given any set $I\subseteq X$ (with $\mathrm{\mid }I\mathrm{\mid }=k$) and any partition $P$ of $X$ into $k$ blocks, there exists $g\in G$ 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 property (with the exception of two classes of 2-homogeneous groups) and the $(k-1,k)$-homogeneous groups (for $2<k\le \lfloor \frac{n+1}{2}\rfloor$). 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 groups that together with any rank $k$ transformation on $X$ generate a regular semigroup (for $1\le k\le \lfloor \frac{n+1}{2}\rfloor$). The paper ends with a number of challenges for experts in number theory, group and/or semigroup theory, linear algebra and matrix theory.