Two generalizations of homogeneity in groups with applications to regular semigroups

Abstract:

Let $X$ be a finite set such that $|X|=n$ and let $i\leqslant j \leqslant n$. A group $G\leqslant \mathcal {S}_{n}$ is said to be $(i,j)$-homogeneous if for every $I,J\subseteq X$, such that $|I|=i$ and $|J|=j$, there exists $g\in G$ such that $Ig\subseteq J$. (Clearly $(i,i)$-homogeneity is $i$-homogeneity in the usual sense.)

A group $G\leqslant \mathcal {S}_{n}$ is said to have the $k$-universal transversal property if given any set $I\subseteq X$ (with $|I|=k$) 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 $2<k\leqslant \lfloor \frac {n+1}{2}\rfloor$). 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 $1\leqslant k\leqslant \lfloor \frac {n+1}{2}\rfloor$).

The paper ends with a number of challenges for experts in number theory, group and/or semigroup theory, linear algebra and matrix theory.