The Classification of Partition Homogeneous Groups with Applications to Semigroup Theory

Let $\lambda=(\lambda_1,\lambda_2,...)$ be a \emph{partition} of $n$, a sequence of positive integers in non-increasing order with sum $n$. Let $\Omega:=\{1,...,n\}$. An ordered partition $P=(A_1,A_2,...)$ of $\Omega$ has \emph{type} $\lambda$ if $|A_i|=\lambda_i$. Following Martin and Sagan, we say that $G$ is \emph{$\lambda$-transitive} if, for any two ordered partitions $P=(A_1,A_2,...)$ and $Q=(B_1,B_2,...)$ of $\Omega$ of type $\lambda$, there exists $g\in G$ with $A_ig=B_i$ for all $i$. A group $G$ is said to be \emph{$\lambda$-homogeneous} if, given two ordered partitions $P$ and $Q$ as above, inducing the sets $P'=\{A_1,A_2,...\}$ and $Q'=\{B_1,B_2,...\}$, there exists $g\in G$ such that $P'g=Q'$. Clearly a $\lambda$-transitive group is $\lambda$-homogeneous. The first goal of this paper is to classify the $\lambda$-homogeneous groups. The second goal is to apply this classification to a problem in semigroup theory. Let $\trans$ and $\sym$ denote the transformation monoid and the symmetric group on $\Omega$, respectively. Fix a group $H\leq \sym$. Given a non-invertible transformation $a\in \trans\setminus \sym$ and a group $G\leq \sym$, we say that $(a,G)$ is an \emph{$H$-pair} if the semigroups generated by $\{a\}\cup H$ and $\{a\}\cup G$ contain the same non-units, that is, $< a,G>\setminus G=< a,H>\setminus H$. Using the classification of the $\lambda$-homogeneous groups we classify all the $\sym$-pairs. This topic involves both group theory and semigroup theory; we have attempted to include enough exposition to make the paper self-contained for researchers in both areas. The paper finishes with a number of open problems on permutation and linear groups.