A Theorem of Siemons and Wagner

Abstract

In 1988 Siemons and Wagner describe a relationship between the lengths of $G$-orbits on subsets of a $G$-set $\Omega$. They highlighted the situation where $\Delta \subset \Omega$ with $|\Delta|=k,$ and $|\Delta^G|>|\Sigma^G|$ for all $(k+1)$-subsets, $\Sigma$ of $\Omega$ where $\Delta\subset\Sigma$. They went on to classify all primitive groups with this property for $k=2$. Here we address some questions about primitive permutation groups satisfying this property when $k=3$ and list all $3$-homogeneous groups satisfying this condition.