Permutation Groups and Orbits on Power Sets

Abstract

Let $G$ be a permutation group of degree $n$ and let $s(G)$ denote the number of set-orbits of $G$. We determine $\inf (\frac{{\log }_{2}s(G)}{n})$ over all groups $G$ that satisfy certain restrictions on composition factors (i.e. $Alt(k),k>4$ cannot be obtained as a quotient of a subgroup of $G$).