Finite permutation groups with few orbits under the action on the power set

Abstract

We study the orbits under the natural action of a permutation group $G\le {\text{S}}_n$ on the powerset $\mathcal{𝒫}\left(\left\{1,\dots ,n\right\}\right)$. The permutation groups having exactly $n+1$ orbits on the powerset can be characterized as set-transitive groups and were fully classified by Beaumont and Peterson in 1955. In this paper, we establish a general method that allows one to classify the permutation groups with $n+r$ set-orbits for a given $r$, and apply it to integers $2\le r\le 15$ with the help of GAP.