# Finite Permutation Groups with Few Orbits Under the Action on the Power Set

## TL;DR

This paper develops a general method to classify permutation groups with a small number of orbits on the power set, extending previous classifications for groups with exactly n+1 orbits to those with n+r orbits for 2 ≤ r ≤ 15.

## Contribution

It introduces a new systematic approach to classify permutation groups with few orbits on the power set, applicable for small values of r, using computational tools.

## Key findings

- Classified permutation groups with n+r orbits for 2 ≤ r ≤ 15.
- Established a general method for such classifications.
- Extended known classifications beyond the case of n+1 orbits.

## Abstract

We study the orbits under the natural action of a permutation group $G \subseteq S_n$ on the powerset $\mathscr{P}(\{1, \dots , n\})$. The permutation groups having exactly $n+1$ orbits on the powerset can be characterized as set-transitive groups and were fully classified in \cite{BP55}. 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 $2 \leq r \leq 15$ using the computer algebra system GAP.