Orbit closed permutation groups, relation groups, and simple groups

Abstract

A permutation group G on a set Omega is called orbit closed if every permutation of Omega preserving the orbits of G in its action on the power set P(Omega ) belongs to G. It is called a relation group if there exists a family R subseteq P(Omega ) such that G is the group of all permutations preserving R. We prove that if a finite orbit closed permutation group G is simple, or is a subgroup of a simple group, then it is a relation group. This result justifies our general conjecture that with a few exceptions every finite orbit closed group is a relation group. To obtain the result, we prove that most of the finite simple permutation groups are relation groups. We also obtain a complete description of those finite simple permutation groups that have regular sets, and prove that (with one exception) if a finite simple permutation group G is a relation group, then every subgroup of G is a relation group.