Subgroups of simple primitive permutation groups defined by unordered relations (Automorphism groups of hypergraphs)

TL;DR

This paper investigates the automorphism groups of unordered relations, showing that most subgroups of finite simple primitive permutation groups (excluding the alternating group) are relation groups, with only four exceptions.

Contribution

It extends the understanding of relation groups by proving that all but four subgroups of finite simple primitive groups (not including A_n) are relation groups.

Findings

Most subgroups of finite simple primitive groups are relation groups.

Only four subgroups of these groups are exceptions.

The result applies to all simple primitive groups except the alternating groups.

Abstract

The problem of describing the invariance groups of unordered relations, called briefly \emph{relation groups}, goes back to classical work by H. Wielandt. In general, the problem turned out to be hard, and so far it has been settled only for a few special classes of permutation groups. The problem have been solved, in particular, for the class of primitive permutation groups, using the classification of finite simple groups and other deep results of permutation group theory. In this paper we show that, if $G$ is a finite simple primitive permutation group other then the alternating group $A_n$, then each subgroup of $G$, with four exceptions, is a relation group.