Symmetry groups of boolean functions: simple groups

TL;DR

This paper characterizes simple permutation groups that serve as symmetry groups of Boolean functions, linking group theory with hypergraph automorphisms, and explores properties like regular sets within these groups.

Contribution

It provides a complete classification of simple groups that are relation groups and analyzes their subgroups and regular sets.

Findings

Characterization of simple relation groups

Complete description of simple groups with regular sets

Most subgroups of these simple groups are also relation groups

Abstract

We consider the problem of characterizing the class of those permutation groups that are the symmetry groups of Boolean functions. These are exactly the automorphism groups of hypergraphs. They are also called the relation groups. In this paper we describe those of them that are simple as abstract groups. This is done by combining results based on the classification of finite simple groups with the description of intransitive actions of simple groups. We also obtain a complete characterization of those simple permutation groups that have regular sets, and prove that (with one exception) if a simple permutation group G is a relation group, then every subgroup of G is a relation group.