On primitive $2$-closed permutation groups of rank at most four

TL;DR

This paper characterizes primitive 2-closed permutation groups of rank at most four that are not automorphism groups of graphs or digraphs, revealing new examples beyond regular groups.

Contribution

It provides the first known examples of non-regular 2-closed groups not arising as automorphism groups of graphs or digraphs, and classifies such groups under certain conditions.

Findings

Only two infinite families of such groups exist for degree at least 2402.

All such groups are subgroups of the affine semilinear group .

The paper identifies specific structural properties of these groups.

Abstract

We characterise the primitive 2-closed groups $G$ of rank at most four that are not the automorphism group of a graph or digraph and show that if the degree is at least 2402 then there are just two infinite families or $G\leqslant \mathrm{A}\Gamma\mathrm{L}_1(p^d)$, the 1-dimensional affine semilinear group. These are the first known examples of non-regular 2-closed groups that are not the automorphism group of a graph or digraph.