On primitive $2$closed permutation groups of rank at most four
Abstract
We characterise the primitive 2closed 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 1dimensional affine semilinear group. These are the first known examples of nonregular 2closed groups that are not the automorphism group of a graph or digraph.
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 arXiv:
 arXiv:2110.07896
 Bibcode:
 2021arXiv211007896G
 Keywords:

 Mathematics  Combinatorics;
 Mathematics  Group Theory;
 20B25;
 05E18