The automorphism groups of small affine rank 3 graphs

TL;DR

This paper investigates the automorphism groups of affine rank 3 graphs, providing explicit descriptions for small cases and reducing the general problem to cases with nonabelian simple socles.

Contribution

It explicitly determines automorphism groups for small affine rank 3 graphs and reduces the broader classification problem to a specific algebraic case.

Findings

Automorphism groups of small affine rank 3 graphs are explicitly found.

The classification reduces to cases with nonabelian simple socles.

Provides partial results towards the full automorphism group classification.

Abstract

A rank 3 graph is an orbital graph of a rank 3 permutation group of even order. Despite the classification of rank 3 graphs being complete, see, e.g., Chapter 11 of the recent monograph 'Strongly regular graphs' by Brouwer and Van Maldeghem, the full automorphism groups of these graphs (equivalently, the 2-closures of rank 3 groups) have not been explicitly described, though a lot of information on this subject is available. In the present note, we address this problem for the affine rank 3 graphs. We find the automorphism groups for finitely many relatively small graphs and show that modulo known results, this provides the full description of the automorphism groups of the affine rank 3 graphs, thus reducing the general problem to the case when the socle of the automorphism group is nonabelian simple.