On the orbital diameter of classical groups in standard action

TL;DR

This paper investigates the orbital diameter of almost simple primitive groups in standard actions, providing bounds and classifying cases with small diameters, enhancing understanding of their permutation properties.

Contribution

It offers a lower bound for the orbital diameter and partially classifies pairs with diameter at most 2 for almost simple groups in standard actions.

Findings

Established a lower bound for the orbital diameter.

Partially classified pairs with orbital diameter ≤ 2.

Analyzed the structure of orbital graphs in primitive groups.

Abstract

Let $G$ be a primitive permutation group acting on a finite set $X$. The orbital diameter $\mathrm{diam}(X,G)$ is defined to be the supremum of the diameters of the (connected) orbital graphs of $G$ after disregarding the directions of all edges in the graphs. This invariant is studied in the case when $G$ is an almost simple group in a standard action. A lower bound is given for $\mathrm{diam}(X,G)$ and we provide a partial classification of pairs $(X,G)$ for which the orbital diameter is at most $2$.