On the Orbital Diameter of Groups of Diagonal Type

TL;DR

This paper establishes explicit bounds on the orbital diameters of primitive groups of simple diagonal type, classifies those with diameter ≤ 4, and explores related properties of finite simple groups.

Contribution

It provides the first explicit bounds for orbital diameters of simple diagonal groups and classifies those with small diameters, advancing understanding of their structure.

Findings

Classified simple diagonal groups with orbital diameter ≤ 4

Bounded the covering number and conjugacy width of certain simple groups

Derived general bounds for groups of Lie type

Abstract

The orbital diameter of a primitive permutation group is the maximal diameter of its orbital graphs. There has been a lot of interest in bounds for the orbital diameter. In this paper we provide explicit bounds on the diameters of groups of simple diagonal type. As a consequence we obtain a classification of simple diagonal groups with orbital diameter less than or equal to 4. As part of this, we classify all finite simple groups with covering number and conjugacy width at most 3. We also prove some general bounds on the covering number and conjugacy width of groups of Lie type.