Bounds for the diameters of orbital graphs of affine groups

TL;DR

This paper establishes bounds on the diameters of orbital graphs in finite affine primitive groups, providing explicit bounds based on the group's parameters without relying on the classification of finite simple groups.

Contribution

It introduces general bounds for orbital diameters in affine primitive groups and constructs families with large diameters, independent of finite simple group classification.

Findings

Orbital diameter bounded by dimension and subgroup size ratios

Construction of infinite families with large orbital diameters

Bounds applicable without classification of finite simple groups

Abstract

General bounds are presented for the diameters of orbital graphs of finite affine primitive permutation groups. For example, it is proved that the orbital diameter of a finite affine primitive permutation group with a nontrivial point stabilizer $ H \leq \mathrm{GL}(V) $, where the vector space $ V $ has dimension $ d $ over the prime field, can be bounded in terms of $ d $ and $ \log |V| / \log |H| $ only. Several infinite families of affine primitive permutation groups with large orbital diameter are constructed. The results are independent from the classification of finite simple groups.