Normalisers of maximal tori and a conjecture of Vdovin

TL;DR

This paper proves a strengthened version of Vdovin's conjecture for finite simple groups of Lie type by combining probabilistic and computational methods to analyze normalizers of maximal tori.

Contribution

It introduces a novel approach using probabilistic and computational techniques to verify a modified form of Vdovin's conjecture for these groups.

Findings

Confirmed the modified Vdovin's conjecture for most finite simple groups of Lie type.

Calculated the base size for the natural action of the groups on cosets of the normalizer.

Demonstrated the normalizer intersections are p-groups under certain conditions.

Abstract

Let $G = O^{p'}(\bar{G}^F)$ be a finite simple group of Lie type defined over a field of characteristic $p$, where $F$ is a Steinberg endomorphism of the ambient simple algebraic group $\bar{G}$. Let $\bar{T}$ be an $F$-stable maximal torus of $\bar{G}$ and set $N = N_G(\bar{T})$. A conjecture due to Vdovin asserts that if $G \not\cong {\rm L}_3(2)$ then $N \cap N^x$ is a $p$-group for some $x \in G$. In this paper, we use a combination of probabilistic and computational methods to calculate the base size for the natural action of $G$ on $G/N$, which allows us to prove a stronger, and suitably modified, version of Vdovin's conjecture.