Normalisers of maximal tori and a conjecture of Vdovin

Let $G=O^{p^{\mathrm{\prime }}}({\bar{G}}^F)$ 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 $G\cong ̸{\mathrm{L}}_3(2)$ 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 $G\mathrm{/}N$, which allows us to prove a stronger, and suitably modified, version of Vdovin’s conjecture.