Normalisers of maximal tori and a conjecture of Vdovin

Abstract

Let G=Op′(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 G¯. Let T¯ be an F-stable maximal torus of G¯ and set N=NG(T¯). A conjecture due to Vdovin asserts that if G≇L3(2) then N∩Nx is a p-group for some x∈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.