Normalisers of maximal tori and a conjecture of Vdovin.

Let G = O p ′ ( 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 = N G (T ¯). A conjecture due to Vdovin asserts that if G ≇ L 3 (2) then N ∩ N x 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. [ABSTRACT FROM AUTHOR]