On the generation of simple groups by Sylow subgroups

Abstract

Let G G be a finite simple group of Lie type and let P P be a Sylow 2 2 -subgroup of G G . In this paper, we prove that for any nontrivial element x ∈ G x \in G , there exists g ∈ G g \in G such that G = ⟨ P , x g ⟩ G = \langle P, x^g \rangle . By combining this result with recent work of Breuer and Guralnick, we deduce that if G G is a finite nonabelian simple group and r r is any prime divisor of | G | |G| , then G G is generated by a Sylow 2 2 -subgroup and a Sylow r r -subgroup.