On the generation of simple groups by Sylow subgroups

Abstract

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

arXiv comment17 pages; to appear in Contemporary Mathematics (Amitsur Centennial volume)