On the generation of simple groups by Sylow subgroups

TL;DR

This paper proves that finite simple groups of Lie type can be generated by a Sylow 2-subgroup and any nontrivial element, extending to generation by Sylow subgroups of different primes.

Contribution

It establishes a new generation property for finite simple groups, showing they can be generated by Sylow subgroups of different primes and a conjugate element.

Findings

Any nontrivial element can be combined with a Sylow 2-subgroup to generate the entire group.

Finite simple groups are generated by Sylow 2-subgroups and Sylow r-subgroups for any prime divisor r.

The result extends previous work on generation properties of simple groups.

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 $G = \langle P, x^g \rangle$. 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.