On the second-largest Sylow subgroup of a finite simple group of Lie type

Abstract

Let $T$ be a finite simple group of Lie type in characteristic $p$, and let $S$ be a Sylow subgroup of $T$ with maximal order. It is well known that $S$ is a Sylow $p$-subgroup except in an explicit list of exceptions, and that $S$ is always `large' in the sense that $|T{|}^{1/3}<|S|⩽|T{|}^{1/2}$. One might anticipate that, moreover, the Sylow $r$-subgroups of $T$ with $r\ne p$ are usually significantly smaller than $S$. We verify this hypothesis by proving that for every $T$ and every prime divisor $r$ of $|T|$ with $r\ne p$, the order of the Sylow $r$-subgroup of $T$ at most $|T{|}^{2\lfloor {\log }_r(4(\ell +1)r)\rfloor /\ell }=|T{|}^{\mathrm{O}({\log }_r(\ell )/\ell )}$, where $\ell$ is the Lie rank of $T$.

9 pages, 3 tables