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

TL;DR

This paper investigates the sizes of Sylow subgroups in finite simple groups of Lie type, showing that Sylow r-subgroups for primes r different from p are significantly smaller than the Sylow p-subgroup, with bounds depending on the Lie rank.

Contribution

It provides explicit bounds on the sizes of Sylow r-subgroups for r ≠ p in finite simple groups of Lie type, confirming they are generally much smaller than the Sylow p-subgroup.

Findings

Sylow r-subgroups are at most |T|^{O(log_r(ℓ)/ℓ)} in size

The size of Sylow r-subgroups is significantly smaller than the Sylow p-subgroup

Bounds depend explicitly on the Lie rank ℓ and prime r

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 $|T|^{1/3} < |S| \leqslant |T|^{1/2}$. One might anticipate that, moreover, the Sylow $r$-subgroups of $T$ with $r \neq p$ are usually significantly smaller than $S$. We verify this hypothesis by proving that for every $T$ and every prime divisor $r$ of $|T|$ with $r \neq p$, the order of the Sylow $r$-subgroup of $T$ at most $|T|^{2\lfloor\log_r(4(\ell+1) r)\rfloor/\ell}=|T|^{{\rm O}(\log_r(\ell)/\ell)}$, where $\ell$ is the Lie rank of $T$.