On the secondlargest 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 $T^{1/3} < S \leqslant T^{1/2}$. 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 $T$.
 Publication:

arXiv eprints
 Pub Date:
 December 2017
 DOI:
 10.48550/arXiv.1712.05899
 arXiv:
 arXiv:1712.05899
 Bibcode:
 2017arXiv171205899G
 Keywords:

 Mathematics  Group Theory;
 20D08;
 20E32;
 20E07
 EPrint:
 9 pages, 3 tables