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 $|T|^{1/3} < |S| \leqslant |T|^{1/2}$. 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 e-prints
- Pub Date:
- December 2017
- DOI:
- 10.48550/arXiv.1712.05899
- arXiv:
- arXiv:1712.05899
- Bibcode:
- 2017arXiv171205899G
- Keywords:
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- Mathematics - Group Theory;
- 20D08;
- 20E32;
- 20E07
- E-Print:
- 9 pages, 3 tables