Let $p$ be a prime, and let $G$ be any of the finite classical groups $SL_n(\mathbb{F}_p)$, $O_n(\mathbb{F}_p)$, or $SP_n(\mathbb{F}_p)$. Let $P$ be a Sylow $p$-subgroup of $G$. What is $P$ as a subgroup of $G$, that is, how can we describe the matrices in $P$? Also, how many Sylow $p$-subgroups are in $G$?

Answers and/or references are welcome. Thanks!

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    $\begingroup$ The Sylow p-subgroup of $SL_n(F_p)$ is the subgroup of upper triangular matrices (this should be well-known). $\endgroup$ – Ralph Mar 25 '13 at 0:03
  • $\begingroup$ You're right; I am aware of this result. Do you know how many conjugates there are? $\endgroup$ – Jared Mar 25 '13 at 0:06

As indicated in the comments, a Sylow subgroup is the group $U$ of unipotent upper triangular matrices in each case. I'll indicate how to compute the normalizer of $U$ in the $SL_n$ case (the others could be handled similarly, I bet). Let $e_1,\dots,e_n$ be the standard basis of $F^n$, where $F$ is a field. We will actually compute the normalizer of the group of unipotent upper triangular matrices in this generality, showing that it is simply the group $B$ of upper triangular matrices of determinant one.

It is obvious that this group normalizes $U$. So now we prove that the normalizer $N$ of $U$ is contained in $B$. The vector $e_1$ is the unique eigenvector with eigenvalue $1$ for all of $U$ (i.e., the intersection of the kernels of $1-u$ for all $u \in U$). It follows that $Ne_1 \subseteq F e_1$. Likewise, the intersection of the kernels of $(1-u)^2$ for $u \in U$ is the span $F \{e_1,e_2 \}$ of $e_1$ and $e_2$, and hence this span is $N$ stable as well. Evidently one can continue in this way to prove that $N \subseteq B$, done. Now you know how many Sylow subgroups there are.


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