What is the number of Sylow $p$-subgroup of $\mbox{SL}_n(\Bbb F_p)$ where $\Bbb F_p$ is finite field of order $p$?
This problem is known for $\mbox{GL}_n(\Bbb F_p)$. By checking the order, strictly upper triangular matrix $P$ is a Sylow $p$-subgroup of $\mbox{GL}_n(\Bbb F_p)$. I know the normalizer of $P$ in $\mbox{GL}_n(\Bbb F_p)$ is a group of upper triangular matrices $T$. Since the order of $T$ is $(p-1)^n p^{1+2+\cdots n-1}$, the number of Sylow $p$-subgroup of $\mbox{ GL}_n(\Bbb F_p)$ is $$n_p = {(p^n-1)(p^n-p)\cdots(p^n-p^{n-1})\over (p-1)^np^{1+2+\cdots+(n-1)}}.$$ Since $P$ is also Sylow $p$-subgroup of $\mbox{SL}_n(\Bbb F_p)$, all I need to do is compute the number of $\det =1$ elements in $N:=N_{\mbox{GL}_n(\Bbb F_p)}(P)$. I tried to use the fact that for any given $A\in N$, I can correct one value in the diagonal to make $\det A =1$. But I don't know how to get further.