$p$-sylow subgroups of $SL(3, \mathbb {Z}_p)$

Question

I wonder how many $p$-sylow subgroups of $SL(3, \mathbb {Z}_p)$ are there. ($p$ is any prime)

Rather than finding generators, I used the fact that |$GL(3, \mathbb {Z}_p)$| = $({p}^3 - 1)({p}^3 - p)({p}^3 - {p}^2)$.

Since $SL(3, \mathbb {Z}_p)$ is a kernel of $ \phi : GL(3, \mathbb {Z}_p) \to {Z}_p^* $, $ \phi (A) = det(A) $,

I got |$SL(3, \mathbb {Z}_p)$| = ${(p - 1)}^2 ({p}^2 + p + 1) {p}^3 (p + 1)$.

Now due to the third sylow theorem, the number of $p$-sylow subgroups is of the form $1 + pk$ and it must divides |$SL(3, \mathbb {Z}_p)$|. ($k$ is a nonnegative integer)

So there are several possibilities due to the factorization of |$SL(3, \mathbb {Z}_p)$|.

But after then, I can't determine the exact number of $p$-sylow subgroups among them. Is there any helpful fact that I can apply to progress?

Answer 1

Hint:

The number of $p$ Sylow subgroups of $SL(3,F_p)$ is $$\frac{\vert SL(3,F_p) \vert}{\vert N(P) \vert}$$ where $P$ is any $p$ Sylow subgroup and $N(P)$ is the normalizer.

Take in particular, $$P=\left\{\begin{pmatrix} 1& x & y\\ 0& 1 & z\\ 0&0&1\end{pmatrix}: x,y,z \in F_p \right\}$$

Try to find $N(P)$ and complete your problem!