$3$-Sylow subgroup of $SL(n, \mathbb{F}_3)$

Question

Find a $3$-Sylow subgroup of $SL(n, \mathbb{F}_3)$.

For $n=3$, how many elements are there in the center of this subgroup?

I tried to understand the order of the $3$-Sylow subgroups, but since we have a general $n$, I couldn't get to anything specific( I know the formula for $|SL(n, \mathbb{F}_3)|$).

Without understanding the order of that sylow subgroup, I can't move any further, as that is the most basic assumption to continue, unless I don't see something.

Any ideas?

Answer 1

We have

$$\left|SL(3,\Bbb F_3)\right|=\frac1{3-1}(3^3-1)(3^3-3)(3^3-3^2)=13\cdot24\cdot18=3^3\cdot2^4\cdot13\cdot$$

and thus you need a subgroup of order $\;3^3=27\;$ (any such subgroup. Why?).

Check the following:

$$H:=\left\{\;\begin{pmatrix} 1&a&b\\ 0&1&c\\ 0&0&1 \end{pmatrix}\;|\; a,b, c\in\Bbb F_3\;\right\}$$