$\newcommand{\sl}{\mathrm{SL}}\newcommand{\gl}{\mathrm{GL}}$Let $G = \sl_2(\Bbb F_p)$ with $p$ and odd prime. Prove that the Sylow $l$-subgroup of $G$ are cyclic for every odd prime $l$ dividing $|G|$. Hint: $\Bbb F^\times_{p^2}\subset\gl_2(\Bbb F_p)$.
My question is basically why the hint is true. Note that $|\sl_2(\Bbb F_p)| = p(p^2-1)$ and if $l = p$ then strictly upper triangular matrices form a cyclic sylow $p$ subgroup. If $l\neq p$ then $l\mid p^2-1 = |\Bbb F_{p^2}^\times|$. Since $\Bbb F^\times_{p^2}$ is a cyclic group, if the hint is true then I can use it somehow.
$$\Bbb F_{p^2}^\times = \langle 1,\alpha\rangle_{\Bbb F_p}\to\gl_2(\Bbb F_p),\quad a+b\alpha\mapsto\begin{pmatrix} a & rb\\ b & a\end{pmatrix},$$ where $\alpha^2 = r$ for some $r\in\Bbb F_p$. So, only the elements of the form $a+b\alpha$ with $a^2-b^2r=1$ get mapped to $\sl_2(\Bbb F_p)$.