Find the number of Sylow $p$-subgroups of $G$.

Question

Let $G=GL(2,\mathbb Z_p)$. Find a Sylow $p$-subgroup of $G$ and find the number of Sylow $p$-subgroups of $G$.

My try:

$|G|=(p^2-1)(p^2-p)=p(p-1)^2(p+1)$. Hence any subgroup of order $p$ is a Sylow $p$-subgroup of $G$. On taking $H=$ \begin{cases}\begin{bmatrix} 1 & x \\ 0 & 1 \end{bmatrix}:x\in \mathbb Z_p\end{cases}

$|H|=p$. Hence $H$ is a Sylow $p$-subgroup of $G$.

Now the number of Sylow $p$-subgroups of $G$ will be equal to the number of conjugates of $H$ in $G$.

But the problem is I can't proceed any more. Please help.

Answer 1

You are on the right track. You should consider the normalizer of $H$ now. Show that :

$$N_G(H)=\{\begin{pmatrix}\lambda&y\\0&\mu\end{pmatrix}\mid \lambda,\mu\in \left(\frac{\mathbb{Z}}{p\mathbb{Z}}\right)^*\text{ and } y\in \frac{\mathbb{Z}}{p\mathbb{Z}}\} $$

by strictly using linear algebra methods and conclude...