prove that $GL_2(\Bbb Z_3)$ is solvable

Question

I need to prove that $GL_2(\Bbb Z_3)$ is solvable

What I tried:

I know that $GL_2(\Bbb Z_3)$ has $(3^2-1)(3^2-3) = 48 = 3 * 2^4$ elements.

I know that $n_3 \in \{1,4,16\}$ and $n_2 \in \{1,3\}$

here I'm stuck .... any help will be appreciated.

Answer 1

Consider $H=SL_2(\mathbb{Z}_3)$, the subgroup of matrices with determinant $1$. Show that $|H|=24$ and is normal subgroup. Then you may try to proceed that $H$ is solvable (and since $G/H$ is also solvable, so is $G$.)

One reason to proceed this way is that, as you were considering number of Sylow subgroups, it is better to work for it in $H$ rather than in $G$; in $G$ no Sylow subgroups are normal. But, in $H$, some Sylow subgroup is normal. It requires some detailed computations.