$\Gamma (n)= \{ \begin{pmatrix} a&b\\c&d \end{pmatrix} \in SL(2,\mathbb{Z})| a,d \equiv 1 \, mod \, n, b,c \equiv 0 \, mod \, n \}$
$\Gamma (n)$ is a normal subgroup of $SL(2,\mathbb{Z})$.
Show that $SL(2,\mathbb{Z}) / \Gamma (n)$ is isomorphic to $SL(2,\mathbb{Z}_n)$.
How can I identify a homomorphism by which these are isomorphic?
My thinking is that since $\Gamma (n) = \begin{pmatrix} Id \end{pmatrix}$ we get that $SL(2,\mathbb{Z})$, which must be isomorphic to $SL(2,\mathbb{Z}_n)$. But such a homomorphism can't be bijective, since the second group must be smaller (as the integers run only to $n$).
I suspect it is possible to use the first isomorphism theorem to conclude that they are isomorphic directly? But for that I need a homomorphism for which $\Gamma (n)$ is the kernel.
I can't see how to get on with this. Help would be much appreciated.