I was asked in an exam
Let $n \in \mathbb N$. Let $S = SL (2, \mathbb Z)$. Let $$ \left\{ A = \begin{pmatrix}a & b\\c&d\end{pmatrix}\in S,\quad \begin{matrix} a \equiv 1 \pmod{n}, & b \equiv 0 \pmod{n}\\c \equiv 0 \pmod{n},& d \equiv 1 \pmod{n} \end{matrix}\right\}$$ Determine whether $A$ is a normal subgroup of $S$.
I have proved this by checking subgroup criterion and then showed that $\forall s \in S , \forall a \in a: \quad s^{-1}as\in A.$ Then I completed the proof.
My question is:
how should I define the homomorphism $\varphi$, so kernel of $\varphi$ will be $A$?
First thought that is from $S = SL (2, \mathbb Z)$ to $S = SL (2, \mathbb Z_n^*)$ would be fine but then I confused if $\varphi$ is really a homomorpism. I tried to check if $\varphi$ is homomorphism by checking where entries of $s_1,s_2 \in S$ goes and where entries of $s_1s_2$ goes. I think that for $s_1 = \begin{pmatrix} x_{1,1}&x_{1,2}\\x_{2,1}&x_{2,2} \end{pmatrix}$, each entry $x_{i,j}$ goes to $x_{i,j}\pmod n$ but this doesn't have to be in $SL (2, \mathbb Z_n^*)$, because $n$ doesn't have to be prime. This is where I stucked.
I have looked that if similar question was asked here and I saw some questions similar but not the same and after seeing that questions I still can't find proper answer.
Any suggestion would be fine. Thanks!