I am trying exercises of Tom M Apostol Modular Functions and Dirichlet series in number Theory Chapter 2 and I cannot think about this problem ( Problem 2.12) .
Its image has many definitions which are useful in question but doubt is only in question 12 .
Adding Image --
I can only think that $\Gamma^n$ will be equal to Identity matrix modulo n here. But I don't know if that can be used to prove the statement.
Can somebody please help. Edit 1 - Can somebody please give hint for this question also.
Question 13 Prove that The index of $\Gamma^n$ in $\Gamma$ is the number of equivalence classes of matrices modulo n.
If $\Gamma=\operatorname{SL}_2(\mathbb{Z})$,the ring homomorphism $$ \mathbb{Z}\to\mathbb{Z}/n\mathbb{Z},\;x\mapsto [x] $$induces a group homomorphism$$ \varphi\colon\operatorname{SL}_2(\mathbb{Z})\to\operatorname{SL}_2(\mathbb{Z}/n\mathbb{Z}),\;\begin{pmatrix} a & b\\ c & d \end{pmatrix}\mapsto\begin{pmatrix} [a] & [b]\\ [c] & [d] \end{pmatrix}. $$By isomorphism theorems, we have$$ \operatorname{SL}_2(\mathbb{Z})/\ker(\varphi)\cong \operatorname{im}(\varphi)\leq\operatorname{SL}_2(\mathbb{Z}/n\mathbb{Z}). $$But you can easily determine the kernel of the map:$$ \ker(\varphi)=\{A=\begin{pmatrix}a & b\\ c& d\end{pmatrix}\in\operatorname{SL}_2(\mathbb{Z}):a\equiv d\equiv 1\mod n,\;b\equiv c\equiv 0\mod n\}=\Gamma^{(n)} $$Since the image is a subgroup of $\operatorname{SL}_2(\mathbb{Z}/n\mathbb{Z})$ and this is finite, we have that the quotient is finite aswell.