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I have the following congruence subgroups: $$\begin{align} \Gamma_1(N)&=\left\{ \begin{bmatrix} a & b \\ c&d \end{bmatrix} \in \operatorname{SL}(2,\mathbb{Z}): \begin{bmatrix} a & b \\ c&d \end{bmatrix} \equiv\begin{bmatrix} 1 & * \\ 0&1 \end{bmatrix}\pmod{N}\right\} \\ \Gamma_0(N)&=\left\{ \begin{bmatrix} a & b \\ c&d \end{bmatrix} \in \operatorname{SL}(2,\mathbb{Z}): \begin{bmatrix} a & b \\ c&d \end{bmatrix} \equiv\begin{bmatrix} * & * \\ 0&* \end{bmatrix}\pmod{N}\right\} \end{align}$$

I want to show that $\Gamma_1(N)$ is a normal subgroup of $\Gamma_0(N)$ so I want to find a homomorphism from $\Gamma_0(N)$ to some group and show that the kernel of this homomorphism is $\Gamma_1(N)$. But I don't seem to find one.

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The group is the multiplicative group of diagonal matrices mod $N.$

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Exercise: Verify that the following $$\phi\colon\ \Gamma_0(N)\mapsto(\mathbb{Z}/N\mathbb{Z})^\times,\quad \phi\begin{pmatrix}a & b\\c & d\end{pmatrix} = d$$ is a group homomorphism with kernel $\Gamma_1(N)$.

Exercise: Show that replacing the above right-hand side with $a$ works as well.

Bonus exercise: Verify that $$\psi\colon\ \Gamma_1(N)\mapsto\mathbb{Z}/N\mathbb{Z},\quad \psi\begin{pmatrix}a & b\\c & d\end{pmatrix} = b$$ is a group homomorphism and find its kernel. Note that the right-hand side's group operation is addition this time.

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