# Congruence subgroups and translations

Let $$\Gamma$$ be a congruence subgroup of $$\operatorname {SL}_2 (\mathbb Z)$$. Let $$N$$ be the smallest positive integer such that $$\begin{pmatrix} 1 & N \\0 &1 \end{pmatrix}\in \Gamma$$.

Is necessarily $$\Gamma(N)\subset \Gamma$$?

• The answer is "yes" if $\Gamma$ is normal in $SL_2(Z)$ and $-1 \in \Gamma$, but this is not at all obvious (it is a special case of Wohlfart's theorem). May 8 '19 at 20:03

No. $$\Gamma$$ can be $$\left\{\begin{pmatrix}a&b\\c&d\end{pmatrix}\in\operatorname {SL}_2 (\mathbb Z):\text{M\mid c and N\mid b}\right\}$$, which contains $$\Gamma(N)$$ if and only if $$M\mid N$$.