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 $\Gamma(N)\subset \Gamma$?
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2$\begingroup$ 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). $\endgroup$– David LoefflerCommented May 8, 2019 at 20:03
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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 $M\mid N$.
answered May 8, 2019 at 1:18