Congruence subgroup

Remarks

Example: $\Gamma_0(N) = \set{\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathrm{SL}_2(\Z) : c \equiv 0 \pmod N}$ is a congruence subgroup, since each matrix of $\Gamma(N)$ has $c \equiv 0 \pmod N$. Another example is $\Gamma_1(N)$, defined by $c \equiv 0$ and $a \equiv d \equiv 1 \pmod N$. Since $\mathrm{SL}_2(\Z / N\Z)$ is finite, $\Gamma(N)$ has finite index, and so does every congruence subgroup. Non-example: the subgroup $\set{\begin{pmatrix} 1 & n \\ 0 & 1 \end{pmatrix} : n \in \Z}$ has infinite index, so it contains no $\Gamma(N)$. For a prime $p$, the index of $\Gamma_0(p)$ in $\mathrm{SL}_2(\Z)$ is $p + 1$. Not every subgroup of finite index in $\mathrm{SL}_2(\Z)$ is a congruence subgroup.