Principal congruence subgroup
= Principal congruence subgroup
{title2=$\Gamma(N)$}
{wiki=Congruence_subgroup#Principal_congruence_subgroups}
The principal congruence subgroup of level $N$ is
$$
\Gamma(N)=\ker\left(SL_n(\mathbb Z)\longrightarrow SL_n(\mathbb Z/N\mathbb Z)\right).
$$
It has finite index because the target is finite.