Congruence subgroup

In the special linear group of integers
A subgroup $$H$$ of a special linear group over integers $$SL(n,\mathbb{Z})$$ is a subgroup defined by congruence conditions on the entries.

More precisely, a congruence subgroup can be defined as a subgroup that contains a principal congruence subgroup, where a principal congruence subgroup is a subgroup obtained as the kernel of the map $$SL(n,\mathbb{Z}) \to SL(n,\mathbb{Z}/k\mathbb{Z})$$ for some positive integer $$k$$.

Note that by definition, any congruence subgroup has finite index, though for $$n \ge 3$$, there exist subgroups of finite index that are not congruence subgroups.