Level of an open subgroup (awaiting review)

The level of an open subgroup $H$ of a matrix group $G$ over $\widehat \Z$ is the least positive integer $N$ for which $H$ is equal to the inverse image of its projection to the reduction of $G$ modulo $N$.

This also applies to open subgroups of matrix groups over $\Z_\ell$, in which case the level is necessarily a power of $\ell$.