A saturation theorem for submonoids of nilpotent groups and the Identity Problem

If $M$ is a submonoid of a finitely generated nilpotent group $G$, and $M{G}^{\mathrm{\prime }}$ is a finite index subgroup of $G$, then $M$ itself is a finite index subgroup of $G$. If $M{G}^{\mathrm{\prime }}=G$, then $M=G$. This generalizes a well-known theorem for subgroups of finitely generated nilpotent groups. As a result, we give an algorithm for the Identity Problem in nilpotent groups.