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 $MG'$ is a finite index subgroup of G, then M itself is a finite index subgroup of G. If $MG'=G$, 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.