Periodic points of algebraic actions of discrete groups

Let $\mathrm{\Gamma }$ be a countable group. A $\mathrm{\Gamma }$-action on a compact abelian group $X$ by continuous automorphisms of $X$ is called Noetherian if the dual of $X$ is Noetherian as a $\mathbb{Z}(\mathrm{\Gamma })$-module. We prove that any Noetherian action of a finitely generated virtually nilpotent group has a dense set of periodic points.