When $\mathbb{Z}^n \rtimes \mathbb{Z}_m$ nilpotent?

Question

For what values of $m,n$, and what kind of actions by $\mathbb{Z}_m$ , $\mathbb{Z}^n \rtimes \mathbb{Z}_m$ nilpotent?

Answer 1

The torsion elements of a finitely generated nilpotent group form a normal subgroup, so this group is nilpotent if and only if it is equal to the direct product ${\mathbb Z}^n \times {\mathbb Z}_m$. That is, if the action is trivial.