I am trying to understand the proof of Lemma $3.1$ of the paper On Some of the Residual Properties of Finitely Generated Nilpotent Groups by Koberda. It states the following:
Let $N$ be a finitely generated torsion-free nilpotent group which is virtually abelian. Then $N$ is abelian.
The first step is to let $N' < N$ be a finite index normal subgroup which is abelian, but I'm not sure why such a subgroup exists. A group is virtually abelian if it has an abelian subgroup of finite index, but it doesn't necessarily have to be a normal subgroup does it?
If $G$ is a virtually abelian group, does it necessarily contain a finite index normal subgroup which is abelian?
If the answer is no, why does such a subgroup exist in this case?