I have a question about the proof of proposition 6.9 of the paper "Rational Subgroups of Biautomatic Groups" by Gersten and Short (available here). The proposition states that a finitely presented nilpotent subgroup $H$ of a biautomatic group contains an abelian subgroups of finite index.
In the second line of the proof the authors claim that they can pass to a torsion free subgroup of $H$ of finite index and then proceed to prove the result for finitely generated, torsion free nilpotent subgroups. They do not justify why they can always find such a finite index subgroup.
I know that in finitely generated nilpotent groups the torsion subgroup $T$ is always finite and that $H/T$ is torsion free but this is not what the authors are claiming.
So my question is: if $H$ is a finitely generated nilpotent group, does there exist a subgroup $H'$ such that $H'$ is torsion free and $[H:H']<\infty$?