# Do finitely generated nilpotent groups contain torsion free subgroups of finite index?

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 $$[H:H']<\infty$$?

• I have seen Proposition 2 in page 2 of D. Segal, Polycyclic groups, CUP, Cambridge, 1983 as a reference for this result (but haven't checked it). It also follows from the fact that $T$ is finite and f.g. nilpotent groups are residually finite. Feb 19, 2020 at 8:35
• @DerekHolt thank you. I will follow up on the reference. Feb 19, 2020 at 10:36

This is an immediate corollary of Theorem 2.1 in

G.Baumslag, "Lectures on nilpotent groups," Regional Conference Series in Mathematics, No. 2 American Mathematical Society, Providence, R.I. 1971.

Theorem (K.A.Hirsch, 1938). Every finitely generated nilpotent group embeds as a finite index subgroup in $$A\times B$$ where $$A$$ is a finite group and $$B$$ is a torsion-free group.

(The result you are after also follows from Corollary 1.21 in Baumslag's book: Residual finiteness of polycyclic groups, which Baumslag also attributes to Hirsch.)