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

  • 3
    $\begingroup$ 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. $\endgroup$
    – Derek Holt
    Feb 19, 2020 at 8:35
  • $\begingroup$ @DerekHolt thank you. I will follow up on the reference. $\endgroup$ Feb 19, 2020 at 10:36

1 Answer 1


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.)


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