Commutator subgroup of a finitely generated nilpotent group.

Question

Let $G$ be a finitely generated nilpotent group.
Is the commutator subgroup $[G,G]$ finitely presented?

Edit: I am also interested in the weaker question: is $[G,G]$ finitely generated?

Answer 1

Let $G=\gamma_1(G) \ge \gamma_2(G) \ge \cdots \ge \gamma_{k+1}(G)=1$ be the lower central series of the finitely generated nilpotent $G$.

Suppose that $G$ is generated by $x_{11},x_{12},\ldots,x_{1n_1}$. Then $\gamma_2(G)/\gamma_3(G)$ is generated by the images of the commutators $x_{21},\ldots,x_{2n_2}$, say, of the elements $x_{1i}$.

Since $\gamma_{m}(G)=[G,\gamma_{m-1}(G)]$, you can show by induction on $m$ that $\gamma_m(G)/\gamma_{m+1}(G)$ is generated by the commutators $x_{m1},\ldots,x_{mn_m}$ of the $x_{1i}$ with $x_{m-1,j}$.

So each subgroup $\gamma_m(G)$ is finitely generated by the elements $x_{li}$ with $l \ge m$, and every element of $G$ can be written in normal form as a product of powers of all these generators. You can get a finite presentation of $G$ by evaluating the normal form words of all commutators of pairs of generators, and making those the relations. For generators $x_{mi}$ that have finite order in $\gamma_m(G)/\gamma_{m+1}(G)$, you also need power relations that express that power in normal form. These realtions enable you to put an arbitrary word in the generators into normal form, which shows that they constitute a presentation of the group.

The easiest nonabelian example is the Heisenberg group ($3 \times 3$ upper unitriangular integer matrices) which, in this notation, has the presentation

$\langle x_{11},x_{12},x_{21} \mid [x_{11},x_{12}]=x_{21}, [x_{11},x_{21}]=[x_{12},x_{21}]=1 \rangle$.

You can also show that any subgroup of $H \le G$ is finitely generated, and hence finitely presented. To do that, assume inductively that $H \gamma_k(G)/\gamma_k(G) \cong H/(H \cap \gamma_k(G))$ is finitely generated, and then observe that $H \cap \gamma(G)$ is a subgroup of the finitely generated abelian group $\gamma_k(G)$ and hence is itself fintiely generated.