Finiteness of a nilpotent group is detected by its abelianization ($|G|<\infty ⟺|G/[G,G]|<\infty$)

A nilpotent group is finite exactly when its abelianization is finite; finite generation is not a separate hypothesis. Induct on the nilpotency class $c$. The abelian case is immediate. Otherwise $G/{\gamma }_c(G)$ has smaller class and the same abelianization, so is finite. Let $H=G$, $K={\gamma }_{c-1}(G)$. Their commutator subgroup $[H,K]={\gamma }_c(G)$ is central. Hence its commutator map is a homomorphism in each variable. Both centralizer quotients are finite because their kernels contain ${\gamma }_c(G)$. Finitely many pairs of coset representatives therefore generate the abelian group $[H,K]$, and its exponent divides the exponent of $H/C_H(K)$. It is finite, completing the induction.