= Finiteness of a nilpotent group is detected by its abelianization
{title2=$|G|<\infty\iff|G/[G,G]|<\infty$}
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.
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