Lower central series of a nilpotent group

Question

Consider $G$ - finitely generated nilpotent group. Is it true that all members of the lower central series $\gamma_i G=[\gamma _{i-1} G, G],\ \gamma_0 = G$ are finitely generated as well?

Answer 1

Yes. You check it by induction on the nilpotency length. Starting the induction is clear. Suppose $G$ $n$-step nilpotent with $n\ge 2$. Using the inductive hypothesis, $G/\gamma_nG$ is an iterated extension of finitely generated abelian groups and hence is finitely presented. Hence $\gamma_nG$ is finitely generated as a normal subgroup, and therefore, being central, $\gamma_nG$ is finitely generated. Since $\gamma_i(G)$ is extension of $\gamma_n(G)$ by $\gamma_i(G/\gamma_nG)$ which are both finitely generated, we conclude that $\gamma_i(G)$ is finitely generated for all $i$.