Upper-central factors of a torsion-free nilpotent group

For each fixed $g\in G$, define ${\varphi }_g:Z_2\mathrm{/}{Z}_1\to Z_1$ by ${\varphi }_g(y{Z}_1)=[y,g]$. The value lies in $Z_1$ by the definition of $Z_2$; multiplying $y$ by a central element does not change it. The product identity, with central values, proves ${\varphi }_g$ is a homomorphism. If $y{Z}_1$ has finite order $m>0$, then ${\varphi }_g(y{Z}_1{)}^m=1$. Torsion-freeness of $Z_1$ implies $[y,g]=1$ for every $g$, hence $y\in Z_1$. Thus $Z_2\mathrm{/}{Z}_1$ is torsion-free.

Induct on an upper-central length $c$. For $c=0$ the group is trivial; for $c=1$ the claim is the assumed torsion-freeness of the center. For $c\ge 2$, the group $\bar{G}=G\mathrm{/}{Z}_1$ has length at most $c-1$ and center $Z_2\mathrm{/}{Z}_1$, torsion-free by step 1.1. Recursion on the definitions gives $Z_j(\bar{G})=Z_{j+1}(G)\mathrm{/}{Z}_1$ for $j\ge 0$: after quotienting this subgroup, its next center is exactly the defining next upper-center factor. The induction hypothesis therefore proves torsion-freeness of $Z_{j+2}(G)\mathrm{/}{Z}_{j+1}(G)$; the first factor is torsion-free by assumption.

If $x^m=1$ with $m>0$, its image in $G\mathrm{/}{Z}_{c-1}$ is torsion and therefore trivial, so $x\in Z_{c-1}$. Repeating down the torsion-free factors gives $x\in Z_0=1$. Factors after $Z_c$ are trivial. This proves both conclusions with the stated ascending indices.