Finite generation of lower-central factors

The $i=1$ quotient is generated by the images of $S$. Suppose ${\gamma }_i\mathrm{/}{\gamma }_{i+1}$ is generated by the $i$-fold simple commutators. The next quotient is generated by $[u,g]{\gamma }_{i+2}$ for $u\in {\gamma }_i,g\in G$, since $[{\gamma }_i,G]={\gamma }_{i+1}$. Expand the two entries in their factor generators. Biadditivity expresses this class as a product of the $(i+1)$-fold simple commutators and their inverses. Their number is at most $\mathrm{\mid }S{\mathrm{\mid }}^{i+1}$ before repetitions. This proves the claim by induction.

Abelianness follows from centrality of ${\gamma }_i\mathrm{/}{\gamma }_{i+1}$. In a nilpotent group start with the empty generating list for ${\gamma }_{c+1}=1$. If ${\gamma }_{i+1}$ has a finite generating list and $u_1,\dots ,u_t$ lift factor generators, then for any $g\in {\gamma }_i$ a word $w$ in the $u_j$ has the same coset, so $w^{-1}g\in {\gamma }_{i+1}$. The union of the two finite lists generates ${\gamma }_i$. Descending induction reaches $i=1$; terms after $c$ are trivial. If $S$ is empty, $G=1$ and all lists are empty.