Central-series comparison theorem
= Central-series comparison theorem
{title2=$\Gamma_{r-i+1}(G)\le G_i\le Z_i(G)$}
For a length-$r$ <central series> $1=G_0\le\cdots\le G_r=G$, induction from the top gives $\Gamma_{r-i+1}(G)\le G_i$, and induction from the bottom gives $G_i\le Z_i(G)$. Consequently the <lower central series> terminates at $\Gamma_{c+1}=1$ exactly when the <upper central series> reaches $Z_c=G$.
// Target: algebra.bigb