  Let $\gamma_i(G)$ and $Z_i(G)$ denote the $i$-th terms of the lower and upper
central series of a group $G$, respectively. P. Hall showed that if
$\gamma_{i+1}(G)$ is finite then the index $|G:Z_{2i}(G)|$ is finite. We prove
that the same result holds under the weaker hypothesis that
$|\gamma_{i+1}(G):\gamma_{i+1}(G)\cap Z_i(G)|$ is finite.

Group Theory