Generalizing a theorem of P. Hall on finite-by-nilpotent groups

TL;DR

This paper extends P. Hall's theorem on the relationship between the finiteness of certain group series and the index of central series subgroups, under weaker conditions.

Contribution

It generalizes Hall's theorem by replacing the finiteness condition with a weaker one involving the intersection of the lower central series with the upper central series.

Findings

The finiteness of $| ext{γ}_{i+1}(G): ext{γ}_{i+1}(G) igcap Z_i(G)|$ implies the finiteness of $|G:Z_{2i}(G)|$.

The result broadens the class of groups where the relationship between the lower and upper central series is understood.

The theorem applies to finite-by-nilpotent groups under weaker hypotheses.

Abstract

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.