On hypercentre-by-polycyclic-by-nilpotent groups

Abstract

If $\{{\gamma }^{s+1}G\}$ and $\{{\zeta }_s(G)\}$ denote respectively the lower and upper central series of the group $G$, $s\ge 0$ an integer, and if ${\gamma }^{s+1}G/({\gamma }^{s+1}\cap {\zeta }_s(G))$ is polycyclic (resp. polycyclic-by-finite) for some $s$, then we prove that $G/{\zeta }_{2s}(G)$ is polycyclic (resp. polycyclic-by-finite). The corresponding result with polycyclic replaced by finite was proved in 2009 by G.A. Fernández-Alcober and M. Morigi. We also present an alternative approach to the latter.