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 $G$, $s\geq 0$ 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.