On hypercentre-by-polycyclic-by-nilpotent groups (Q2313578)

Summary: 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 \textit{G. A. Fernández-Alcober} and \textit{M. Morigi} [Proc. Am. Math. Soc. 137, No. 2, 425--429 (2009; Zbl 1165.20026)]. We also present an alternative approach to the latter.