On hypercentre-by-polycyclic-by-nilpotent groups

Wehrfritz, B.A.F.

Rendiconti del Seminario Matematico della Università di Padova, Tome 141 (2019), pp. 155-164.

If $\left\{{\gamma }^{s+1}G\right\}$ and $\left\{{\zeta }_s\left(G\right)\right\}$ denote respectively the lower and upper central series of the group $G$, $s\ge 0$ an integer, and if ${\gamma }^{s+1}G/({\gamma }^{s+1}\cap {\zeta }_s\left(G\right))$ is polycyclic (resp. polycyclic-by-finite) for some $s$, then we prove that $G/{\zeta }_{2s}\left(G\right)$ 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.

Classification : 20, 00
Mots-clés : Upper central series, lower central series, polycyclic group