On the upper central series of infinite groups

Abstract:

Two relevant theorems by R. Baer and P. Hall show that a group is finite over a term with finite ordinal type of its upper central series if and only if it is finite-by-nilpotent. Extending these results, we prove here that if $G$ is any group, the hypercentre factor group $G/\bar Z(G)$ is finite if and only if $G$ contains a finite normal subgroup $N$ such that $G/N$ is hypercentral (where the hypercentre $\bar Z(G)$ of $G$ is defined as the last term of its upper central series).