On a generalization of Baer Theorem

Abstract:

R. Baer has proved that if the factor-group $G/\zeta _n(G)$ of a group $G$ by the member $\zeta _n(G)$ of its upper central series is finite (here $n$ is a positive integer), then the member $\gamma _{n+1}(G)$ of the lower central series of $G$ is also finite. In particular, in this case, the nilpotent residual of $G$ is finite. This theorem admits the following simple generalization, which has been published very recently by M. de Falco, F. de Giovanni, C. Musella and Ya. P. Sysak: “If the factor-group $G/Z$ of a group $G$ modulo its upper hypercenter $Z$ is finite, then G has a finite normal subgroup $L$ such that $G/L$ is hypercentral.” In the current article we offer a new, simpler, very short proof of this theorem and specify it substantially. In fact, we prove that if $|G/Z| = t$, then $|L|\leq t^k$, where $k = \frac {1}{2}(log_pt+1)$ and $p$ is the least prime divisor of $t$.