  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 that 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|\leqt^{k}, where k=(1/2)(log_{p}t+1), and p is the least prime divisor of t.
