On Some Properties of the Upper and Lower Central Series

R. Baer has proved that if in a group G the factor-group G / zeta(n) (G) by a member zeta(n) (G) of an upper central series is finite for some positive integer n, 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. The following two questions are in a logical connection with the Baer theorem. (a) Let | G / zeta k (G)| = t. Is there the function beta(1) such that | gamma(n + 1)(G)| = beta 1(t, k)? (b) Suppose that a group G has finite central length and the upper hypercenter of G has finite index m. Is there the function beta(2) such that the order of nilpotent residual is at most beta(2)(m, zl(G)) (here zl(G) is the length of the upper central series of G)? The current article provides the answer on these questions.