Schur’s classical theorem states that for a group G, if G/Z(G) is finite, then G is finite. Baer extended this theorem for the factor group G/Zn(G), inwhich Zn(G) is the n-th term of the upper central series of G. Hekster proved a converse of Baer’s theorem as follows: If G is a finitely generated group such that γn+1(G) is finite, then G/Zn(G) is finite where γn+1(G) denotes the (n+1)st term of the lower central series of G. In this paper, we generalize this result by obtaining the same conclusion under the weaker hypothesis that G/Zn(G) is finitely generated. Furthermore, we show that the index of the subgroup Zn(G) is bounded by a precisely determined function of the order of γn+1(G). Moreover, we prove that the mentioned theorem of Hekster is also valid under a weaker condition that Z2n(G)/Zn(G) is finitely generated. Although in this case the bound for the order of γn+1(G) is not achieved.

Keywords