A converse of Baer’s theorem

Abstract

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 $G/Z_n(G)$, in which $Z_n(G)$ 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 $\gamma _{n+1}(G)$ is finite, then $G/Z_n(G)$ is finite where $\gamma _{n+1}(G)$ 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 $G/Z_n(G)$ is finitely generated. Furthermore, we show that the index of the subgroup $Z_n(G)$ is bounded by a precisely determined function of the order of $\gamma _{n+1}(G)$. Moreover, we prove that the mentioned theorem of Hekster is also valid under a weaker condition that $Z_{2n}(G)/Z_{n}(G)$ is finitely generated. Although in this case the bound for the order of $\gamma _{n+1}(G)$ is not achieved.