New variations on the theme of Baer's theorem

Let ${\gamma }_s(G)$ and $Z_s(G)$ denote the $s$-th terms of the lower and upper central series of a group $G$, respectively. A classical theorem by R. Baer states that if $Z_s(G)$ has finite index $n$ in $G$, then ${\gamma }_{s+1}(G)$ is also finite. In this paper, we prove that if $G$ is a generalized soluble group such that the quotient ${\gamma }_s(G)\mathrm{/}({\gamma }_s(G)\cap Z_t(G))$ has finite rank $r$ for some $s,t$, then the rank of ${\gamma }_{s+t}(G)$ is finite and $(r,s,t)$-bounded. Moreover, a corresponding result replacing the finite-rank assumption by the condition to be a Chernikov group of bounded size is also obtained. These results extend recent generalizations of the classical Baer’s theorem.