Hyper-Abelian groups with finite co-central rank.
As is well known, a group \(G\) has finite (Prüfer) rank if there is an integer \(r\geq 0\) such that every finitely generated subgroup of \(G\) can be generated by \(r\) elements. Here the author studies a more general rank restriction: a group \(G\) has finite co-central rank if there is an integer \(r\geq 0\) such that for every finitely generated subgroup \(H\) the quotient group \(H/Z(H)\) can be generated by \(r\) elements. In a previous work the author had shown that a locally soluble-by-finite group with finite co-central rank is hyper-(Abelian-by-finite). The converse of this result is proved here, so that the following holds. Theorem A. Let \(G\) be a group with finite co-central rank. Then \(G\) is locally soluble-by-finite if and only if it is hyper-(Abelian-by-finite). In addition it is shown that, not unexpectedly, the chief factors of the groups in Theorem A are finite, while their maximal subgroups have finite index.
scientific article; zbMATH DE number 2105406