Groups with finite co-central rank (Q2747316)

The authors consider the following generalization of the well-known concept of finite rank in the sense of Prüfer. A group \(G\) is said to have finite co-central rank \(s\) if \(s\) is the least integer such that every finitely generated subgroup \(H\) can be generated by at most \(s\) elements modulo the centre of \(H\). It is shown that torsionfree locally nilpotent groups and \(p\)-groups with finite co-central rank have finite Prüfer rank modulo their centre. Every locally soluble-by-finite group \(G\) with finite co-central rank is hyperabelian-by-finite, and if \(G\) is periodic it has an Abelian normal subgroup \(A\) such that the factor group \(G/A\) is residually finite and has finite Prüfer rank. Furthermore, every periodic locally graded group with finite co-central rank is locally finite.