On soluble groups in which centralizers are finitely generated (Q1356827)

The author considers the following open question: if \(G\) is a soluble group in which the centralizer of every finitely generated subgroup is finitely generated, is \(G\) polycyclic? The following results are obtained. Theorem A: A soluble group of finite rank is polycyclic if all centralizers of finitely generated subgroups are finitely generated. Theorem B: If \(G\) is a nilpotent-by-polycyclic group in which the centralizer of every polycyclic subgroup is finitely generated, then \(G\) is polycyclic. -- The second of these results implies the first.