Groups with Finitely Many Homomorphic Images of Finite Rank

A group is called a Cernikov group if it is abelian-by-finite and satisfies the minimal condition on subgroups. A new characterization of Cernikov groups is given here, by proving that in a suitable large class of generalised soluble groups they coincide with the groups having only finitely many homomorphic images of finite rank (up to isomorphisms) and admitting an ascending normal series whose factors have finite rank.