Centralizer of Engel Elements in a Group

In this paper we show that some finiteness properties on a centralizer of a particular subgroup can be inherited by the whole group. Among other things, we prove the following characterization of polycyclic groups: a soluble group G is polycyclic if and only if it contains a finitely generated subgroup H, formed by bounded left Engel elements, whose centralizer C-G(H) is polycyclic. In the context of Cernikov groups we obtain a more general result: a radical group is a Cernikov group if and only if it contains a finitely generated subgroup, formed by left Engel elements, whose centralizer is a Cernikov group. The aforementioned results generalize a theorem by Onishchuk and Zaitsev about the centralizer of a finitely generated subgroup in a nilpotent group.