Abstract
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 CG(H) is polycyclic. In the context of Černikov groups we obtain a more general result: a radical group is a Černikov group if and only if it contains a finitely generated subgroup, formed by left Engel elements, whose centralizer is a Černikov group. The aforementioned results generalize a theorem by Onishchuk and Zaĭtsev about the centralizer of a finitely generated subgroup in a nilpotent group.
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