Soluble groups with their centralizer factor groups of bounded rank

For a group class image, a group G is said to be a image-group if the factor group image for all gset membership, variantG, where CG(gG) is the centralizer in G of the normal closure of g in G. For the class image of groups of finite order less than or equal to f, a classical result of B.H. Neumann [Groups with finite classes of conjugate elements, Proc. London Math. Soc. 1 (1951) 178–187] states that if image, the commutator group G′ belongs to image for some f′ depending only on f. We prove that a similar result holds for the class image, the class of soluble groups of derived length at most d which have Prüfer rank at most r. Namely, if image, then image for some r′ depending only on r. Moreover, if image, then image for some r′ and f′ depending only on r,d and f.