Soluble groups with their centralizer factor groups of bounded rank

For a group class XX, a group GG is said to be a CXCX-group if the factor group G/CG(gG)∈XG/CG(gG)∈X for all g∈Gg∈G, where CG(gG)CG(gG) is the centralizer in GG of the normal closure of gg in GG. For the class FfFf of groups of finite order less than or equal to ff, 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 G∈CFfG∈CFf, the commutator group G′G′ belongs to Ff′Ff′ for some f′f′ depending only on ff. We prove that a similar result holds for the class Sr(d), the class of soluble groups of derived length at most dd which have Prüfer rank at most rr. Namely, if G∈CSr(d), then G′∈Sdr′(d) for some r′r′ depending only on rr. Moreover, if G∈C(Sr(d)Ff), then G′∈Sr′(d+3)Ff′ for some r′r′ and f′f′ depending only on r,dr,d and ff.