Soluble groups with their centralizer factor groups of bounded rank

AbstractFor a group class X, a group G is said to be a CX-group if the factor group G/CG(gG)∈X for all g∈G, where CG(gG) is the centralizer in G of the normal closure of g in G. For the class Ff 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 G∈CFf, the commutator group G′ belongs to Ff′ for some f′ depending only on f. We prove that a similar result holds for the class Sr(d), the class of soluble groups of derived length at most d which have Prüfer rank at most r. Namely, if G∈CSr(d), then G′∈Sdr′(d) for some r′ depending only on r. Moreover, if G∈C(Sr(d)Ff), then G′∈Sr′(d+3)Ff′ for some r′ and f′ depending only on r,d and f