On finite-by-nilpotent profinite groups

Let γn = [x1, …, xn] be the nth lower central word. Suppose that G is a profinite group where the conjugacy classes xγn(G) contains less than 2ℵ0 elements for any x ∈ G. We prove that then γn+1 (G) has finite order. This generalizes the much celebrated theorem of B. H. Neumann that says that the commutator subgroup of a BFC-group is finite. Moreover, it implies that a profinite group G is finite-by-nilpotent if and only if there is a positive integer n such that xγn(G) contains less than 2ℵ0 elements, for any x ∈ G.