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 x^γn(g) contains less than 2^ℵ0 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 x γn(g) contains less than 2^ℵ0 elements, for any x ∈ g.