on neumann’s bfc-theorem and finite-by-nilpotent profinite groups

Abstract. let γn = [x1, . . . , xn] be the nth lower central word and xn(g) the set of γn-values in a group g. suppose that g is a profinite group where, for each g ∈ g, there exists a positive integer n = n(g) such that the set gxn(g) = {gy | y ∈ xn(g)} contains less than 2ℵ0 elements. we prove that g is a finite-by-nilpotent group.