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.