ON FINITE-BY-NILPOTENT PROFINITE GROUPS

Let gamma(n) = [x(1), ... , x(n)] be the nth lower central word. Suppose that G is a profinite group where the conjugacy classes x(gamma n(G)) contains less than 2(aleph 0) elements for any x is an element of G. We prove that then gamma(n+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(gamma n(G)) contains less than 2(aleph 0) elements, for any x is an element of G.