On finite-by-nilpotent profinite groups

Let $\gamma_n=[x_۱,\ldots,x_n]$ be the $n$th lower central word‎. ‎Suppose that $G$ is a profinite group‎ ‎where the conjugacy classes $x^{\gamma_n(G)}$ contains less than $۲^{\aleph_۰}$‎ ‎elements‎ ‎for any $x \in G$‎. ‎We prove that then $\gamma_{n+۱}(G)$ 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 $۲^{\aleph_۰}$‎ ‎elements‎, ‎for any $x\in G$‎.

Keywords:

‎Conjucagy classes‎ , ‎verbal subgroups‎ , ‎profinite groups‎ , ‎FC-groups