On Neumann’s BFC-theorem and finite-by-nilpotent profinite groups

Let \gamma_{n}=[x_{۱},\ldots,x_{n}] be the nth lower central word and X_{n}(G) the set of \gamma_{n}-values in a group G. Suppose that G is a profinite group where, for each g\in G, there exists a positive integer n=n(g) such that the set g^{X_{n}(G)}=\{g^{y}\,|\,y\in X_{n}(G)\} contains less than ۲^{\aleph_{۰}} elements. We prove that G is a finite-by-nilpotent group.

Keywords:

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