On Finite-by-Nilpotent Groups

Abstarct Let γ n = [ x 1 ,…, x n ] be the n th lower central word. Denote by X n the set of γ n -values in a group G and suppose that there is a number m such that |g^Xₙ| ≤ m for each g ∈ G . We prove that γ n+ 1 ( G ) has finite ( m, n ) -bounded order. This generalizes the much-celebrated theorem of B. H. Neumann that says that the commutator subgroup of a BFC-group is finite.