On finite-by-nilpotent groups

Abstract

Let ${\gamma }_n=[x_1,\dots ,x_n]$ be the $n$th lower central word. Denote by $X_n$ the set of ${\gamma }_n$-values in a group $G$ and suppose that there is a number $m$ such that $|g^{X_n}|\le m$ for each $g\in G$. We prove that ${\gamma }_{n+1}(G)$ 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.

arXiv admin note: text overlap with arXiv:1803.04202