Finite-by-Nilpotent Groups and a Variation of the BFC-Theorem

Abstract

For a group G and an element $a\in G$, let $|a|_k$ denote the cardinality of the set of commutators $[a,x_1,\dots ,x_k]$, where $x_1,\dots ,x_k$ range over G. The main result of the paper states that a group G is finite-by-nilpotent if and only if there are positive integers k and n, such that $|x|_k\le n$ for every $x\in G$. More precisely, if $|x|_k\le n$ for every $x\in G$, then $\gamma _{k+1}(G)$ has finite (k, n)-bounded order. Furthermore, in any group G, the set $FC_k(G)=\{x\in G;\ |x|_k<\infty \}$ is a subgroup and $\gamma _{k+1}(FC_k(G))$ is locally normal.