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.