On $n$-Kappe groups

Abstract

Let $G$ be an infinite group and $nin {3‎, ‎6}cup{2^k| kin mathbb{N}}$‎. ‎In this paper‎, ‎we prove that $G$ is an $n$-Kappe group if and only if for any two infinite subsets $X$ and $Y$ of $G$‎, ‎there exist $xin X$ and $yin Y$ such that $[x^n‎, ‎y‎, ‎y]=1$‎.