Boundedly finite conjugacy classes of tensors

Abstract

Let $n$ be a positive integer and let $G$ be a group. We denote by $\nu (G)$ a certain extension of the non-abelian tensor square $G\otimes G$ by $G\times G$. Set $T_{\otimes }(G)=\{g\otimes h\mid g,h\in G\}$. We prove that if the size of the conjugacy class $\left|x^{\nu (G)}\right|\le n$ for every $x\in T_{\otimes }(G)$, then the second derived subgroup $\nu (G{)}^{\prime \prime }$ is finite with $n$-bounded order. Moreover, we obtain a sufficient condition for a group to be a BFC-group.