Boundedly finite conjugacy classes of tensors

Let n be a positive integer and let G be a group. We denote by ν(G) a certain extension of the non-abelian tensor square G⊗G by G×G. Set T⊗(G)={g⊗h∣g,h∈G}. We prove that if the size of the conjugacy class ∣∣xν(G)∣∣≤n for every x∈T⊗(G), then the second derived subgroup ν(G)′′ is finite with n-bounded order. Moreover, we obtain a sufficient condition for a group to be a BFC-group. Keywords