Probabilistic properties of the relative tensor degree of finite groups

Denoting by H⊗KH⊗K the nonabelian tensor product of two subgroups HH and KK of a finite group GG, we investigate the relative tensor degree d⊗(H,K)=|{(h,k)∈H×K|h⊗k=1}||H||K| of HH and KK. The case H=K=GH=K=G has been studied recently. Here we deal with arbitrary subgroups HH and KK, showing analogies and differences between d⊗(H,K)d⊗(H,K) and the relative commutativity degree d(H,K)=|{(h,k)∈H×K|[h,k]=1}||H||K|, which is a generalization of the probability of commuting elements, introduced by Erdős.