Results Concerning the Automorphism Group of the Tensor Product \(G\otimesK_n\)

It is straightforward to show that the full automorphism group of \(G \otimes K_n\) contains the Cartesian cross product of \(\text{Aut}(G)\) and \(S_n\). If \(\text{Aut}(G \otimes K_n)\) properly contains this cross product, then we will say that \(G \otimes K_n\) has a “rich” automorphism group. First, several conditions on \(G\) that ensure that \(G \otimes K_n\) has a rich automorphism group are given. Then, it is shown that these conditions are both necessary and sufficient for \(G \otimes K_n\) to have a rich automorphism group.