Results Concerning the Automorphism Group of the Tensor Product $G\otimes K_n$

Abstract

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.