Prime-valent Symmetric Cayley Graphs of Characteristically Simple Groups

Let $\text{Ga}$ be a connected prime-valent $X$-arc-transitive Cayley graph of a finite characteristically simple group $G\cong T^k$, where $k⩾2$. We obtain a precise structural characterization of such graphs and their arc-transitive automorphism groups. In the cubic case, every connected symmetric Cayley graph of $T^k$, where $T$ is a finite nonabelian simple group, is normal.