On Automorphisms of Direct Products of Cayley Graphs on Abelian Groups

Let $X$ and $Y$ be connected Cayley graphs on abelian groups, such that no two distinct vertices of $X$ have exactly the same neighbours, and the same is true about $Y$. We show that if the number of vertices of $X$ is relatively prime to the number of vertices of $Y$, then the direct product $X \times Y$ has only the obvious automorphisms (namely, the ones that come from automorphisms of its factors $X$ and $Y$). This was not previously known even in the special case where $Y = K_2$ has only two vertices. The proof of this special case is short and elementary. The general case follows from the special case by standard arguments.