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 have exactly the same neighbours, is true about $Y$. We show if number relatively prime to $Y$, then direct product $X \times Y$ has only obvious automorphisms (namely, ones come from its factors $Y$). This was not previously known even in special case where $Y = K_2$ vertices. The proof this short elementary. general follows by standard arguments.