  Let $X$ be a connected Cayley graph on an abelian group of odd order, such
that no two distinct vertices of $X$ have exactly the same neighbours. We show
that the direct product $X \times K_2$ (also called the "canonical double
cover" of $X$) has only the obvious automorphisms (namely, the ones that come
from automorphisms of its factors $X$ and $K_2$). This means that $X$ is
"stable". The proof is short and elementary. The theory of direct products
implies that $K_2$ can be replaced with members of a much more general family
of connected graphs.
