  Given a permutation group $G$, the derangement graph $\Gamma_G$ of $G$ is the
Cayley graph with connection set the set of all derangements of $G$. We prove
that, when $G$ is transitive of degree at least $3$, $\Gamma_G$ contains a
triangle.
  The motivation for this work is the question of how large can be the ratio of
the independence number of $\Gamma_G$ to the size of the stabilizer of a point
in $G$. We give examples of transitive groups where this ratio is maximum.
