  Let $G$ be a finite group and let $k$ be a field whose characteristic $p$
divides the order of $G$. Freyd's generating hypothesis for the stable module
category of $G$ is the statement that a map between finite-dimensional
$kG$-modules in the thick subcategory generated by $k$ factors through a
projective if the induced map on Tate cohomology is trivial. We show that if
$G$ has periodic cohomology then the generating hypothesis holds if and only if
the Sylow $p$-subgroup of $G$ is $C_2$ or $C_3$. We also give some other
conditions that are equivalent to the GH for groups with periodic cohomology.
