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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 modulecategory of $G$ is the statement that a map between finite-dimensional$kG$-modules in the thick subcategory generated by $k$ factors through aprojective 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 ifthe Sylow $p$-subgroup of $G$ is $C_2$ or $C_3$. We also give some otherconditions that are equivalent to the GH for groups with periodic cohomology.

Freyd's generating hypothesis for groups with periodic cohomology