The generating hypothesis for the stable module category of a p-group

Freydʹs generating hypothesis, interpreted in the stable module category of a finite p-group G, is the statement that a map between finite-dimensional kG-modules factors through a projective if the induced map on Tate cohomology is trivial. We show that Freydʹs generating hypothesis holds for a non-trivial finite p-group G if and only if G is either C2 or C3. We also give various conditions which are equivalent to the generating hypothesis.