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Freyd's generating hypothesis for the stable module category of a non-trivialfinite group G is the statement that a map between finitely generatedkG-modules that belongs to the thick subcategory generated by k factors througha projective if the induced map on Tate cohomology is trivial. In this paper weshow that Freyd's generating hypothesis fails for kG when the Sylow p-subgroupof G has order at least 4 using almost split sequences. By combining this withour earlier work, we obtain a complete answer to Freyd's generating hypothesisfor the stable module category of a finite group. We also derive someconsequences of the generating hypothesis.

Freyd's generating hypothesis with almost split sequences