Abstract. Let P be an abelian p-group, E a cyclic -group acting freely on P and k an algebraically closed field of characteristic . In this work, we prove that every self-equivalence of the stable module category of comes from a self-equivalence of the derived category of . Work of Puig and Rickard allows us to deduce that if a block B with defect group P and inertial quotient E is Rickard equivalent to , then they are splendidly Rickard equivalent. That is, Brous original conjecture implies Rickards refinement of the conjecture in this case. All of this follows from a general result concerning the self-equivalences of the thick subcategory generated by the trivial module.