On Equivalences between Blocks of Group Algebras: Reduction to the Simple Components

A conjecture of Michel Brou e states that if D is an abelian Sylow p-subgroup of a nite group G, and H = N G (D), then the principal blocks of G and H are Rickard equivalent. The structure of groups with abelian Sylow p-subgroups, as determined by P. Fong and M.E. Harris, raises the following question: assuming that Brou e's conjecture holds for the simple components of G, under what conditions does it hold for G itself? Due to the structure of G, this problem requires mainly the lifting of Rickard complexes to p 0-extensions of the simple components and the construction of complexes over wreath products. We give here these reduction steps, which may be regarded as a \Cliiord theory" of tilting complexes.