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

A conjecture of Michel Broué states that ifDis an abelian Sylowp-subgroup of a finite groupG, andH=NG(D), then the principal blocks ofGandHare Rickard equivalent. The structure of groups with abelian Sylowp-subgroups, as determined by P. Fong and M. E. Harris, raises the following question: Assuming that Brouéʹs conjecture holds for the simple components ofG, under what conditions does it hold forGitself? Due to the structure ofG, this problem requires mainly the lifting of Rickard complexes top′-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 “Clifford theory” of tilting complexes.