Modular Representation Theory of Blocks with Trivial Intersection Defect Groups

We show that Uno’s refinement of the projective conjecture of Dade holds for every block whose defect groups intersect trivially modulo the maximal normal p-subgroup. This corresponds to the block having p-local rank one as defined by Jianbei An and Eaton. An immediate consequence is that Dade’s projective conjecture, Robinson’s conjecture, Alperin’s weight conjecture, the Isaacs– Navarro conjecture, the Alperin–McKay conjecture and Puig’s nilpotent block conjecture hold for all trivial intersection blocks. Mathematics Subject Classification (2000): Primary 20C20.