A reduction theorem for Dade's projective conjecture

In this paper, we propose a strengthening of Dade's Conjecture. This version, called the Character Triple Conjecture, once assumed for quasisimple groups, is shown to imply Dade's Projective Conjecture for all finite groups. In particular Dade's Projective Conjecture holds for a group whose nonabelian simple sections have only covering groups satisfying the Character Triple Conjecture. We verify the new conjecture for some classes of quasisimple groups.