The group of splendid morita equivalences of principal 2-blocks with dihedral and generalised quaternion defect groups

Abstract

Let k be an algebraically closed field of characteristic 2, let G be a finite group and let B be the principal 2-block of kG with a dihedral or a generalised quaternion defect group P. Let also \calT(B) denote the group of splendid Morita auto-equivalences of B. We show that \calT(B)≅\OutP(A)⋊\Out(P,\calF),where \Out(P,\calF) is the group of outer automorphisms of P which stabilize the fusion system \calF of G on P and \OutP(A) is the group of algebra automorphisms of a source algebra A of B fixing P modulo inner automorphisms induced by (AP)×.