THE GROUP OF SPLENDID MORITA EQUIVALENCES OF PRINCIPAL 2-BLOCKS WITH DIHEDRAL AND GENERALISED QUATERNION DEFECT GROUPS

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 T (B) denote the group of splendid Morita auto-equivalences of B. We show that T (B) congruent to Out P (A) (sic) Out(P, F), where Out(P, F) is the group of outer automorphisms of P which stabilize the fusion system F of G on P and Out(P) (A) is the group of algebra automorphisms of a source algebra A of B fixing P modulo inner automorphisms induced by (A(P))(x).