Broué's isotypy conjecture for the sporadic groups and their covers and automorphism groups

Abstract

Let B be a p-block of a finite group G with abelian defect group D such that S ≤ G ≤ Aut (S), S′ = S and S/Z(S) is a sporadic simple group. We show that B is isotypic to its Brauer correspondent in N G(D) in the sense of Broué. This has been done by Rouquier for principal blocks and it remains to deal with the non-principal blocks.