Brouéʼs abelian defect group conjecture holds for the sporadic simple Conway group [formula omitted

Abstract

In the representation theory of finite groups, there is a well-known and important conjecture due to M. Broué. He conjectures that, for any prime p, if a p-block A of a finite group G has an abelian defect group P, then A and its Brauer corresponding block A N of the normaliser N G ( P ) of P in G are derived equivalent (Rickard equivalent). This conjecture is called Strong Version of Brouéʼs Abelian Defect Group Conjecture. In this paper, we prove that the strong version of Brouéʼs abelian defect group conjecture is true for the non-principal 2-block A with an elementary abelian defect group P of order 8 of the sporadic simple Conway group Co 3 . This result completes the verification of the strong version of Brouéʼs abelian defect group conjecture for all primes p and for all p-blocks of Co 3 .