Brou\'e's abelian defect group conjecture holds for the sporadic simple Conway group Co_3

Abstract

In the representation theory of finite groups, there is a well-known and important conjecture due to M. Brou\'e. 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 Brou\'e's Abelian Defect Group Conjecture. In this paper, we prove that the strong version of Brou\'e's 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 Brou\'e's abelian defect group conjecture for all primes p and for all p-blocks of Co_3.