Broué’s Abelian defect group conjecture holds for the Janko simple group J4

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 B of the normalizer NG(P) of P in G are equivalent (Rickard equivalent). This conjecture is called Broué’s Abelian defect group conjecture. We prove in this paper that Broué’s Abelian defect group conjecture is true for a non-principal 3-block A with an elementary Abelian defect group P of order 9 of the Janko simple group J4. It then turns out that Broué’s Abelian defect group conjecture holds for all primes p and for all p-blocks of the Janko simple group J4.