Broué’s Abelian defect group conjecture holds for the Janko simple group J4J4

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 pp, if a pp-block AA of a finite group GG has an Abelian defect group PP, then AA and its Brauer corresponding block BB of the normalizer NG(P)NG(P) of PP in GG 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 AA with an elementary Abelian defect group PP of order 9 of the Janko simple group J4J4. It then turns out that Broué’s Abelian defect group conjecture holds for all primes pp and for all pp-blocks of the Janko simple group J4J4.