Broué’s Abelian defect group conjecture holds for the Janko simple 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 B of the normalizer N G ( 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 J 4 . 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 J 4 .