The inductive Alperin-McKay and blockwise Alperin weight conditions for blocks with cyclic defect groups

We verify the inductive blockwise Alperin weight (BAW) and the inductive Alperin-McKay (AM) conditions introduced by the second author for blocks of finite quasisimple groups with cyclic defect groups. Furthermore we establish a criterion that describes conditions under which the inductive AM condition for blocks with abelian defect groups implies the inductive BAW condition for those blocks

Brou\'e's abelian defect group conjecture holds for the Harada-Norton sporadic simple group HNHN

In 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 $B$ of the normaliser $N_G(P)$ of $P$ in $G$ are derived equivalent (Rickard equivalent). This conjecture is called Brou\'e's abelian defect group conjecture. We prove in this paper that Brou\'e'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 Harada-Norton simple group $HN$. It then turns out that Brou\'e's abelian defect group conjecture holds for all primes $p$ and for all $p$-blocks of the Harada-Norton simple group $HN$.Comment: 36 page

The principal 2-blocks of finite groups with abelian Sylow 2-subgroups

Let G be a finite group, p a prime number and B a p-block of G with defect group D. There is an important problem in representation theory of finite groups that is to give a description of B when the strucure of D is given. ..