Frobenius Group Actions And Blocks

Broue's abelian defect group conjecture is very important in the modular rep-resentation theory of group.Let p be a prime number,G a finite group and b a p-block of G.Assuming that the defect group of b is abelian,Broue conjectured that b and the Brauer correspondence of it in the normalizer of the defect group in G are derived equivalent.It is well known that Broue's abelian defect group conjecture implies Alperin's weight conjecture for blocks with abelian defect groups.At the international congress of mathematicians in 1998,Rickard put forward an open ques-tion:how to generalize Broue's abelian defect group conjecture to blocks with non abelian defect groups,such that the generalization also can imply Alperin's weight conjecture.Rouquier partly answered the question of Rickard.Assuming that the hyperfocal subgroup is abelian,he conjectured that b and the Brauer correspondence in the normalizer of the hyperfocal subgroup are derived equivalence.It is well known that derived equivalence of blocks implies perfect isometry.In chapter 3,we always assume that the hyperfocal subgroup is abelian.Under some appropriate conditions,we prove that b and the Brauer correspondence in the normalizer of the defect group are perfectly isometric.With some additional conditions,we also get that b and the Brauer correspondence are derived equivalent.In chapter 4,we prove that the principal block,with hyperfocal subgroup of the form C2n × C2n,and the Brauer correspondence in the normalizer of the hyperfocal subgroup are perfectly isometric.