Murai’s Conjecture On General Linear Groups And Exceptional Groups G₂（q）

The famous Brauer’s k(B)-conjecture is one of the most important problems in the modular representation theory of finite groups.Up to now,it keeps wide open.About 40 years ago,Murai put forward his conjecture for the case of principal blocks of Brauer’s k(B)-conjecture when he investigated Frobenius’ conjecture.So far there are few results on Mura’s conjecture.In receently,it is known to be true for finite groups with a cyclic Sylow psubgroup.Based on this result,we further investigate Murai’s conjecture.Indeed,the main purpose of this thesis is to show that Murai’s conjecture is true for the finite general linear groups GL(n,q)with 2 ≤n≤4,for GL(n,q)with p | q(q±1),and for the finite exceptional Chevalley groups G2(q).