Abstract: Let G be a finite group. Brauer’s Problem 2 which asks the following question:‘when do non-isomorphic groups have isomorphic group algebras?’This question can be paraphrased as follows: What does CG know about G? Classical resultof I.M. Isaacs showed that if CG is isomorphic to CH and H is nilpotent, then G is also nilpotent. Although it is not necessarily true that G is isomorphic to H as exemplified by the quaternion and dihedral groups both of order 8. For nonabelian simple groups,the situation is quite different. It was proved by Kimmerle that if FG is isomorphic to FH for all fields F, then G is isomorphic to H, where H is a nonabelian simple group.

In this talk, I will outline the proof of a result saying that if CG is isomorphic to CH, where H is a nonabelian simple group, then G is isomorphic to H. This significantly improved the result of Kimmerle above. Several related results and open problems in this areas will be discussed in this talk.