The Schur Subgroup of the Brauer Group

In this final chapter we apply the techniques of group cohomology to the representation theory of finite groups. Given G a finite group we know that ℙ(G) is semi-simple for any field of characteristic zero. Consequently, from the Wedderburn theorems there is a decomposition $$F(G) = \sum {{M_{ni}}({D_i})} $$ (0.1) where the D i run over central simple division algebras with center Ki a finite cyclotomic extension of F. The question that we answer here is the determination of all the classes {D i } ∊ B(F) which arise in this way, that is to say, which division algebras occur in the simple components of the group ring of a finite group.