A filtration of the modular representation functor

Let and be algebraically closed fields of characteristics p>0 and 0, respectively. For any finite group G we denote by the modular representation algebra of G over where is the Grothendieck group of finitely generated -modules with respect to exact sequences. The usual operations induction, inflation, restriction, and transport of structure with a group isomorphism between the finitely generated modules of group algebras over induce maps between modular representation algebras making an inflation functor. We show that the composition factors of are precisely the simple inflation functors where C ranges over all nonisomorphic cyclic p′-groups and V ranges over all nonisomorphic simple -modules. Moreover each composition factor has multiplicity 1. We also give a filtration of .