Composition factors of functors

Abstract

Let X be a family of finite groups satisfying certain conditions and K be a field. We study composition factors, radicals, and socles of biset and related functors defined on X over K . For such a functor M and for a group H in X , we construct bijections between some classes of maximal (respectively, simple) subfunctors of M and some classes of maximal (respectively, simple) K Out ( H ) -submodules of M ( H ) . We use these bijections to relate the multiplicity of a simple functor S H , V in M to the multiplicity of V in a certain K Out ( H ) -module related to M ( H ) . We then use these general results to study the structure of one of the important biset and related functors, namely the Burnside functor B K which assigns to each group G in X its Burnside algebra B K ( G ) = K ⊗ Z B ( G ) where B ( G ) is the Burnside ring of G. We find the radical and the socle of B K in most cases of X and K . For example, if K is of characteristic p > 0 and X is a family of finite abelian p-groups, we find the radical and the socle series of B K considered as a biset functor on X over K . We finally study restrictions of functors to nonfull subcategories. For example, we find some conditions forcing a simple deflation functor to remain simple as a Mackey functor. For an inflation functor M defined on abelian groups over a field of characteristic zero, we also obtain a criterion for M to be semisimple, in terms of the images of inflation and induction maps on M.