Let k be a field of positive characteristic p. One considers some categories, whose objects are given classes of finite p-groups, and morphisms are given classes of k-virtual bisets, i.e., linear combinations of bisets with coefficients in k. The category of k-linear functors from such a category to the category of k-vector spaces is abelian, and one can try to classify and describe its simple objects, or its projective objects. By specific subfunctors of the Burnside functor, which have a unique simple quotient S-Q,S-k, one will get some estimates on the k-dimension of the evaluations of these simple functors. These evaluations are equalities for abelian p-groups, and for such groups P the result is even stronger, since it provides some explicit k-bases for the evaluations S-Q,S-k(P).
