Kernels, inflations, evaluations, and imprimitivity of Mackey functors

AbstractLet M be a Mackey functor for a finite group G. By the kernel of M we mean the largest normal subgroup N of G such that M can be inflated from a Mackey functor for G/N. We first study kernels of Mackey functors, and (relative) projectivity of inflated Mackey functors. For a normal subgroup N of G, denoting by PH,VG the projective cover of a simple Mackey functor for G of the form SH,VG we next try to answer the question: how are the Mackey functors PH/N,VG/N and PH,VG related? We then study imprimitive Mackey functors by which we mean Mackey functors for G induced from Mackey functors for proper subgroups of G. We obtain some results about imprimitive Mackey functors of the form PH,VG, including a Mackey functor version of Fong's theorem on induced modules of modular group algebras of p-solvable groups. Aiming to characterize subgroups H of G for which the module PH,VG(H) is the projective cover of the simple KN¯G(H)-module V where the coefficient ring K is a field, we finally study evaluations of Mackey functors