Kernels, inflations, evaluations, and imprimitivity of Mackey functors (Q2425426)

The author defines and studies the concept of kernel of a Mackey functor \(M\) for a finite group \(G\). He defines the kernel of \(M\) as the largest normal subgroup \(N\) of \(G\) such that \(M\) can be inflated from a Mackey functor for \(G/N\). An imprimitive Mackey functor for \(G\) is a Mackey functor for \(G\) induced from a Mackey functor for a proper subgroup of \(G\). For a normal subgroup \(N\) of \(G\), denote by \(P^G_{H,V}\) the projective cover of a simple Mackey functor for \(G\) of the form \(S_{H,V}^G\). The author obtains some results about imprimitive Mackey functors of the form \(P^G_{H,V}\), including a Mackey functor version of Fong's theorem on induced modules of modular groups algebras of \(p\)-solvable groups. Finally, evaluations of Mackey functors are analyzed.