The socle of a projective Mackey functor for a p-group

We study the simple subfunctors of indecomposable projective Mackey functors for a p-group P. Unlike the case of group algebras, the Mackey algebra is not in general self-injective. Thus, the socle of an indecomposable projective functor is not in general simple. We first show that the simple subfunctors of a projective functor PH,k, where H⩽P, are indexed by the normalizer in H of a subgroup of H. We then study the socle of a specific projective Mackey functor, namely the Burnside functor BP, and we focus on the case where P is abelian. In particular, our study enables us to determine the socle of an indecomposable projective Mackey functor indexed by a cyclic p-group, an abelian p-group of rank 2 and an elementary abelian p-group of rank 3.