Socles and radicals of Mackey functors

Abstract

We study the socle and the radical of a Mackey functor M for a finite group G over a field K (usually, of characteristic p > 0). For a subgroup H of G, we construct bijections between some classes of the simple subfunctors of M and some classes of the simple K over(N, -)G (H)-submodules of M (H). We relate the multiplicity of a simple Mackey functor SH, V G in the socle of M to the multiplicity of V in the socle of a certain K over(N, -)G (H)-submodule of M (H). We also obtain similar results for the maximal subfunctors of M. We then apply these general results to some special Mackey functors for G, including the functors obtained by inducing or restricting a simple Mackey functor, Mackey functors for a p-group, the fixed point functor, and the Burnside functor BK G. For instance, we find the first four top factors of the radical series of BK G for a p-group G, and assuming further that G is an abelian p-group we find the radical series of BK G. © 2009 Elsevier Inc. All rights reserved.