Biset functors and genetic sections for p-groups

In this note I show that if F is a biset functor defined over finite p-groups, then for each finite of p-group P, there is a direct summand of F(P) admitting a natural direct sum decomposition indexed by the irreducible rational representations of P, or equivalently, by the equivalence classes of origins in P, or also by equivalence classes of genetic sections of P. This leads to a description of the torsion part of the group of relative syzygies in the Dade group of P, and to a conjecture on the structure of the torsion part of the whole Dade group of P. (C) 2004 Elsevier Inc. All rights reserved.