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Let R be a (unital) commutative ring, and G be a finite group with orderinvertible in R. We introduce new idempotents in the double Burnside algebraRB(G,G), indexed by conjugacy classes of minimal sections of G, i.e. pairs(T,S) of subgroups of G, where S is a normal subgroup of T contained in theFrattini subgroup of T. These idempotents are orthogonal, and their sum isequal to the identity. It follows that for any biset functor F over R, theevaluation F (G) splits as a direct sum of specific R-modules indexed byminimal sections of G, up to conjugation. The restriction of theseconstructions to the biset category of p-groups, where p is a prime numberinvertible in R, leads to a decomposition of the category of p-biset functorsover R as a direct product of categories F\_L indexed by atoric p-groups L upto isomorphism. We next introduce the notions of L-enriched biset andL-enriched biset functor for an arbitrary finite group L, and show that for anatoric p-group L, the category F\_L is equivalent to the category of L-enrichedbiset functors defined over elementary abelian p-groups. Finally, the notion ofvertex of an indecomposable p-biset functor is introduced (when p is invertiblein R), and, when R is a field of characteristic different from p, the objectsof the category F\_L are characterized in terms of vertices of theircomposition factors.

Idempotents of double Burnside algebras, L-enriched bisets, and decomposition of p-biset functors