  We give a new, unexpected characterization of saturated fusion systems on a
p-group S in terms of idempotents in the p-local double Burnside ring of S that
satisfy a Frobenius reciprocity relation, and reformulate fusion-theoretic
phenomena in the language of idempotents. Interpreting our results in stable
homotopy, we answer a long-standing question on stable splittings of
classifying spaces of finite groups, and generalize the Adams--Wilkerson
criterion for recognizing rings of invariants in the cohomology of an
elementary abelian p-group. This work is partly motivated by a conjecture of
Haynes Miller which proposes retractive transfer triples as a purely
homotopy-theoretic model for p-local finite groups. We take an important step
toward proving this conjecture by showing that a retractive transfer triple
gives rise to a p-local finite group when two technical assumptions are made,
thus reducing the conjecture to proving those two assumptions.
