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.Comment: Added citation to Puig; clarified discussion of Miller's conjecture
on the homotopy characterization of p-local finite group
Let P be a p-group and [Formula: see text] a fusion system on P. The aim of this paper is to give necessary and sufficient conditions on a subgroup Q of P for the normalizer [Formula: see text] to be [Formula: see text] itself. This generalizes a result of Gilotti and Serena on finite groups. As an application we find some classes of resistant p-groups, which are p-groups P such that the normalizer [Formula: see text] is equal to [Formula: see text], for any fusion system [Formula: see text] on P.
We show that every fusion system on a p-group S is equal to the fusion system associated to a discrete group G with the property that every p-subgroup of G is conjugate to a subgroup of S.
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