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Abstract. We show that the abelian monoid of isomorphism classes of G-stable finite S-sets is free for a finite group G with Sylow p-subgroup S; here a finite S-set is called Gstable if it has isomorphic restrictions to G-conjugate subgroups of S. These G-stable Ssets are of interest, e.g., in homotopy theory. We prove freeness by constructing an explicit (but somewhat non-obvious) basis, whose elements are in one-to-one correspondence with the G-conjugacy classes of subgroups in S. As a central tool of independent interest, we give a detailed description of the embedding of the Burnside ring for a saturated fusion system into its associated ghost ring.
Abstract. We show that every saturated fusion system F has a unique minimal Fcharacteristic biset ΛF . We examine the relationship of ΛF with other concepts in plocal finite group theory: In the case of a constrained fusion system, the model for the fusion system is the minimal F-characteristic biset, and more generally, any centric linking system can be identified with the F-centric part of ΛF as bisets. We explore the grouplike properties of ΛF , and conjecture an identification of normalizer subsystems of F with subbisets of ΛF .
This paper interprets Hesselholt and Madsen's real topological Hochschild homology functor THR in terms of the multiplicative norm construction. We show that THR satisfies cofinality and Morita invariance, and that it is suitably multiplicative. We then calculate its geometric fixed points and its Mackey functor of components, and show a decomposition result for groupalgebras. Using these structural results we determine the homotopy type of THR(F p ) and show that its bigraded homotopy groups are polynomial on one generator over the bigraded homotopy groups of H F p . We then calculate the homotopy type of THR(Z) away from the prime 2, and the homotopy ring of the geometric fixed-points spectrum Φ Z 2 THR(Z). * preserves monoids with anti-involution.Proof. We use the Bousfield-Kan formula to express the homotopy colimits as geometric realizations of simplicial spaces, and we show that the diagram of the statement commutes simplicially.
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