Vol. 9, No. 10, 2015

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The abelian monoid of fusion-stable finite sets is free

Sune Precht Reeh

Vol. 9 (2015), No. 10, 2303–2324

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 G-stable if it has isomorphic restrictions to G-conjugate subgroups of S. These G-stable S-sets are of interest, e.g., in homotopy theory. We prove freeness by constructing an explicit (but somewhat nonobvious) 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.

Fusion systems, Burnside rings, finite groups
Mathematical Subject Classification 2010
Primary: 20D20
Secondary: 20J15, 19A22
Received: 3 December 2014
Revised: 31 August 2015
Accepted: 8 October 2015
Published: 16 December 2015
Sune Precht Reeh
Department of Mathematics
Massachusetts Institute of Technology
Cambridge, MA 02139
United States