The abelian monoid of fusion-stable finite sets is free

Reeh, Sune

doi:10.2140/ant.2015.9.2303

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The abelian monoid of fusion-stable finite sets is free

Sune Precht Reeh

Vol. 9 (2015), No. 10, 2303–2324

DOI: 10.2140/ant.2015.9.2303

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 $G$ -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.

Keywords

Fusion systems, Burnside rings, finite groups

Mathematical Subject Classification 2010

Primary: 20D20

Secondary: 20J15, 19A22

Milestones

Received: 3 December 2014

Revised: 31 August 2015

Accepted: 8 October 2015

Published: 16 December 2015

Authors

Sune Precht Reeh
Department of Mathematics Massachusetts Institute of Technology Cambridge, MA 02139 United States

Vol. 9, No. 10, 2015