The abelian monoid of fusion-stable finite sets is free

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.