The abelian monoid of fusion-stable finite sets is free

TL;DR

This paper proves that the abelian monoid of G-stable finite S-sets for a finite group G with Sylow p-subgroup S is free, providing an explicit basis and insights into the Burnside ring embedding, with applications in homotopy theory.

Contribution

It establishes the freeness of the monoid of G-stable S-sets and constructs an explicit basis, advancing understanding of fusion systems and Burnside rings.

Findings

The monoid of G-stable finite S-sets is free.

An explicit basis for the monoid is constructed.

The embedding of the Burnside ring into its ghost ring is described.

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 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.