Burnside Rings of Fusions Systems and their Unit Groups

TL;DR

This paper investigates the Burnside ring associated with fusion systems on p-groups, providing criteria for membership, relating it to group Burnside rings, and analyzing the unit groups, especially for abelian cases.

Contribution

It introduces criteria for elements of the Burnside ring of a fusion system, links it to group Burnside rings, and characterizes the unit group for abelian p-groups.

Findings

Criteria for membership in B(𝓕) based on automorphism groups.

Image of B(G) restriction equals B(𝓕) for finite groups.

Complete determination of the unit group B(𝓕)^× for abelian S.

Abstract

For a saturated fusion system $\mathcal F$ on a $p$-group $S$, we study the Burnside ring of the fusion system $B(\mathcal F)$, as defined by Matthew Gelvin and Sune Reeh, which is a subring of the Burnside ring $B(S)$. We give criteria for an element of $B(S)$ to be in $B(\mathcal F)$ determined by the $\mathcal F$-automorphism groups of essential subgroups of $S$. When $\mathcal F$ is the fusion system induced by a finite group $G$ with $S$ as a Sylow $p$-group, we show that the restriction of $B(G)$ to $B(S)$ has image equal to $B(\mathcal F)$. We also show that for $p=2,$ we can gain information about the fusion system by studying the unit group $B(\mathcal F)^\times.$ When $S$ is abelian, we completely determine this unit group.