Burnside Rings of Fusions Systems and their Unit Groups

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.

14 pages