Burnside Rings of Fusions Systems and their Unit Groups

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 $\mathcal F$isthefusionsysteminducedbyafinitegroup$G$with$S$asaSylow$p$-group,weshowthattherestrictionof$B(G)$to$B(S)$hasimageequalto$B(\mathcal F)$\mathrm{.}Wealsoshowthatfor$p=2,$ we can gain information about the fusion system by studying the unit group $B(\mathcal{F}{)}^{\times }\mathrm{.}$ When $S$ is abelian, we completely determine this unit group.