Glauberman's and Thompson's theorems for fusion systems

We prove analogues of results of Glauberman and Thompson for fusion systems. Namely, given a (saturated) fusion system $\mathcal{F}$ on a finite $p$-group $S$, and in the cases where $p$ is odd or $\mathcal{F}$ is $S_4$-free, we show that $\mathrm{Z}(\mathrm{N}_{\mathcal{F}}(\mathrm{J}(S))) =\mathrm{Z}(\mathcal{F})$ (Glauberman) and that if $\mathrm{C}_{\mathcal{F}} (\mathrm{Z}(S))=\mathrm{N}_{\mathcal{F}}(\mathrm{J}(S))=\mathcal{F}_S(S)$, then $\mathcal{F}=\mathcal{F}_S(S)$ (Thompson). As a corollary, we obtain a stronger form of Frobenius' theorem for fusion systems, applicable under the above assumptions and generalizing another result of Thompson.