A further generalization of the Glauberman-Thompson $p$-nilpotency criterion in fusion systems

Let $p$ be a prime and $\mathcal{F}$ be a saturated fusion system over a finite $p$-group $P$. The fusion system $\mathcal{F}$ is said to be nilpotent if $\mathcal{F}=\mathcal{F}_{P}(P)$. We provide new criteria for a saturated fusion system $\mathcal{F}$ to be nilpotent, which may be viewed as extending the Glauberman-Thompson $p$-nilpotency criterion to fusion systems.