Criteria for nilpotency of fusion systems

Let [Formula: see text] be a prime and [Formula: see text] be a fusion system over a finite [Formula: see text]-group [Formula: see text]. A fusion system [Formula: see text] is said to be nilpotent if [Formula: see text]. In this paper, we provide new criteria for saturated fusion systems to be nilpotent, which can be viewed as extension of the [Formula: see text]-nilpotency theorem of Glauberman and Thompson for fusion systems attributed to Kessar and Linckelmann.