ZJ-theorems for fusion systems

For p an odd prime, we generalise the Glauberman-Thompson pnilpotency theorem (Gorenstein, 1980) to arbitrary fusion systems. We de. ne a notion of Qd(p)-free fusion systems and show that if F is a Qd(p)-free fusion system on some finite p-group P, then F is controlled by W( P) for any Glauberman functor W, generalising Glauberman's ZJ-theorem (Glauberman, 1968) to arbitrary fusion systems.