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 F on a finite p-group S, and in the cases where p is odd or F is S-4-free, we show that Z(N-F(J(S))) = Z(F) (Glauberman) and that if C-F(Z(S)) = N-F(J(S)) = F-S(S), then F = 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.