We prove analogues of results of Glauberman and Thompson for fusion systems. Namely, given a (saturated) fusion system $\CF$ on a finite $p$-group $S$, and in the cases where $p$ is odd or $\CF$ is $S_4$-free, we show that $\Z(\N_\CF(\J(S)))=\Z(\CF)$ (Glauberman), and that if $\C_\CF(\Z(S))=\N_\CF(\J(S))=\CF_S(S)$, then $\CF=\CF_S(S)$ (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.
First published in Proceedings of the American Mathematical Society in 137, 2009, published by the American Mathematical Society. Copyright 2008, American Mathematical Society