Abstract. In this article we study several classes of 'small' 2-groups: we complete the classification, started in [26], of all saturated fusion systems on metacyclic p-groups for all primes p. We consider Suzuki 2-groups, and classify all center-free saturated fusion systems on 2-groups of 2-rank 2. We end by classifying all possible F -centric, F -radical subgroups in saturated fusion systems on 2-groups of 2-rank 2.

We prove analogues of results of Tate and Yoshida on control of transfer for fusion systems. This requires the notions of p-group residuals and transfer maps in cohomology for fusion systems. As a corollary we obtain a p-nilpotency criterion due to Tate.

Abstract. 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.

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