We show that every saturated fusion system can be realized as a full subcategory of the fusion system of a finite group. The result suggests the definition of an 'exoticity index' and raises some other questions which we discuss.

We answer the gluing problem of blocks of finite groups (Linckelmann (2004) [7, 4.2]) for tame blocks and the principal p-block of PSL 3 (p) for p odd. In particular, we show that the gluing problem for the principal p-block of PSL 3 (p) does not have a unique solution when p ≡ 1 mod 3.

Abstract. We develop the fundamentals of Mackey functors in the setup of fusion systems including an acyclicity condition as well as a parametrization and an explicit description of simple Mackey functors. Using this machinery we extend Dwyer's sharpness results to exotic fusion systems F on a finite p-group S with an abelian subgroup of index p.

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.