Local control in fusion systems of p-blocks of finite groups

If p is an odd prime, b a p-block of a finite group G such that SL(2,p) is not involved in NG(Q,e)/CG(Q) for any b-subpair (Q,e), then NG(Z(J(P))) controls b-fusion, where P is a defect group of b. This is a block theoretic analogue of Glauberman's ZJ-Theorem. Several results of general interest about fusion and blocks are also proved.