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 N-G (Q, e)/C-G (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. (C) 2002 Elsevier Science (USA). All rights reserved.