The Influence Of Minimal Subgroups On Saturated Fusion Systems

Let F be a saturated fusion system over a p-group S. In this paper, we investigate the influence of the minimal subgroups in foc(F), the focal subgroup of T. Our main result is that if for each cyclic subgroup P <= foc(F) of order p (of order 2 and 4 if p = 2) and each phi is an element of Hom (F)(P, foc(T)), phi extends to an automorphism of S, then S is normal in F. We also give several applications of this result.