Normal subsystems of fusion systems

If we look at Frobenius’s theorem, this suggests that we should focus on the subgroups of a fixed Sylow p-subgroup P , and all of the conjugation maps cg for the various g ∈ G between the subgroups of P . Often, if g ∈ G and Q 6 P , Q · cg = Q does not lie inside P , and so we simply ignore those maps, and only keep the ones where both the domain and the image are inside P . This leads us to define the fusion system of a finite group to be just this.