A morphism Φ of a fusion system is a natural notion: since a fusion system is a category, this map Φ should be a functor on the category. It should also be a group homomorphism on the underlying group P. It turns out that, whenever Q is a strongly ℱ-closed subgroup of P, there is a surjective morphism of fusion systems with kernel Q, and this morphism of fusion systems is determined uniquely by the group homomorphism (and hence by Q). The theory of morphisms of fusion systems is very satisfactory, and might be said to be complete, in a quite reasonable sense.

However, there is one aspect of the theory that is distinctly less appealing: while the kernel of a morphism is a subgroup, there is no nice and obvious way to get a subsystem on this kernel. There is an obvious way to get a ‘kernel subsystem’, but this subsystem does not have the ‘nice’ properties that we want such a subsystem to have. We clearly want this subsystem to be ‘normal’ in some sense; we will introduce the concept of weak normality here, and normality in Chapter 8.

In this chapter, we introduce another important notion: a normal subgroup. A normal subgroup is simply a subgroup Q such that ℱ = Nℱ(Q): we have seen some results about this type of subgroup already, and will see more in this chapter and Chapter 7.