The local theory of fusion systems – analysing the normalizers and centralizers of a fusion system to get information about the system itself – is mostly the product of a series of papers by Michael Aschbacher [Asc08a], [Asc08b], [Asc10], [Asc11]. In this chapter we will give an exposition of some of this theory, although many of the more difficult theorems are beyond the scope of this work.

In the first section, we give the definition of a normal subsystem. This differs from a weakly normal subsystem by the inclusion of an axiom on extending certain automorphisms beyond the strongly ℱ-closed subgroup on which the normal subsystem is defined. We give an example of a weakly normal subsystem that is not normal.

We continue by proving some of the basic facts for normal subsystems in Section 8.1, before considering the relationship between weakly normal and normal subsystems in Section 8.2. In Section 8.3, we examine intersections of weakly normal and normal subsystems. We do not have the space to prove the main theorem in this direction, namely that if ε1 and ε2 are normal subsystems on T1 and T2 respectively, then there is a normal subsystem ε on T = T1 ∩ T2, with ε < ε1 ∩ ε2. However, we will use this to develop a general theory of intersections of weakly normal subsystems, as well as of normal subsystems.