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Upon replacing α by αr for some appropriate r, we can assume it has p-power order. Choose R ≤ S which is F-conjugate to NS (P ) and fully normalized in F, and fix ϕ ∈ IsoF (NS (P ), R). 1 again), R is fully automized by assumption. Hence there is χ ∈ AutF (R) such that χϕ α ∈ AutS (R) ∈ Sylp (AutF (R)). Let g ∈ NS (R) be such that cg = χϕα. In particular, g∈ / R. Set Q = χϕ(P ) R. Then g ∈ NS (Q) since α(P ) = P , and so R, g ≤ NS (Q). But then |NS (Q)| > |R| = |NS (P )|, which contradicts the assumption P was fully normalized.

The local theory of fusion systems Michael Aschbacher Part II of the book is intended to be an introduction to the local theory of saturated fusion systems. By the “local theory of fusion systems” we mean an extension of some part of the local theory of finite groups to the setting of saturated fusion systems on finite p-groups. 4. One can ask: Why deal with saturated fusion systems rather than p-local finite groups? There are two reasons for this choice. First, it is not known whether to each saturated fusion system there is associated a unique p-local finite group.

Proof. 4. 3]. 6. Invariant subsystems are fairly natural and have many nice properties. 4. Assume E is F-invariant and D is a subsystem of F on the subgroup D of S. Then (a) E ∩ D is a D-invariant subsystem of D on T ∩ D. (b) If D is F-invariant then E ∩ D is F-invariant on T ∩ D. Proof. 1. On the other hand invariant subsystems have the big drawback that they need not be saturated. 5. Assume T is strongly closed in S with respect to F. Define E to be the subsystem of F on T such that for each P, Q ≤ T , HomE (P, Q) = HomF (P, Q); that is E is the full subcategory of F whose objects are the subgroups of T .

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