CHAPTER 1 Background on fusion systems A saturated fusion system over a p-group S is a category F whose objects are the subgroups of S, and where for each P, Q ≤ S, MorF(P, Q) is a set of injective homomorphisms from P to Q which includes all morphisms induced by conjugation in S, and which satisfies a set of axioms which are described, for example, in [AKO, § I.2], [BLO2, Definition 1.2], or [Cr, Definition 4.11]. We write HomF(P, Q) = MorF(P, Q) to emphasize that the morphisms are all homomorphisms. The following terminology for subgroups in a fusion system will be used fre- quently. Recall that a subgroup H G is strongly p-embedded if p |H|, and p |H ∩ g H| for g ∈ G H. Definition 1.1. Fix a prime p, a p-group S, and a saturated fusion system F over S. Let P ≤ S be any subgroup. • Let P F denote the set of subgroups of S which are F-conjugate (isomorphic in F) to P . Similarly, gF denotes the F-conjugacy class of an element g ∈ S. • P is fully normalized in F (fully centralized in F) if |NS(P )| ≥ |NS(R)| (|CS(P )| ≤ |CS(R)|) for each R ∈ P F . • P is fully automized in F if AutS(P ) ∈ Sylp(AutF(P )). • P is F-centric if CS(P ) = Z(P ) for all P which is F-conjugate to P . • P is F-essential if P is F-centric and fully normalized in F, and OutF(P ) contains a strongly p-embedded subgroup. Let EF denote the set of all F-essential subgroups of S. • P is central in F if every morphism ϕ ∈ HomF(Q, R) in F extends to a morphism ϕ ∈ HomF(PQ, PR) such that ϕ|P = IdP . • P is normal in F if every morphism ϕ ∈ HomF(Q, R) in F extends to a morphism ϕ ∈ HomF(PQ, PR) such that ϕ(P ) = P . • For any ϕ ∈ Aut(S), ϕ F denotes the fusion system over S defined by HomϕF(P, Q) = ϕ ◦ HomF(ϕ−1(P ),ϕ−1(Q)) ◦ ϕ−1 all P, Q ≤ S By analogy with finite groups, the maximal normal p-subgroup of a saturated fusion system F is denoted Op(F). Also, for any P ≤ S, NF(P ) ⊆ F is the largest fusion subsystem over NS(P ) in which P is normal. If P is fully normalized in F, then NF(P ) is a saturated fusion system by, e.g., [AKO, Theorem I.5.5]. Since we will have frequent need to refer to the “Sylow axiom” and the “ex- tension axiom” for a saturated fusion system, we state them here in the form of a 5

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