1. BACKGROUND ON FUSION SYSTEMS 7 Theorem I.4.9(a)], there is a finite group M (a “model” for NF(Q)) which satisfies the above conditions (and also FN S (Q) (M) NF(Q)). The following lemma on automorphisms will also be useful. Lemma 1.5. Let F be a fusion system over a p-group S. Let Q P S be a pair of subgroups both fully normalized in F, such that Q is F-centric and normalized by AutF(P ). Set Out(P, Q) = NAut(P ) (Q)/Inn(P ) = Aut(P ) | α(Q) = Q}/Inn(P ) , and let R: Out(P, Q) −−−− −−−→ NOut(Q)(OutP (Q))/OutP (Q) be the homomorphism R ( [α] ) = [α|Q] · OutP (Q) . Here, [α] Out(P ) denotes the class of α Aut(P ). Then the following hold. (a) R sends OutF(P ) isomorphically to NOut F (Q) (OutP (Q))/OutP (Q). (b) Assume that p = 2, and that either Z(Q) has exponent 2 and P/Q acts freely on some basis of Z(Q), or that |Z(Q)| = |P/Q| = 2. If Γ Out(P, Q) is any subgroup such that R(Γ) = NOut F (Q) (OutP (Q))/OutP (Q) and OutS(P ) Sylp(Γ), then Γ = OutF(P ). Proof. By [OV, Lemma 1.2], R is well defined and Ker(R) H1(P/Q Z(Q)). In particular, Ker(R) is a p-group since Z(Q) is a p-group. Also, OutF(P ) Out(P, Q) since AutF(P ) normalizes Q. (a) By the extension axiom (and since CS(Q) Q and Q is fully normal- ized), R sends OutF(P ) onto NOut F (Q) (OutP (Q))/OutP (Q). Also, OutS(P ) Sylp(OutF(P )) since P is fully normalized, so Ker(R|Out F (P ) ) OutS(P ) since Ker(R) is a p-group. Hence if α AutF(P ) and [α] Ker(R), then α = cg for some g NS(P ), g PCS(Q) = P since [α|Q] OutP (Q) and Q is F-centric, and thus [α] = 1 in OutF(P ). So R|Out F (P ) is injective. (b) If Z(Q) has exponent 2, and the conjugation action of P/Q permutes freely some basis for Z(Q), then R is injective by [OV, Corollary 1.3], and the result is immediate. If |P/Q| = |Z(Q)| = 2, then each element in Ker(R) is represented by some α Aut(P ) such that α|Q = Id, and α(g) gZ(Q) for all g P Q. Thus |Ker(R)| 2, and in particular, Ker(R) Z(Out(P, Q)). By (a), R sends OutF(P ) isomorphically onto NOut F (Q) (AutP (Q))/OutP (Q). By a similar argument, for Γ Out(P, Q) as in (b), R sends Γ isomorphically onto NOut F (Q) (AutP (Q))/OutP (Q). Since Ker(R) is central, Out(P, Q) = Ker(R) × OutF(P ) = Ker(R) × Γ. In particular, OutF(P ) and Γ have the same p -elements. By assumption, OutS(P ) is a Sylow p-subgroup of both OutF(P ) and Γ, and hence OutF(P ) = Γ. 1.1. Essential subgroups in fusion systems Recall that EF denotes the set of F-essential subgroups of a fusion system F. We begin with Alperin’s fusion theorem for fusion systems, in the form originally proven by Puig.
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