By J. W. S. Cassels, A. Frohlich

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Therefore Nφ = NP (Q) and, since Q is fully normalized, there is an extension φ¯ of φ to the whole of NP (Q). Since φ has p-power order, we may assume that φ¯ has p-power order (by raising φ¯ to a suitable power). Let ψ be a map in F from NP (Q) such that its image, R, is fully normalized. We see ¯ ψ is a p-element of AutF (R). By maximal choice of Q, we see that AutP (R) is a that (φ) ¯ ψ may be conjugated into AutP (R); hence we may Sylow p-subgroup of AutF (R), and so (φ) ¯ ψ ∈ AutP (R). Thus there is some g ∈ NP (R) such that x(φ) ¯ ψ = xg for choose ψ so that (φ) all x ∈ R.

Let Q be a subgroup of P . Then Q is normal in F if and only if Q is strongly F-closed and Q is contained in every member of F frc . Proof: Suppose that Q is normal in F. 19, Q is contained within every F-centric, F-radical subgroup. Thus one direction of the proof is clear. ) Thus suppose that Q is strongly F-closed and contained within every member of F frc , and let φ : R → S be an isomorphism in F. We need to prove that there is some map φ¯ : QR → P extending φ, such that φ¯ restricts to an automorphism of Q.

Ii) If F is saturated, then CF (Q) is saturated whenever Q is fully centralized, and NF (Q) is saturated whenever Q is fully normalized. Proof: To prove (i), we will check the axioms for a fusion system, noting that the objects in the respective categories are correct. Thus let R and S be subgroups of CP (Q). For g ∈ CP (Q), if θg : R → S is a conjugation map it clearly extends to a map QR → QS that acts trivially on Q. Thus the first axiom of a fusion system is satisfied by both CF (Q). If φ : R → S is a map in CF (Q), then it extends to a map φ¯ : QR → QS that acts trivially on Q, and so clearly the isomorphism map R → Rφ also has this condition; thus CF (Q) satisfies the second axiom of a fusion system.

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