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28 Proof: Firstly, suppose that Q is fully normalized, and let R be a subgroup of P that is F-isomorphic with Q such that R is fully centralized. Let φ : R → Q be an isomorphism in F; by the extension axiom the map φ extends to an injective map φ¯ : R CG (R) → P . The image of φ must be contained within Q CG (Q), and so Q is fully centralized, as claimed. Now suppose that Q is fully normalized, but that AutP (Q) is not a Sylow p-subgroup of AutF (Q). Choose Q to be of maximal order with this property; certainly Q is not equal to P .
The first axiom simply requires that the morphism sets in L should behave in the same way as those for LcP (G); in that case we had that HomL (Q, R)/Z(Q) ∼ = HomF (Q, R), and Z(Q) acts freely on the maps by composition, and so it seems natural to require this in general. The second axiom is equally important, since it makes sure that the map δ was chosen to match up with the map π, in that the automorphism of Q given by δ(x) is the same as the automorphism given by θx . The third axiom is more complicated: it is essentially there because proofs demand it to be.
12 Let b be a block idempotent of kG. Then G acts transitively on the set of maximal b-Brauer pairs, and if (D, e) is such a pair, then NG (D, e)/D CG (D) is a p -group. 13 Let G be a finite group and let k be a field of characteristic p. Let b be a block idempotent of kG, and (D, eD ) denote a maximal b-Brauer pair. Denote by F = F(D,eD ) (G, b) the category whose objects are all subgroups of D, and whose morphisms sets are described below. Let Q and R be subgroups of D, and eQ and eR be the unique block idempotents such that (Q, eQ ) G such that (Q, eQ )x (D, eD ) and (R, eR ) (D, eD ).