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12 A specially interesting case where we ﬁnd the above situation concerns the group algebra kG but considered as an H-algebra with respect to some subgroup H of G . 1. 2, 3. The Frobenius P -category of a block 45 and, for another element x ∈ G such that (Q, f ) ⊂ (P, e)x , x x−1 determines a natural isomorphism between both inclusion functors. All this framework will be widely employed in chapter 11. 13 A last remark; whenever H = NG (Q, f ) where (Q, f ) is a Brauer (b, G)-pair and K a subgroup of Aut(Q) , assuming that Q is fully K-norK malized in F(b,G) and denoting by ν the point of NG (Q, f ) on kGb determined by f (cf.
14 in ); denote by (P, eA ) one of them and let us describe the P -category F(α,G) associated with A and α . 9 Let Q and R be a pair of subgroups of P ; once again, recall that there are unique primitive idempotents fA of Z A(Q) and gA of Z A(R) fulﬁlling (cf. 2; the transitivity of the inclusion between Brauer (α, G)-pairs guarantees that the sets F(α,G) (Q, R) deﬁne a subcategory F(α,G) of Gr over the set of subgroups of P and it is clear that all these group homomorphisms are injective. Moreover, for any u ∈ P , we still have (Q, fA )u ⊂ (P, eA ) and therefore if R ⊂ Qu then we get (R, gA ) ⊂ (Q, fA )u (cf.
7), it is clear that U = uU u−1 = uλ(Q ) is also fully centralized and fully normalized in F , and it suﬃces to consider λ = uλ and the isomorphism IF (U ) ∼ = IF (U ) determined by the conjugation by u . 6 such that the subgroups U = ζ(Q) and U = ζ (Q ) are both fully normalized and fully centralized in F (cf. 7); in this case, we have the equalities IF (U ) = U and IF (U ) = U , and therefore the existence of an F-morphism σ : U → U fulﬁlling the conditions above is clear. 2, 331-339] forces the set of subgroups H considered there to include all the F-intersected subgroups.