The hyperfocal subcategory of a Frobenius P-category

Abstract

13.1 Let P be a finite p-group and F a Frobenius P-category. Denoting by $\mathfrak{i}_F$ the inclusion functor from F to the category $\mathfrak{G}\mathfrak{r}$ of groups, it is clear that we have a canonical surjective homomorphism from P to the direct limit of $\mathfrak{i}_F$ , and we call F-focal subgroup of P the kernel F _F of this homomorphism

$$ P \to \underrightarrow {\lim }{\text{ }}\mathfrak{i}_F $$

(13.1.1)

actually, it easily follows from Corollary 5.14 that F _F is generated by the union of the sets {u^-1σ(u)}_u∈Q where Q runs over the set of subgroups of P and σ over F(Q). In the case we consider the Frobenius P-category F _G associated with a finite group G having P as a Sylow p-subgroup (cf. 1.8), it is well-known (for instance, see Theorems 3.1 and 4.1 in [28, Ch.7]) that $P/F_{F_G }$ is canonically isomorphic to the maximal Abelian p-quotient of G.