Narrowing the basic F^sc-locality

Abstract

23.1 Let P be a finite p-group and F a Frobenius P-category. As in chapter 4, denote by F^sc the full subcategory of F over the set of F-selfcentralizing subgroups of P and by $\tilde F^{sc}$ its exterior quotient (cf. 1.3). In this chapter, we exhibit some significant quotients of the full subcategory L^b,sc of L^b over the set of F-selfcentralizing subgroups of P — namely, the polycentral and the reduced F^sc-localities — which are necessary candidates to contain any perfect F^sc-locality, as we show in the last chapter; in particular, we give a sufficient condition for the existence of a perfect F^sc-locality (cf. 17.4 and 17.13). We freely use the notation introduced in the previous chapter and abundantly employ the terminology and the results of chapter 6.