Exterior quotient of a Frobenius P-category over the selfcentralizing objects

Abstract

6.1 Let P be a finite p-group and F a Frobenius P-category. Denote by F^sc the full subcategory of F over the set of all the F-selfcentralizing subgroups of P (cf. 4.8) and consider the exterior quotient $\tilde {F}^{sc}$ † of F^sc (cf. 1.3); that is to say, for any pair of F-selfcentralizing subgroups Q and R of P, $\tilde {F}(Q,R)$ is the the set of Q-conjugacy classes in F(Q, R). Although Corollary 5.14 supplies a suitable decomposition for any morphism in $\tilde {F}^{sc}$, Proposition 6.7 below leads to a more precise description of the structure of this category inside its additive cover (cf. A2.7)

$$ \mathfrak{a}\mathfrak{c}(\tilde {F}^{sc} ) = \mathfrak{p}\mathfrak{r}((\tilde {F}^{sc} )^ \circ )^ \circ $$

(6.1.1.)

Called the centric orbit category in [13], whereas their orbit category is our $\tilde {F}$ (cf. 1.3).