The Frobenius P-categories via the Alperin condition

In [Lluís Puig, Frobenius Categories Versus Brauer Blocks, Progr. Math., vol. 274, Birkhäuser, Basel, 2009], we introduce the Frobenius P-categories giving two quite different definitions of them. In this paper, we exhibit a third equivalent definition based on the form of the old Alperin Fusion Theorem; this theorem can be reformulated in our abstract setting, and ultimately depends on the behavior of the so-called F-essential subgroups of P: we call “Alperin condition” a sufficient form of this behavior. Then, we prove that a divisible P-category F is a Frobenius P-category if and only if all the partial normalizers of a suitable set of representatives for the F-isomorphism classes of subgroups of P fulfill both the Sylow and the Alperin conditions.