2010/04/09 by Lluis Puig, Puig, Lluis
Mathematics · #FOS: Mathematics #Group Theory (math.GR) #math.GR
paper · pdf · doi:10.48550/arxiv.1004.1620
arxiv created 2010/04/09 · arxiv updated 2010/04/12
In "Frobenius Categories versus Brauer Blocks", Progress in Math. 274, 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.