Frobenius P-categories via the Alperin condition
Group Theory
2010-04-12 v1
Abstract
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.
Cite
@article{arxiv.1004.1620,
title = {Frobenius P-categories via the Alperin condition},
author = {Lluis Puig},
journal= {arXiv preprint arXiv:1004.1620},
year = {2010}
}