Full Description
I1 More than one hundred years ago, Georg Frobenius [26] proved his remarkable theorem a?rming that, for a primep and a ?nite groupG, if the quotient of the normalizer by the centralizer of anyp-subgroup ofG is a p-group then, up to a normal subgroup of order prime top,G is ap-group. Ofcourse,itwouldbeananachronismtopretendthatFrobenius,when doing this theorem, was thinking the category - notedF in the sequel - G where the objects are thep-subgroups ofG and the morphisms are the group homomorphisms between them which are induced by theG-conjugation. Yet Frobenius' hypothesis is truly meaningful in this category. I2 Fifty years ago, John Thompson [57] built his seminal proof of the nilpotencyoftheso-called Frobeniuskernelofa FrobeniusgroupGwithar- ments - at that time completely new - which might be rewritten in terms ofF; indeed, some time later, following these kind of arguments, George G Glauberman [27] proved that, under some - rather strong - hypothesis onG, the normalizerNofasuitablenontrivial p-subgroup ofG controls fusion inG, which amounts to saying that the inclusionN?G induces an ? equivalence of categoriesF =F .
Contents
General notation and quoted results.- Frobenius P-categories: the first definition.- The Frobenius P-category of a block.- Nilcentralized, selfcentralizing and intersected objects in Frobenius P-categories.- Alperin fusions in Frobenius P-categories.- Exterior quotient of a Frobenius P-category over the selfcentralizing objects.- Nilcentralized and selfcentralizing Brauer pairs in blocks.- Decompositions for Dade P-algebras.- Polarizations for Dade P-algebras.- A gluing theorem for Dade P-algebras.- The nilcentralized chain k*-functor of a block.- Quotients and normal subcategories in Frobenius P-categories.- The hyperfocal subcategory of a Frobenius P-category.- The Grothendieck groups of a Frobenius P-category.- Reduction results for Grothendieck groups.- The local-global question: reduction to the simple groups.- Localities associated with a Frobenius P-category.- The localizers in a Frobenius P-category.- Solvability for Frobenius P-categories.- A perfect F-locality from a perfect Fsc -locality.- Frobenius P-categories: the second definition.- The basic F-locality.- Narrowing the basic Fsc-locality.- Looking for a perfect Fsc-locality.
