Introduction

Abstract

I1 More than one hundred years ago, Georg Frobenius [26] proved his remarkable theorem affirming that, for a prime p and a finite group G, if the quotient of the normalizer by the centralizer of any p-subgroup of G is a p-group then, up to a normal subgroup of order prime to p , G is a p-group. Of course, it would be an anachronism to pretend that Frobenius, when doing this theorem, was thinking the category — noted F_G in the sequel — where the objects are the p-subgroups of G and the morphisms are the group homomorphisms between them which are induced by the G-conjugation. Yet Frobenius’ hypothesis is truly meaningful in this category.