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...