Affirmative answer to a question of Feit

Abstract

As he points out, if P is abelian the affirmative answer follows from Dade’s classification [4]. Our purpose here is to show that the general case reduces to the abelian one, which answers affirmatively Feit’s question. More generally, in [4] Dade determines the cap group associated with a finite abelian p-group showing in particular that this group is finitely generated: our main result here (cf. Theorem 2.2 below) implies that finite generation is still true in the non-abelian case (cf. Corollary 2.4 below). This result and the main steps of a proof were already announced in 17, Section81 and here we are just discharging ourselves of an old debt. Actually, as we said in [7, Section81, our result would follow from a complete classification of endo-permutation modules.