Commutative Group Algebras of Summable p-Groups

Abstract

Let G be an Abelian group whose p-component G p is summable and let F be a field of characteristic p. Let S(FG) be the normalized unit p-group of the group algebra FG. The main results of the present article are that G p is a direct factor of S(FG) with totally projective complement, provided G p is of countable length and F is perfect; and, in particular, under these circumstances S(FG) is summable if and only if G p is summable. Moreover, if FG≅ FH as F-algebras for some group H and if G p is summable, then H p is summablale. These achievements improve results due to Hill and Ullery (1997) and also their generalizations given by Danchev (2000ab20012004).