Commutative Group Algebras of Direct Sums of Countable Abelian Groups

Abstract

Suppose G is an abelian group with p-primary component Gp and F is a field of charF = p > 0. The main goal, motivating the present paper, is to show that if G is a direct sum of groups whose p-components are of countable lengths direct sums of countable groups, then Gp is a direct factor of the groupS(FG) of all normed p-units in the group algebra FG with a direct sum of countables complementary factor or equivalently S(FG)=Gp is a direct sum of countables. In particular, when G is p-mixed, it is proved that G is a direct factor of the groupV(FG) of all normalized units in FG with a direct sum of p-countables complementary factor or in other words V(FG)=G is a direct sum of p-countables. Moreover, for an arbitrary group H the F-isomorphism FH = FG implies that Hp = Gp, and that there exists a direct sum of p-countables T with the property that H T = G T provided G is p-mixed.These facts on the group structure and on the isomorphism problem generalize statements due to Hill-Ullery (Commun. Algebra, 1997) and to the author (Compt. rend.Acad. bulg. Sci., 1995 and Hokkaido Math. J., 2000). Besides, the claims on the isomorphism partially settle a question raised by May (Proc. Amer. Math. Soc., 1988) and extend an other author's result in this theme (Proc. Amer. Math. Soc., 1997).