Suppose G is an abelian group with p-primary component

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

is a direct factor of the group S(FG) of all normed p-units in the group algebra FG with a direct sum of countables complementary factor or equivalently

is a direct sum of countables. In particular, when G is p-mixed, it is proved that G is a direct factor of the group V(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

implies that

, and that there exists a direct sum of p-countables T with the property that

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).