Abstract. Suppose $G$ is an abelian $p$-group and $F$ is a field. The Structure, Direct Factor and Isomorphism Theorems are proved for the group algebra $FG$ in the class of all restricted direct products of totally projective groups with $p^{\omega +n}$-projective groups ($n\in {\msbm N}_0={\msbm N}\cup \{0\} $) under some additional conditions on $F$. Specifically, we establish the following: Let $G$ be a separate $p^{\omega +1}$-totally projective $p$-group and ${\msbm F}_p$ the finite $p$-element field. Then $G$ is a direct factor of $V({\msbm F}_pG)$ such that $V({\msbm F}_pG)/G$ is separate $p^{\omega +1}$-totally projective and ${\msbm F}_pG$ as an ${\msbm F}_p$-algebra determines $G$ up to isomorphism. The results obtained strengthen assertions due to W. May (Proc. Amer. Math. Soc., 1979 and 1988) as well as due to the author (Compt. rend. Acad. bulg. Sci., 2001) and (Acta Math. Vietnamica, 2004).

AMS Subject Classification (1991): 16S34, 16U60, 20C07, 20K10, 20K15, 20K20, 20K21

Keyword(s): p^{\omega +n}, totally projective groups, -projective groups, elongations, units, isomorphisms

Received November 25, 2005, and in revised form November 3, 2006. (Registered under 15/2005.)