NOTES ON THE ISOMORPHIC MODULAR GROUP ALGEBRAS OF P-SPLITTING AND P-MIXED ABELIAN GROUPS

Theorem C. Suppose that G is a coproduct of countable abelian groups and K is perfect. Then S(KG)/Gp is a coproduct of countable abelian groups and Gp is a direct factor of S(KG). Thus S(KG) is a coproduct of countable abelian groups. Moreover, if KH ≅ KG as K-algebras for some group H, then Hp ≅ Gp. In particular, if G is in addition p-mixed, G is a direct factor of V(KG) with the same complementary factor. Thus V(KG) is a coproduct of countable abelian groups. Moreover, if KH ≅ KG as K-algebras for some group H, there exists a coproduct of countable abelian p-groups T such that H× T ≅ G× T. Thus H is a coproduct of p-mixed countable abelian groups. With this at hand, we obtained in [5] the following isomorphism claim.