The solution to W. May's problem for isomorphism of commutative group algebras of mixed groups over finite fields

Suppose G is an abelian group whose p-component G(p) is totally projective (or more generally, is an S-group) and F-p is the simple field of p-elements. It is shown that the F-p-isomorphism FpH congruent to F(p)G for any group H implies H-p congruent to G(p). This almost completely solves a problem of Warren May (1988) from the Proceedings of the American Mathematical Society posed, however, for an arbitrary field F with char(F) = p > 0. In particular, H congruent to G provided G is a p-mixed Warfield group of torsion-free rank one. The last consequence enlarges in some aspect a result of May (1988) published in the same Proceedings for the case when G is a p-local Warfield group.