On a question of W. L. May concerning the isomorphism of modular group algebras

In 1988, Warren Lee May has asked in the Proceedings of the American Mathematical Society the following: Let G be an abelian group whose p-component G(p) is simply presented and let F be a field of char F = p > 0. Then whether the F-isomorphism FH congruent to FG for any group H implies that H-p congruent to G(p)? The first actual positive answer is given by us in 1997 again in these Proceedings for the particular case when length G(p) < Omega. Here we shall establish an elegant positive solution of the question when length G(p) < Omega (2) but only over the field of p-elements F-p, utilizing a new criterion for simply presentedness with length Omega in the terms of summable p-groups.