COMMUTATIVE GROUP ALGEBRAS OF ABELIAN GROUPS WITH UNCOUNTABLE POWERS AND LENGTHS

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• <p>Let F be a field of char(F) = p > 0 and G an abelian groupwith p-component G_p of cardinality at most ℵ₁ and length at most ω₁. The main affirmation on the Direct Factor Problem is that S(FG)/G_p is totally projective whenever F is perfect. This extends results due to May (Contemp. Math., 1989) and Hill-Ullery (Proc. Amer. Math. Soc., 1990). As applications to the Isomorphism Problem, suppose that for any group H the F-isomorphism FH ≅ FG holds. Then if G_p is totally projective, H_p ≅ G_p. This partially solves a problem posed by May (Proc. Amer. Math. Soc., 1988). In particular, H ≅ G provided G isp-mixed of torsion-free rank one so that G_p is totally projective. The same isomorphism H ≅ G is fulfilled when G is p-local algebraically compact too. Besides if F_p is the simple field with p-elements and G_p F_pH is a coproduct of torsion complete groups, F_pH ≅ F_pG as F_p F_p-algebras implies H_p ≅ G_p. This expands the central theorem obtained by us in (Rend. Sem. Mat. Univ. Padova, 1999) and partly settles the generalized version of a question raised by May (Proc. Amer. Math. Soc.,1979) as well. As a consequence, when G_p is torsion complete and G is p-mixed of torsion-free rank one, H ≅ G. Moreover, if G is a coproduct of p-local algebraically compact groups then H ≅ G. The lastattainment enlarges an assertion of Beers-Richman-Walker (Rend. Sem. Mat. Univ. Padova, 1983).Each of the reported achievements strengthens our statements in this direction (Southeast Asian Bull. Math., 2001-2002) and also continues own studies in this aspect (Hokkaido Math. J., 2000) and (Kyungpook Math. J., 2004).</p>

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