An Application of Set Theory to ω + n-Totally p ω+n -Projective Primary Abelian Groups

Abstract

Several equivalent descriptions are given of the class of primary abelian groups whose separable subgroups are all direct sums of cyclic groups; such groups are called ω-totally Σ-cyclic. This establishes the converse of a theorem due to Megibben. For n < ω, this is generalized to a consideration of the class of primary abelian groups whose pω+n-bounded subgroups are all pω+n-projective. The question of whether there are such groups that are proper in the sense that they are neither pω+n-projective nor ω-totally Σ-cyclic is shown to be logically equivalent to a natural question about the structure of valuated vector spaces. Finally, it is shown that both of these statements are independent of ZFC.