An application of set theory to \(\omega+n\)-totally \(p^{\omega+n}\)-projective primary Abelian groups. (Q662627)

Recall that a group is called \(\omega\)-totally \(\Sigma\)-cyclic if it is a primary Abelian group whose separable subgroups are all direct sums of cyclic groups. The two authors study the class of such groups and their main result is Theorem 2.6. that contains a number of equivalent descriptions of the groups in this class. As a consequence the converse of a theorem due to Megibben is obtained. As a natural generalization the authors then consider the notion of \(\omega+n\)-totally \(p^{\omega+n}\)-projective groups and translate the existence of some \(\omega+n\)-totally \(p^{\omega+n}\)-projective group that is not \(\omega\)-totally \(\Sigma\)-cyclic into a question on valuated vector spaces. This is Theorem 3.11. Finally they prove that these statements are undecidable in ZFC.