Purity and Self-Small Groups

Let A be an abelian group. A group B is A-solvable if the natural map Hom(A, B) ⊗ E(A) A → B is an isomorphism. We study pure subgroups of A-solvable groups for a self-small group A of finite torsion-free rank. Particular attention is given to the case that A is in 𝒢, the class of self-small mixed groups G with G/tG≅ ℚ n for some n < ω. We obtain a new characterization of the elements of 𝒢, and demonstrate that 𝒢 differs in various ways from the class 𝒯 ℱ of torsion-free abelian groups of finite rank despite the fact that the quasi-category ℚ 𝒢 is dual to a full subcategory of ℚ 𝒯 ℱ.

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