Submission date: 4 October 2018

Abstract:

We study limit models in the class of abelian groups with the subgroup relation and in the class of torsion-free abelian groups with the pure subgroup relation. We show:
Theorem
(1) If G is a limit model of cardinality λ in the class of abelian groups with the subgroup relation, then G ≅ ℚ^{(λ)} ⊕ (⊕_{p} ℤ(p^∞)^{(λ)}).
(2) If G is a limit model of cardinality λ in the class of torsion-free abelian groups with the pure subgroup relation, then:
* If the length of the chain has uncountable cofinality, then G ≅ ℚ^{(λ)} ⊕ Π_{p} \overline{ℤ_{(p)}^{(λ)}}.
* If the length of the chain has countable cofinality, then G is not algebraically compact.

We also study the class of finitely Butler groups with the pure subgroup relation, we show that it is an AEC, Galois-stable and (<ℵ_0)-tame and short.

Mathematics Subject Classification: 03C48, 03C45, 20K20

Keywords and phrases:

Full text arXiv 1810.02203: pdf, ps.