Algebraic description of limit models in classes of abelian groups

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: $\textbf{Theorem}$ (1) If $G$ is a limit model of cardinality $\lambda$ in the class of abelian groups with the subgroup relation, then $G \cong (\oplus_{\lambda}\mathbb{Q}) \oplus \oplus_{p \text{ prime}} (\oplus_{\lambda} \mathbb{Z}(p^\infty))$. (2) If $G$ is a limit model of cardinality $\lambda$ 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 \cong (\oplus_{\lambda} \mathbb{Q} ) \oplus \Pi_{p\text{ prime}} \overline{(\oplus_{\lambda} \mathbb{Z}_{(p)})}$. * 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 $(<\aleph_0)$-tame and short.