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 $G \cong \mathbb{Q}^{(\lambda)} \oplus (\oplus_{p} \mathbb{Z}(p^\infty)^{(\lambda)})$. (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 $G \cong \mathbb{Q}^{(\lambda)} \oplus \Pi_{p} \overline{\mathbb{Z}_{(p)}^{(\lambda)}}$. * 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. 查看全文>>