Algebraic description of limit models in classes of abelian groups

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: $\mathbf{\text{Theorem}}$ (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 $(<{\mathrm{\aleph }}_0)$-tame and short.