  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.
