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 0.1.(1) If G is a limit model of cardinality λ in the class of abelian groups with the subgroup relation, then G ∼ = (⊕ λ Q) ⊕ ⊕ p prime (⊕ λ Z(p ∞ )).(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• 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.2 For a more detailed introduction to the theory of AECs we suggest the reader to look at [Gro02], [Bal09] or [BoVas17] (this only covers tame AECs, but the AECs that we will study in this paper are all tame). 3 We say that K is Galois-superstable if there is µ < (2 LS(K) ) + such that K is λ-Galois-stable for every λ ≥ µ.Under the assumption of joint embedding, amalgamation, no maximal models and LS(K)-tameness (which hold for all the classes studied in this paper, except perhaps the one introduced in the last section) by [GrVas17] and [Vas18] the definition of the previous line is equivalent to any other definition of Galois-superstability given in the context of AECs. 4Recall that H is a pure subgroup of G if for every n ∈ N it holds that nG ∩ H = nH.

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