Algebraic description of limit models in classes of abelian groups

2018/10/04 by Marcos Mazari-Armida, Mazari-Armida, Marcos
Mathematics · #03C45 #03C48 #20K20 #FOS: Mathematics #Group Theory (math.GR) #Logic (math.LO) #math.GR #math.LO #msc:03C45 #msc:03C48 #msc:20K20

paper · pdf · doi:10.48550/arxiv.1810.02203

16 pages

arxiv created 2019/08/19 · arxiv updated 2019/08/20

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: Theorem (1) If G is a limit model of cardinality λ in the class of abelian groups with the subgroup relation, then G ≅ (⊕_λℚ) ⊕ ⊕_{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 G ≅ (⊕_λ ℚ ) ⊕ Π_{p prime} (⊕_λ 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 (<ℵ₀)-tame and short.

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