Submission date: 14 May 2020

Abstract:

Problem 5.1 in page 181 of [Fuc15] asks to find the cardinals λ such that there is a universal abelian p-group for purity of cardinality λ, i.e., an abelian p-group U_λ of cardinality λ such that every abelian p-group of cardinality ≤ λ purely embeds in U_λ. In this paper we use ideas from the theory of abstract elementary classes to show:
Theorem. Let p be a prime number. If λ^{ℵ_0}=λ or ∀ μ < λ ( μ^{ℵ_0} < λ), then there is a universal abelian p-group for purity of cardinality λ. Moreover for n ≥ 2, there is a universal abelian p-group for purity of cardinality ℵ_n if and only if 2^{ℵ_0} ≤ ℵ_n.
As the theory of abstract elementary classes has barely been used to tackle algebraic questions, an effort was made to introduce this theory from an algebraic perspective.

Mathematics Subject Classification: 20k30, 03C48, 03C45, 03C60, 13L05

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Full text arXiv 2005.07120: pdf, ps.