A model theoretic solution to a problem of László Fuchs

Problem 5.1 in page 181 of [7] 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 Uλ of cardinality λ such that every abelian p-group of cardinality ≤λ purely embeds in Uλ. In this paper we use ideas from the theory of abstract elementary classes to show: Theorem 0.1. Let p be a prime number. If λℵ0=λ or ∀μ<λ(μℵ0<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 ℵn if and only if 2ℵ0≤ℵn. 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.