A model theoretic solution to a problem of László Fuchs

Abstract

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. Ifλℵ0=λor∀μ<λ(μℵ0<λ), then there is a universal abelian p-group for purity of cardinality λ. Moreover forn≥2, there is a universal abelian p-group for purity of cardinalityℵnif and only if2ℵ0≤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.