A model theoretic solution to a problem of László Fuchs

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 $\leq λ$ purely embeds in $U_λ$. In this paper we use ideas from the theory of abstract elementary classes to show: $\textbf{Theorem.}$ Let $p$ be a prime number. If $λ^{\aleph_0}=λ$ or $\forall μ< λ( μ^{\aleph_0} < λ)$, then there is a universal abelian $p$-group for purity of cardinality $λ$. Moreover for $n\geq 2$, there is a universal abelian $p$-group for purity of cardinality $\aleph_n$ if and only if $2^{\aleph_0} \leq \aleph_n$. 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.

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