A model theoretic solution to a problem of L\'{a}szl\'{o} Fuchs

Problem 5.1 in page 181 of [Fuc15] asks to find the cardinals $\lambda$ such that there is a universal abelian $p$-group for purity of cardinality $\lambda$, i.e., an abelian $p$-group $U_{\lambda }$ of cardinality $\lambda$ such that every abelian $p$-group of cardinality $\le \lambda$ purely embeds in $U_{\lambda }$. In this paper we use ideas from the theory of abstract elementary classes to show: $\mathbf{\text{Theorem.}}$ Let $p$ be a prime number. If ${\lambda }^{{\mathrm{\aleph }}_0}=\lambda$ or $\forall \mu < \lambda( \mu^{\aleph_0} < \lambda)$,thenthereisauniversalabelian$p$-group for purity of cardinality $\lambda$. Moreover for $n\ge 2$, there is a universal abelian $p$-group for purity of cardinality ${\mathrm{\aleph }}_n$ 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.