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

Abstract

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 $\leq \lambda$ purely embeds in $U_\lambda$. In this paper we use ideas from the theory of abstract elementary classes to show: $\textbf{Theorem.}$ Let $p$ be a prime number. If $\lambda^{\aleph_0}=\lambda$ or $\forall \mu < \lambda( \mu^{\aleph_0} < \lambda)$, then there is a universal abelian $p$-group for purity of cardinality $\lambda$. 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.