A model theoretic solution to a problem of László 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 $\le \lambda$ purely embeds in $U_{\lambda }$. In this paper we use ideas from the theory of abstract elementary classes to show: $\text{Theorem.}$ 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 $n\ge 2$, there is a universal abelian $p$-group for purity of cardinality ${\aleph }_n$ if and only if $2^{{\aleph }_0}\le {\aleph }_n$. 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.

10 pages