Universal abelian groups
  We examine the existence of universal elements in classes of infinite abelian groups. The main method is using group invariants which are defined relative to club guessing sequences. We prove, for example:   Theorem: For $n\ge 2$, there is a purely universal separable $p$-group in $\aleph_n$ if, and only if, $\cont\le \aleph_n$. 
Menachem Kojman and Saharon Shelah