Let $\kappa$ be an infinite cardinal. How many non-isomorphic abelian groups of order (cardinality) $\kappa$ are there?
For finite $\kappa,$ we can use the classification theorem and obtain the number of non-isomorphic abelian groups of this order in finite time. I don't think there is a classification theorem for general infinite abelian groups, but I think this cannot be a difficult problem, since cardinality questions are generally easier for infinite sets than for finite sets. Nevertheless, I have no idea how to solve it.
EDIT If this is actually difficult, perhaps this question isn't:
For a cardinal number $\kappa,$ is there always a cardinal $\lambda$ such that there are more non-isomorphic abelian groups of order $\lambda$ than those of order $\kappa\,?$