Cardinalities in which abelian group theory is categorical?

Question

I want to find all cardinalities in which abelian group theory is categorical. It is clear for some finite cardinalities (how to determine them?) by fundamental theorem of finite abelian groups, but what about infinite ones?

Answer 1

For any infinite set $X$, the groups $(\mathbb{Z}/2\mathbb{Z})^{\oplus{X}}$ and $(\mathbb{Z}/3\mathbb{Z})^{\oplus{X}}$ are two non-isomorphic abelian groups having the same cardinality as $X$.

For a finite cardinality $n$, all abelian groups of order $n$ are isomorphic if and only if $n$ is square-free.