# Classification of countably infinite Abelian groups?

There’s a pretty simple classification of finitely generated Abelian groups, and there’s a relatively understandable classification of countably infinite Abelian $$p$$-groups for any prime $$p$$. But my question is, what is the general classification upto isomorphism of countably infinite Abelian groups?

How complicated is it to state? Is there a sentence we can write, like “Two countably infinite Abelian groups are isomorphic if and only if ...”?

• Even classifying, for prime $p$, subgroups of $\mathbf{Z}[1/p]^2$ up to isomorphism, is quite complicated. Note that there are continuum many. – YCor Nov 29 '20 at 23:01

Another result along the same line is a result of Thomas, which says that if you look at torsion-free countable abelian groups of finite rank, there is no Borel map that takes invariants that work for rank $$n$$ groups and produces invariants that work for rank $$n+1$$ groups. Thus the complexity problems gets strictly harder as the rank increases.