Groups isomorphics to $\bigoplus_{\alpha\in\mathbb Q}\mathbb Z$

Question

Are $\bigoplus_{\alpha\in\mathbb Q}\mathbb Z$ and $\bigoplus_{\alpha\in\mathbb Q}\mathbb Z\oplus\mathbb Z$ to some famous group?

Answer 1

Note that $\mathbb{Q}$ is just an indexing set here, and the only thing that matters is the cardinality.

Both groups are just free abelian with countable basis (free abelian groups having bases of equal cardinality are isomorphic). These are isomorphic to sequences of integers with finitely many non-zero terms, if it is more "famous" to you.