The existence of group isomorphism between Euclidean space.

Hint: For vector spaces over $\mathbb{Q}$, you can see the underlying group homomorphism as a $\mathbb{Q}$-linear map. This reduces the problem to seeing if dimensions of the vector spaces over $\mathbb{Q}$ are equal as this is equivalent to the underlying groups being isomorphic. In turn, it is known (facts section) that the dimension of an infinite dimensional vector space over $\mathbb{Q}$ is the cardinality of the vector set.