TITLE: The existence of group isomorphism between Euclidean space.
QUESTION [5 upvotes]: Is there any group isomorphism for addition $\mathbb{R}^n$ to $\mathbb{R}^m$ where $n\neq m$? I could prove that there exists any vector space isomorphism or smooth map, but I still could not know that if we consider only abelian group structure for the addition of them, then the group isomorphism between them exits or not.

REPLY [6 votes]: 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.