TITLE: Can $f :(\Bbb R, +) \to (\Bbb R, +)$ be a homomorphism?
QUESTION [8 upvotes]: QUESTION: Is there a homomorphism
$$f :(\Bbb R, +) \to (\Bbb R, +)$$
that does not have the form $x \mapsto ax$, where $a\in\mathbb{R}$?

REPLY [13 votes]: Yes, using the Axiom of Choice.
The abelian group $\mathbb R$ is an infinite-dimensional vector space over the field $\mathbb Q$. Unfortuately, one cannot give an explicit basis of this vector space, but the linear map of "swapping" two of the basis vectors, say, is a nontrivial automorphism of $\mathbb R$ as abelian group that is not of the form $x\mapsto ax$.