Find all the endomorphisms of the multiplicative group $\mathbb R^+$.

Question

I am looking for the set $End(\mathbb R^+)$i.e. the set of all endomorphisms from $\mathbb R^{+}$ to itself where the operation is multiplication.Can someone help me to find them explicitly.I doubt whether they can be found explicitly or are existential.Do we require Zorn's Lemma here?

Answer 1

$\mathbb R^+$ is isomorphic to the additive group $\mathbb R$ via the logarithm. This is a $\mathbb Q$-vector space. Any endomorphism of this as a group will be $\mathbb Q$-linear, since it is $\mathbb Z$-linear. Thus the set of endomorphisms is all $\mathbb Q$-linear maps. You won't find an explicit description of all of these since we can't even write $\mathbb R$ as a $\mathbb Q$-vector space explicitly, but this is at least a complete description of them.