Sunday, November 8, 2015

abstract algebra - Is the group isomorphism $exp(alpha x)$ from the group $(mathbb{R},+)$ to $(mathbb{R}_{>0},times)$ unique?




I'm having a problem trying to find the simplest way of proving this, which has most probably been solved a hundred of times but I am unable to find a good reference.




I have two groups, $(\mathbb{R},+)$ and $(\mathbb{R}_{>0},\times)$. I am trying to prove that the only class of isomorphisms between them is the class $F = \{f: f(x) = \exp(\alpha x), $ for all $\alpha \in \mathbb{R}_{>0}\}$. Existence is easy to prove: what I'm having trouble with is a clean algebraic uniqueness proof.



Does anyone know the proof or a reference containing this proof?



Thanks in advance!


Answer



Wait, it may not be true.



Consider $(\Bbb R,+)$ as an infinite (continuum) dimension vector space over $\Bbb Q$, and fix a basis (Hamel basis).




Then, any automorphism of this vector space (for example permuting the basis) will be an automorphism of $(\Bbb R,+)$, and you can compose this with any exponential.


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