What is $\operatorname{Aut}(\mathbb{R},+)$?
I was solving some exercises about automorphisms. I was able to show that $\operatorname{Aut}(\mathbb{Q},+)$ is isomorphic to $\mathbb{Q}^{\times}$. The isomorphism is given by $\Psi(f)=f(1)$, but when I try to do the same thing with $\operatorname{Aut}(\mathbb{R},+)$ I got stuck.
My question is: What is $\operatorname{Aut}(\mathbb{R},+)$?
I would appreciate your help.
Solution 1:
First note that for every $\alpha\in\mathbb R^\times$ we have that $x\mapsto\alpha\cdot x$ is an automorphism of $(\mathbb R,+)$.
Also note that if $f(x+y)=f(x)+f(y)$ then $f(2)=f(1)+f(1)$, and by induction $f(n)=n\cdot f(1)$ for $n\in\mathbb N$, equally $f(k)=k\cdot f(1)$ for $k\in\mathbb Z$. This carries to rationals as well, so $f\left(\frac{p}{q}\right)=\frac{p}{q}\cdot f(1)$.
Now, if $f$ is continuous then for every $x\in\mathbb R$ we have $f(x)=x\cdot f(1)$. So setting $\alpha=f(1)$ gives us that the continuous solutions are the solutions defined by $\mathbb R^\times$, therefore $\mathrm{Aut}(\mathbb R,+)\cap\{f\in\mathbb R^\mathbb R\mid f\text{ continuous}\}\cong(\mathbb R^\times,\cdot\ )$.
The existence of non-continuous solutions requires some axiom of choice, since such solutions generate Lebesgue non-measurable sets. So if we assume that every set is Lebesgue measurable (e.g. Solovay's model of models of Determinacy) then indeed there are no other solutions.
However, assuming the axiom of choice we can generate a basis for the vector space $\mathbb R$ over $\mathbb Q$. Note that every permutation of this basis can be extended to an automorphism of the vector space, namely $(\mathbb R,+)$.
The cardinality of such basis (known as Hamel basis) is $2^{\aleph_0}$, we have $2^{2^{\aleph_0}}$ many non-continuous solutions if we assume that such basis exists.
For further reading:
- Is there a non-trivial example of a $\mathbb Q-$endomorphism of $\mathbb R$?
- Horst Herrlich, The Axiom of Choice. Springer, 2006. (In particular section 5.1)