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Exhibit some automorphisms of the rational numbers

Prove that for each fixed nonzero $k \in \mathbb{Q}$ the map $\varphi : \mathbb{Q} \rightarrow \mathbb{Q}$ given by $q \mapsto kq$ is an automorphism.

Solution: First note that, for all $p,q \in \mathbb{Q}$, $$\varphi(p+q) = k(p+q) = kp + kq = \varphi(p) + \varphi(q),$$ so that $\varphi$ is a homomorphism.

Suppose $q \in \mathbb{Q}$; then $\varphi(q/k) = q$. Thus $\varphi$ is surjective. Suppose $\varphi(p) = \varphi(q)$; then we have $kp = kq$ and thus $p = q$. So $\varphi$ is injective. Thus $\varphi$ is an automorphism of $\mathbb{Q}$.


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