Is it true that every non trivial group homomorphism from $\mathbb Q$ to $\mathbb Q$ is a group isomorphism. The trivial homomorphism being the map that sends every rational to $0$.
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HINT: Show that if $h:\Bbb Q\to\Bbb Q$ is a group homomorphism, then $h(q)=qh(1)$ for every $q\in\Bbb Q$. You might want to begin by showing it for $q\in\Bbb Z^+$.