# Is every group homomorphism of the rationals an isomorphism

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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What have you tried? –  Chris Eagle Oct 27 '12 at 14:37
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 $q\in\Bbb Z^+$.
so if $h(1)\not = 0$ then $h$ is obviously a group isomorphism. Does this generalize to every group homomorphism $\mathbb Q\oplus \mathbb Q\rightarrow \mathbb Q\oplus \mathbb Q$? –  palio Oct 27 '12 at 14:55
@palio: Such a homomorphism is completely determined by $h(\langle 0,1\rangle)$ and $h(\langle 1,0\rangle)$, but it can collapse the range to a copy of $\Bbb Q$: consider $h(\langle p,q\rangle)=\langle p,0\rangle$. –  Brian M. Scott Oct 27 '12 at 15:00