2
$\begingroup$

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 $0$.

1 Answers 1

2

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 $q\in\Bbb Z^+$.

  • 0
    @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 $h(\langle p,q\rangle)=\langle p,0\rangle$.2012-10-27