TITLE: Example of group homomorphism $f: G \to G$ that is injective, but not surjective.
QUESTION [5 upvotes]: Can someone give me an example of an group endomorphism that is injective, but not surjective?

REPLY [2 votes]: Another example: $$f:\mathbb Q_+\to\mathbb Q_+, f(q) = q^2$$ where $\mathbb Q_+$ is the group of positive rationals under multiplication.