Example of group homomorphism $f: G \to G$ that is injective, but not surjective.

Question

Can someone give me an example of an group endomorphism that is injective, but not surjective?

Answer 1

$f : (\mathbb{Z},+) \to (\mathbb{Z}, +), \quad x\mapsto 2x$

Answer 2

If $f: G \rightarrow G$ is injective but not surjective, then $f(G)$ is a proper subgroup of $G$ and $f(G) \cong G$.

Furthermore, if $H$ is a proper subgroup of $G$ and $H \cong G$, then there exists an isomorphism $\phi: G \rightarrow H$. Since $H$ is a proper subgroup, $\phi$ is a homomorphism $G \rightarrow G$ that is injective but not surjective.

Thus finding an example of a homomorphism $f: G \rightarrow G$ that is injective but not surjective is equivalent to finding a proper subgroup $H$ such that $H \cong G$. In azimut's answer, you have the example $\mathbb{Z} \cong 2\mathbb{Z}$.

Answer 3

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