Let $f : G \to G'$ be a homomorphism of groups such that $\ker f =\{e\}$. Then $f$ is one-to-one or onto.
My confusion: if $\ker f = 0$ then it is one-to-one, but here if $e = 0$ (identity) it must be one-to-one but $\dim \ker f = 1$ hence it is onto. Please clear my doubt, what is the correct answer?
$f:2\mathbb{Z}\to\mathbb{Z}$ defined by $f(2n)=2n$ is one-to-one but not onto and a group homomorphisim under addition.
In particular take any proper subgroup $G$ of larger group $G'$ then the inclusion map $f:G \to G'$ definted by $f(g)=g$ is a group homomorphism that is one-to-one but not onto.