Kernel of Homomorphism on Cyclic Group

Theorem
Let $G = \gen g$ be a cyclic group with generator $g$.

Let $H$ be a group.

Let $\phi: G \to H$ be a (group) homomorphism.

Let $\map \ker \phi$ denote the kernel of $\phi$.

Let $\Img G$ denote the homomorphic image of $G$ under $\phi$.

Then:
 * $\map \ker \phi = \gen {g^m}$

where:
 * $m = 0$ if $\Img \phi$ is an infinite cyclic group
 * $m = \order {\Img \phi}$ if $\Img \phi$ is a finite cyclic group.