Mapping from Additive Group of Integers to Powers of Group Element is Homomorphism

Theorem
Let $\struct {G, \circ}$ be a group.

Let $g \in G$.

Let $\struct {\Z, +}$ denote the additive group of integers.

Let $\phi_g: \struct {\Z, +} \to \struct {G, \circ}$ be the mapping defined as:
 * $\forall k \in \Z: \map {\phi_g} k = g^k$.

Then $\phi_g$ is a (group) homomorphism.

Proof
Let $k, l \in \Z$.

thus proving that $\phi_g$ is a homomorphism as required.