Order of Homomorphic Image of Group Element

Theorem
Let $G$ and $H$be groups whose identities are $e_G$ and $e_H$ respectively.

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

Let $g \in G$ be of finite order.

Then:
 * $\forall g \in G: \order {\map \phi g} \divides \order g$

where $\divides$ denotes divisibility.

If $\phi$ is injective, then:
 * $\order {\map \phi g} = \order g$

Proof
Let $\phi: G \to H$ be a homomorphism.

Let $\order g = n, \order {\map \phi g} = m$.

It follows from Element to Power of Multiple of Order is Identity that $m \divides n$.

Now suppose $\phi: G \to H$ is injective.

So $g^m = e$, as $\phi$ is injective.

From the definition of order of group element, that means $n \le m$ since $n$ is the smallest such power.

Thus $m = n$ and the result holds.