Order of Product of Commuting Group Elements of Coprime Order is Product of Orders

Theorem
Let $G$ be a group.

Let $g_1, g_2 \in G$ be commuting elements such that:

where $\order {g_1}$ denotes the order of $g_1$ in $G$.

Let $n_1$ and $n_2$ be coprime.

Then:
 * $\order {g_1 g_2} = n_1 n_2$

Also see

 * Unique Composition of Group Element whose Order is Product of Coprime Integers, a converse of this