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

Proof
Let $g_1 g_2 = g_2 g_1$.

We have:
 * $\paren {g_1 g_2}^{n_1 n_2} = e$

Thus:
 * $\order {g_1 g_2} \le n_1 n_2$

Suppose $\order {g_1 g_2}^r = e$.

Then:

Thus $r$ is divisible by both $m$ and $n$.

The result follows.

Also see

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