Product of Orders of Abelian Group Elements Divides LCM of Order of Product

Theorem
Let $G$ be an abelian group.

Let $a, b \in G$.

Then:
 * $\order {a b} \divides \lcm \set {\order a, \order b}$

where:
 * $\order a$ denotes the order of $a$
 * $\divides$ denotes divisibility
 * $\lcm$ denotes the lowest common multiple.

Proof
Let $\order a = m, \order b = n$.

Let $c = \lcm \set {m, n}$.

Then:

So: