# 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:

 $\displaystyle c$ $=$ $\displaystyle r m$ for some $r \in \Z$ $\displaystyle$ $=$ $\displaystyle s n$ for some $s \in \Z$

So:

 $\displaystyle \paren {a b}^c$ $=$ $\displaystyle a^c b^c$ Power of Product of Commuting Elements in Semigroup equals Product of Powers $\displaystyle$ $=$ $\displaystyle a^{r m} b^{s n}$ $\displaystyle$ $=$ $\displaystyle \paren {a^m}^r \paren {b^n}^s$ $\displaystyle$ $=$ $\displaystyle e^r e^s$ Definition of Order of Group Element $\displaystyle$ $=$ $\displaystyle e$ $\displaystyle \leadsto \ \$ $\displaystyle \order {a b}$ $\divides$ $\displaystyle c$ Element to Power of Multiple of Order is Identity

$\blacksquare$