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

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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:

\(\ds c\) \(=\) \(\ds r m\) for some $r \in \Z$
\(\ds \) \(=\) \(\ds s n\) for some $s \in \Z$


So:

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

$\blacksquare$


Sources