Commuting elements with coprime orders, $|xy| = |x||y|$?

Question

Let $G$ be a group. If $x, y \in G$ commute and $\text{gcd}(|x|, |y|) = 1$, does it follow that $|xy| = |x||y|$?

EDIT: Progress so far. Let $C = |xy|$; then $x^C = (y^{-1})^C$. I am not sure what do from here though.

@Qudit with the condition that $\gcd(|x|,|y|)=1$ along with the condition that $xy=yx$ it does. — JMoravitz, Sep 09 '15 at 03:33

score 2 · Accepted Answer · edited Sep 13 '15 at 14:19

2

Let $d$ be the order of $xy$. Now $(xy)^{mn}=e$. Hence $d|mn$.

Now $(xy)^d = e \implies x^d y^d = e \implies x^{dn} = e \implies m|nd$.

Since $m$ does not divide $n$, $m|d$.

Similarly $n|d$ since $gcd(m,n)=1$. Thus, $mn|d$.

edited Sep 13 '15 at 14:19

Cloudscape

answered Sep 09 '15 at 03:44

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