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This question occurred to me while studying order of elements in finite group theory. The question which I was solving was: if $|x|=12, |y|=15$ in some Abelian group, what would be the order of $x^{8} y^{9}$. Based on it, the following generalization was developed.

Let $x, y\in G$ be of finite orders and $xy=yx$. Show order of $x^r y^s$ is the least common multiple of $|x^r|$ and $|y^s|$.

I tried to prove it but not getting any clue. Any help would be appreciated. In case if any modification needed for correcting the question, you are most welcome.

KON3
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    This problem reduces to the case $r=s=1$. Can you prove it in that case? Can you prove that under your hypotheses, $\forall n \in \Bbb N~((xy)^n=x^ny^n)$? The latter statement is easily proved by induction on $n$. – Robert Shore Apr 22 '23 at 15:31
  • You will probably need to impose some additional condition on $x$ and $y$ in the general statement, since the statement won’t necessarily work if $y=x^{-1}$ (for instance). – Kenanski Bowspleefi Apr 22 '23 at 15:34
  • @RobertShore I think yes I can. Since $n$ is a positive integer, by using induction principle we can show the identity. – KON3 Apr 22 '23 at 15:34
  • This is a better duplicate, sorry. – Shaun Apr 22 '23 at 15:42

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