Let $\sigma$ and $\tau$ be distinct(means relative prime) permutations of order $m$ and $n$, respectively(need not be $\gcd(m,n)=1$) in certain symmetric group.
Now, suppose that $\sigma\circ\tau=\tau\circ\sigma$.
Then, which one is true?:
(i) the order of $\sigma\circ\tau$ is $\textrm{lcm}(m,n)$.
(ii) the order of $\sigma\circ\tau$ divides $mn$.
(iii) the order of $\sigma\circ\tau$ divides $\textrm{lcm}(m,n)$.
Can anyone help me? Thank you!