Suppose f,g∈S_{n} are disjoint cycles, o(f)=r, and o(g)=s. Show that o(fg)= the least common multiple of r and s.
Since $\displaystyle f,g$ are disjoint it means $\displaystyle fg=gf$. And so $\displaystyle (fg)^k = f^k g^k$. Now you are trying to make $\displaystyle (fg)^k = 1\implies f^k g^k = 1$. Again, since there are disjoint this only happens when $\displaystyle f^k=1$ and $\displaystyle g^k = 1$. But since $\displaystyle f^r = 1$ and $\displaystyle g^s = 1$ you want the least common multiple for $\displaystyle k$.