# Math Help - how to prove that o(fg)=lcm(r,s)

1. ## how to prove that o(fg)=lcm(r,s)

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.

2. Originally Posted by hzhang610
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 $f,g$ are disjoint it means $fg=gf$. And so $(fg)^k = f^k g^k$. Now you are trying to make $(fg)^k = 1\implies f^k g^k = 1$. Again, since there are disjoint this only happens when $f^k=1$ and $g^k = 1$. But since $f^r = 1$ and $g^s = 1$ you want the least common multiple for $k$.