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

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    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.
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    Quote Originally Posted by hzhang610 View Post
    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.
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