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Disjoint cycles and least common multiple

Symmetric groups: G = Sn.

(i) Let g1, g2 belong G be two disjoint cycles, and let g = g1g2. Prove that o(g) = lcm
{ o( g1), o(g2)}, where lcm stands for the least common multiple.

(ii) Let g= g1g2 ... gr belong G, where g1,g2, ... gr are disjoint cycles. Prove that o(g) = lcm {o(g1), o(g2), ... o(gr)}.

Can you tell me how to start, and step by step guide?

Solution Preview

Proof:
(i) Since g1 and g2 are disjoint cycles, then g1*g2=g2*g1. They are commutative.
g=g1*g2. Let o(g)=n, then g^n=(g1*g2)^n=(g1^n)*(g2^n)=e. This implies that
g1^n=e and g2^n=e. So o(g1) divides n and o(g2) divides n.
...

Solution Summary

There are two proofs here regarding least common multiple of disjoint cycles.

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