# Suppose α is a single cycle of length m and β is a single cycle of length n. What can be said about the order of the element αβ if α and β are disjoint cycles? If the cycles α and β share a single element in common? If the cycles α and β share two elements in common? (There are two different answers to this last question!)

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1. Suppose α is a single cycle of length m and β is a single cycle of length n. What can be said about the order of the element αβ if α and β are disjoint cycles? If the cycles α and β share a single element in common? If the cycles α and β share two elements in common? (There are two different answers to this last question!)

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Step 1

The proof of third part is long. So, if you need the proof, you may re-post the particular question along with your querry.

Suppose α is a single cycle of length m and β is a single cycle of length n.

(i)   First we will observe,  that what can be said about the order of the element αβ if α and β are disjoint cycles.

Step 2

Let us denote,

Step 3

(ii)   If the cycles α and β s...

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