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!)
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.
Let us denote,
(ii) If the cycles α and β s...
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