Suppose we have a permutation . It can be broken up into a product of disjoint cycles: . If are the lengths of each cycle, it can be shown that the order of is .
So, if the order of is a prime , then .
I will leave it to you to conclude.
Let and let p be a prime number. Show that if and only if the cycles of have lengths 1 or p.
I realise that this is an if and only if question, so it will need to be shown in the two directions.
For showing that if the cycles all have lengths 1 or p, then I have have shown it for length 1 but can't see how to show it for p. I can't think of where to start for the proof in the other direction. Help anyone?