In the symmetric group on a finite set, the conjugacy class of an element is determined exactly by its cycle type.
More precisely, two elements of are conjugate in if and only if they have the same cycle type.
Suppose has the cycle type ; write Let .
Then , where is taken to be .
Therefore which has the same cycle type as did.
Suppose so that has the same cycle type as the from the previous direction of the proof.
Then define , and insist that does not move any other elements.
Now by the final displayed equation of the previous direction of the proof, so and are conjugate.
This result makes it rather easy to list the conjugacy classes of .