Conjugacy class is cycle type in symmetric group

Written by Patrick Stevens last updated

Summaries

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.

Proof

Same conjugacy class implies 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.

Same cycle type implies same conjugacy class

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.

Example

This result makes it rather easy to list the conjugacy classes of .