Conjugacy in the Symmetric Group
Two permutations in are conjugate if and only if they have the same cycle type.
Theorem 8.20 (Conjugacy in )
Proof
Write as disjoint cycles . If has the same cycle type, write its corresponding cycles as and define . Then .
Conversely, if , replacing each by gives a disjoint-cycle expression for with the same cycle lengths as .
Examples
In , the conjugacy classes have representatives and sizes
Their sizes sum to .
Related
Stated in
- Theorem 8.20 (Conjugacy in )§8.3 Conjugacy in and