---
tag: 0J438GAC8K
kind: statement
role: theorem
name: Conjugacy classes of the symmetric group correspond to cycle types
msc: [20B30]
uses: [0BCHJ30REG, 0D1VWQGZZ7, 0RJMNEN3BJ]
sources:
  - url: https://en.wikipedia.org/wiki/Symmetric_group
    locator: Conjugacy classes
  - url: https://en.wikipedia.org/wiki/Permutation
    locator: Conjugating permutations
---

## Statement

Let $\sigma, \tau \in S_n$.
Then $\sigma$ and $\tau$ are [[conjugate|0DXD4QSZ8A]] in $S_n$ if and only if they have the same [[cycle type|0J3K4Y1KPD]].

## Remarks

The key formula is $\pi (a_1\ a_2\ \dots\ a_k) \pi^{-1} = (\pi(a_1)\ \pi(a_2)\ \dots\ \pi(a_k))$ for each $\pi \in S_n$.
So conjugation keeps the cycle type, and a suitable $\pi$ sends the cycles of $\sigma$ onto the cycles of $\tau$.
Thus the number of conjugacy classes of $S_n$ is the number of partitions of $n$.
For example, $S_4$ has $5$ conjugacy classes.

## Explanations

### L1 human

Conjugation by $\pi$ renames the points, but it keeps the shape of each cycle.
Conversely, two permutations of the same shape differ only by a renaming of the points.
So in $S_n$, conjugate means "same shape".