---
tag: Y20PEX049W
kind: example
name: Orders of the elements of S3
msc: [20D60]
instance_of: [0WE2NHZJ7B]
illustrates: [08V7JHS3QZ]
uses: [0515A6D2Y1, 0JPYSD5E0B]
sources:
  - url: https://en.wikipedia.org/wiki/Order_(group_theory)
    locator: Example
---

## Example

The [[symmetric group|0515A6D2Y1]] $S_3$ has order $6$.
The identity has [[order|0JPYSD5E0B]] $1$.
Each transposition $(1\,2)$, $(1\,3)$ and $(2\,3)$ has order $2$.
Each $3$-cycle $(1\,2\,3)$ and $(1\,3\,2)$ has order $3$.
Each of the orders $1$, $2$ and $3$ divides $6$, as the [[result on element orders|08V7JHS3QZ]] states.

## Remarks

No element has order $6$, so $S_3$ is not cyclic.