---
tag: 0P8CY9V437
kind: example
name: Action by conjugation
msc: ["20E45", 20B05]
instance_of: [0WCY42EV9V]
uses: [0YSE802YCQ, 0D1VWQGZZ7, 0Q0Z1BEXTF, 0S7Y9Q990Q]
sources:
  - url: https://en.wikipedia.org/wiki/Conjugacy_class
    locator: Conjugation as a group action, centralizers, and the class equation
---

## Example

Let $G$ be a [[group|G5S1M9]].
The map $G \times G \to G$, $(g, x) \mapsto gxg^{-1}$, is an [[action|0WCY42EV9V]] of $G$ on the set $G$.
Indeed, $exe^{-1} = x$ and $g(hxh^{-1})g^{-1} = (gh)x(gh)^{-1}$.
The [[orbits|0YQPZB5Q1G]] of this action are the [[conjugacy classes|0DEBXAJ5SA]] of $G$.
The [[stabiliser|0AYMP41THN]] of $x$ is the [[centraliser|02YYV36BFM]] $C_G(x)$.
The [[fixed points|0RRWNMYRXC]] are the elements of the [[centre|0DZ93493VN]] $Z(G)$.
For an [[abelian group|082GHNPJ53]], each element is a fixed point, and each orbit has one element.