---
tag: 05BM304NRX
kind: statement
role: proposition
name: Stabilisers of points in the same orbit are conjugate
msc: [20B05]
uses: [0RGGF8ZS6D, 00F2JFA41J, 0D6WHTTH7T]
sources:
  - url: https://en.wikipedia.org/wiki/Group_action
    locator: Fixed points and stabilizer subgroups
---

## Statement

Let a [[group|G5S1M9]] $G$ [[act|0WCY42EV9V]] on a set $X$, let $x \in X$, and let $g \in G$.
Then the [[stabiliser|0AYMP41THN]] of $g \cdot x$ is $G_{g \cdot x} = g G_x g^{-1}$.

## Remarks

An element $h$ fixes $g \cdot x$ if and only if $g^{-1}hg$ fixes $x$.
So the stabilisers of the points of one [[orbit|0YQPZB5Q1G]] are [[conjugate subgroups|0H11ZPX4ND]].
In particular, they all have the same order.

## Explanations

### L1 human

If $g$ moves $x$ to $y$, then the elements that fix $y$ are the elements that fix $x$, seen through $g$.
To fix $y$, you can move $y$ back to $x$, fix $x$, and move $x$ to $y$ again.
So the two stabilisers are conjugate.