---
tag: 08JF1K4DW1
kind: definition
name: Fixed points of an action
msc: [20B05]
defines: 0RRWNMYRXC
uses: [0YSE802YCQ, 00F2JFA41J, 0RGGF8ZS6D]
sources:
  - url: https://en.wikipedia.org/wiki/Group_action
    locator: Fixed points and stabilizer subgroups
---

## Definition

Let a [[group|G5S1M9]] $G$ [[act|0WCY42EV9V]] on a set $X$.
A point $x \in X$ is a fixed point of the action when $g \cdot x = x$ for each $g \in G$.
The set of all fixed points is written $X^G$.
For $g \in G$, the set of points fixed by $g$ is $X^g = \{x \in X : g \cdot x = x\}$.

## Remarks

A point $x$ is a fixed point if and only if its [[orbit|0YQPZB5Q1G]] is $\{x\}$, if and only if its [[stabiliser|0AYMP41THN]] is $G$.
For the action of $G$ on itself by conjugation, the fixed points are the elements of the [[centre|0DZ93493VN]] $Z(G)$.
The sets $X^g$ appear in [[Burnside's lemma|0Z47M1B2VS]].

## Explanations

### L1 human

A fixed point is a point that no element of the group moves.
For the rotations of a square about its centre, the centre is a fixed point.
The corners are not fixed points, because a quarter turn moves them.