Fixed points of an action

Definition

Let a group $G$ act 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 is $\{x\}$, if and only if its stabiliser is $G$. For the action of $G$ on itself by conjugation, the fixed points are the elements of the centre $Z(G)$. The sets $X^g$ appear in Burnside's lemma.

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.