Fixed-Point Set

The subset of points fixed by all group elements in an action

Fixed-Point Set

Let a group action of a group $G$ on a set $X$ be given. The fixed-point set of the action is

X^G := \{x\in X : g\cdot x = x \text{ for all } g\in G\}.

Equivalently, $x\in X^G$ iff its stabilizer is all of $G$ , i.e. $G_x=G$ . Fixed points often detect “invariants” of the action; for example, fixed points of conjugation encode central elements.

Examples:

For the trivial action $g\cdot x=x$ for all $g,x$ , one has $X^G=X$ .
For the conjugation action of $G$ on itself, the fixed-point set is the center $Z(G)$ .
For the action of $C_n$ on the vertices of a regular $n$ -gon by rotation, the fixed-point set is empty if $n>1$ (no vertex is fixed by all rotations).