Stabilizer (Group)

definitiongroup-theory

Definition

For $G\circlearrowleft X$ (a group $G$ acting on a set $X$) and $x\in X$, the stabilizer of $x$ with respect to $G$ is,

$${\mathrm{stab}}_G(x)=\{g\in G:g\bullet x=x\}.$$

Examples

1. Say $\tau \in D_4$ is the reflection across the $x$-axis. $D_4$ acts on a square (where the origin is the center of the square) $\tau$ stabilizes $(1,0)$ and $(-1,0)$ on the $x$-axis since reflecting across the $x$-axis does not change $(1,0)$ and $(-1,0)$. $\tau \in \mathrm{stab}((1,0))$ $\tau \notin \mathrm{stab}((0,1))$

See

• Note - Orbits and Stabilizers