stabilizer

Let $G$ be a group, $X$ a set, and $\cdot :G\times X⟶X$ a group action. For any subset $S$ of $X$, the stabilizer of $S$, denoted $\mathrm{Stab}(S)$, is the subgroup

$$\mathrm{Stab}(S):=\{g\in G\mid g\cdot s\in S\text{for all }s\in S\}.$$

The stabilizer of a single point $x$ in $X$ is often denoted $G_x$.