Definition:Stabilizer

Theorem
Let $G$ be a group.

Let $X$ be a set.

Let $*: G \times X \to X$ be a group action.

For each $x \in X$, the stabilizer of $x$ by $G$ is defined as:
 * $\Stab x := \set {g \in G: g * x = x}$

where $*$ denotes the group action.

Also denoted as
Some authors use $G_x$ for the stabilizer of $x$ by $G$.

Also known as
The stabilizer of $x$ is also known as the isotropy group of $x$.

That it is in fact a group, thus justifying its name, is demonstrated in Stabilizer is Subgroup.

Also see

 * Stabilizer is Subgroup