Definition:Stabilizer

Theorem
Let $G$ be a group which acts on a set $X$.

For each $x \in X$, the stabilizer of $x$ by $G$ is defined as:
 * $\operatorname{Stab} \left({x}\right) := \left\{{g \in G: g * x = x}\right\}$

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