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.

Alternative Notation
Some authors use $G_x$ for the stabilizer of $x$ by $G$.

Linguistic Note
The English spelling for it is stabiliser.