Stabiliser (of a group action)

Let the group $G$ act on the set $X$. Then for each element $x\in X$, the stabiliser of $x$ under $G$ is ${Stab}_G\left(x\right)=\{g\in G:g(x)=x\}$. That is, it is the collection of elements of $G$ which do not move $x$ under the action.

The stabiliser of $x$ is a subgroup of $G$, for any $x\in X$. (Proof.)

A closely related notion is that of the orbit of $x$, and the very important Orbit-Stabiliser theorem linking the two.