Stabiliser (of a group action)

[summary: The stabilizer of an element $x$ under the action of a group $G$ is the set (actually a subgroup) of elements of $G$ which leave $x$ unchanged. ]

Let the Group $G$ act on the set $X$. Then for each element $x \in X$, the stabiliser of $x$ under $G$ is $\mathrm{Stab}_G(x) = \{ g \in G: g(x) = x \}$. 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.