# 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 $$\mathrm{Stab}_G(x) = \{ g \in G: g(x) = x \}$$. 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.

Parents:

• Group action

“Groups, as men, will be known by their actions.”