# 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.”