Stabiliser (of a group action)

by Patrick Stevens Jun 20 2016 updated Jun 28 2016

If a group acts on a set, it is useful to consider which elements of the group don't move a certain element of the set.

[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.