Stabiliser

Definition

Let a group $G$ act on a set $X$, and let $x \in X$. The stabiliser of $x$ is the set $G_x = \{g \in G : g \cdot x = x\}$.

Remarks

The stabiliser $G_x$ is a subgroup of $G$, but in general it is not normal. For the action of $S_n$ on $\{1, \dots, n\}$, the stabiliser of $n$ is a subgroup isomorphic to $S_{n-1}$. For the action of $G$ on itself by conjugation, the stabiliser of $a$ is the centraliser $C_G(a)$.

Explanations

L1 human

The stabiliser of a point collects the group elements that keep this point in place. They may move other points. For the rotations of a square about its centre, only the identity keeps a corner in place.