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Let $G$ be a group.

Let $X$ be a set.

Let $*: G \times X \to X$ be a group action.

For each $x \in X$, the stabilizer of $x$ by $G$ is defined as:

$\Stab x := \set {g \in G: g * x = x}$

where $*$ denotes the group action.

Also denoted as

Some authors use $G_x$ for the stabilizer of $x$ by $G$.

Also known as

The stabilizer of $x$ is also known as the isotropy group of $x$.

That it is in fact a group, thus justifying its name, is demonstrated in Stabilizer is Subgroup.

Also see

  • Results about stabilizers can be found here.

Linguistic Note

The British English spelling for stabilizer is stabiliser.