# Definition:Stabilizer

## Theorem

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**.

## Sources

- 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 5.6$. Stabilizers - 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $2$: The Sylow Theorems: $\S 54$ - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): Chapter $6$: An Introduction to Groups: Exercise $5$ - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): $\S 3.3$: Group actions and coset decompositions - 1996: John F. Humphreys:
*A Course in Group Theory*... (previous) ... (next): $\S 10$: Definition $10.8$