Definition:Stabilizer
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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): Chapter $10$: The Orbit-Stabiliser Theorem: Definition $10.8$