Definition

Orbit and Stabilizer

Lie Groups and Lie Algebras · lie-groups.tex

Let $G$ act on $X$ (Definition~Group Action). For $x \in X$, the orbit and stabilizer of $x$ are \[ G \cdot x := \{ g \cdot x \mid g \in G \}, \qquad G_x := \{ g \in G \mid g \cdot x = x \}. \] Note that $G_x \leq G$ is a subgroup in the sense of Definition~Subgroup and Normal Subgroup.

Depends on

Group Action
Subgroup and Normal Subgroup

Used in

Orbit-Stabilizer Theorem

Dependency Graph

flowchart LR classDef current fill:#6366f1,color:#fff,stroke:#4f46e5 nf7184fc4["Group Action"] n3d67b716["Subgroup and Normal Subgroup"] n1641e614["Orbit and Stabilizer"]:::current n28571139["Orbit-Stabilizer Theorem"] nf7184fc4 --> n1641e614 n3d67b716 --> n1641e614 n1641e614 --> n28571139 click nf7184fc4 "../objects/f7184fc4.html" "_self" click n3d67b716 "../objects/3d67b716.html" "_self" click n28571139 "../objects/28571139.html" "_self"