Theorem

Orbit-Stabilizer Theorem

Lie Groups and Lie Algebras · lie-groups.tex

For any $x \in X$ with orbit and stabilizer as in Definition~Orbit and Stabilizer, \[ |G \cdot x| = [G : G_x], \] where $[G : G_x]$ is the index of the subgroup $G_x \leq G$ (Definition~Subgroup and Normal Subgroup), equivalently the cardinality of the quotient $G/G_x$ (Definition~Quotient Group). In particular, if $G$ is finite then $|G| = |G \cdot x| \cdot |G_x|$.

Depends on

Orbit and Stabilizer
Subgroup and Normal Subgroup
Quotient Group

Dependency Graph

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