The Orbit–Stabilizer Theorem

In the setting of Groups: Standing Notation, let $G$ be a group, let $X$ be a set, let $(g,y)\mapsto g\cdot y$ be an action of $G$ on $X$, and let $x\in X$.

(i) Stabilizers are subgroups. $G_{x}$ is a subgroup of $G$.

(ii) Each point lies in its orbit. $y\in Gy$ for every $y\in X$.

(iii) Equal orbits. For all $y,z\in X$, $z\in Gy$ if and only if $Gz=Gy$.

(iv) Distinct orbits are disjoint. For all $y,z\in X$ with $Gy\neq Gz$, $Gy\cap Gz=\emptyset$.

(v) Orbit and cosets of the stabilizer. Let $G/G_{x}$ be the set of left cosets of $G_{x}$, a subgroup by (i). There is exactly one function $\varphi$ from $G/G_{x}$ to $Gx$ with

$$\varphi(gG_{x})=g\cdot x\qquad\text{for every }g\in G,$$

and $\varphi$ is a bijection.

(vi) Counting. If $G$ is finite, then $Gx$ is finite and, with $\#$ the number of elements and $|G_{x}|$ the order of the subgroup $G_{x}$ of (i),

$$|G|=\#(Gx)\cdot|G_{x}|;$$

in particular $\#(Gx)\mid|G|$, with $\mid$ as in Divisibility of Integers (i).