= Orbit-stabilizer theorem
{title2=$|Gx|=[G:G_x]$}
{wiki}
= Orbit-stabiliser theorem
{synonym}
For a <group action> of any <group> $G$ on a <set> $X$, the <cardinality> of the <orbit of a group action> through $x$ is the <index of a subgroup> of its <stabilizer subgroup>:
$$
|Gx|=[G:G_x],\qquad G_x=\{g\in G:g\cdot x=x\}.
$$
In particular, if $G$ is a <finite group>,
$$
|G|=|Gx|\,|G_x|,\qquad |Gx|=\frac{|G|}{|G_x|}.
$$
For a fixed <finite group>, a larger <stabilizer subgroup> means a smaller <group orbit>: more transformations repeat the same image of $x$. The <set> $X$ itself need not be finite.
The underlying <bijection> uses left <cosets>, without requiring $G_x$ to be a <normal subgroup>:
$$
\Phi:\{gG_x:g\in G\}\longrightarrow Gx,\qquad gG_x\longmapsto g\cdot x.
$$
To check it briefly, replacing $g$ by $gh$ with $h\in G_x$ preserves $g\cdot x$, so $\Phi$ is well-defined. Every member of the <group orbit> has this form. Finally, $g\cdot x=k\cdot x$ implies $k^{-1}g\in G_x$, hence $gG_x=kG_x$, proving the <bijection>. This is the construction in https://ocw.mit.edu/courses/18-703-modern-algebra-spring-2013/96221ce2d5b1077b3e5d3c5e08c88afe_MIT18_703S13_pra_l_23.pdf[MIT's group-action notes, lemma 23.5]. For infinite <groups>, the <bijection> and the <cardinality> statement $|Gx|=[G:G_x]$ still hold; division of infinite <cardinalities> does not supply the finite formula.
For the eight-element <dihedral group> $D_4$ acting on the <square> described under <group action>, moving a marked <vertex of a polygon> gives $8=4\cdot2$. Moving a marked <side of a polygon> also gives $8=4\cdot2$; moving one unoriented <diagonal of a polygon> gives $8=2\cdot4$; moving the <centroid> gives $8=1\cdot8$. These are four different <group actions>, each with one <group orbit>. In the first <group action>, the four left <cosets> of $H=G_A=\{e,s\}$ are
$$
H=\{e,s\},\quad rH=\{r,rs\},\quad r^2H=\{r^2,r^2s\},\quad r^3H=\{r^3,r^3s\}.
$$
They correspond respectively to $A,B,C,D$. Counting all eight transformations would count each position of the marked <vertex of a polygon> twice.
\Image[/orbit-stabilizer-cosets.png]
{title=<Cosets> explain the <orbit-stabilizer theorem> for a <square>}
{description=The left panel matches four left <cosets> of a <stabilizer subgroup> to four images of a <vertex of a polygon>. The right panel compares the <group orbits> for separate <group actions> on <vertices of a polygon>, <sides of a polygon>, <diagonals of a polygon>, and the <centroid>; each <group orbit> size times the corresponding <stabilizer subgroup> <group order> equals eight.}
For a second computed example, let the <symmetric group> $S_3$ act on the <Cartesian product> $X=\{1,2,3\}^2$ by $\sigma\cdot(i,j)=(\sigma(i),\sigma(j))$. There are two <orbits of a group action>:
$$
S_3\cdot(1,1)=\{(1,1),(2,2),(3,3)\},\qquad (S_3)_{(1,1)}=\{e,(23)\},
$$
and
$$
S_3\cdot(1,2)=\{(1,2),(1,3),(2,1),(2,3),(3,1),(3,2)\},\qquad (S_3)_{(1,2)}=\{e\}.
$$
The first <stabilizer subgroup> can exchange $2$ and $3$ while fixing $1$; the second must fix both $1$ and $2$, so also fixes $3$. The <orbit-stabilizer theorem> gives $6=3\cdot2$ and $6=6\cdot1$, and the two <group orbit> sizes account for all nine <ordered pairs>. The general reason for exactly these two <group orbits> is <equality-pattern orbits of ordered pairs>.
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