# Statements > [!NOTE] Statement (Orbit-Stabilizer) > Let $G$ be a [[Finite Group|finite group]] [[Group action|acting]] on a finite set $X.$ For all $x\in X,$ $|O_{x}||S_{x}| = |G|$where $O_{x}$ denotes the [[Orbit under Group Action|orbit]] of $x$ and $x$ denotes the [[Stabilizer under Group Action|stabilizer]] of $x.$ **Notation:** We may write $\text{Orb}_{G}(x)$ and $\text{Stab}_{G}(x)$ instead of $O_{x}$ and $S_{x}$ respectively. # Proofs ###### Proof using Lagrange's Theorem We know that $S_{x}$ [[Stabilizer is Subgroup|is a subgroup]] of $G$ so let $G/S_{x}$ denote the [[Coset space|left coset space]] of $G$ modulo $S_{x}$. Let $\begin{align} \phi: O_{x} &\to G/S_{x} \\ g*x &\mapsto gS_{x} \end{align}$where $*$ is the group action. We must show that $\phi$ is well-defined. Suppose $gx=hx$ then $h^{-1}g x=x$ and so $h^{-1}g\in S_{x}.$ Then $gS_{x} = hS_{x}$ by [[Necessary Condition for Equality of Cosets]]. This shows also that that $\phi$ is injective. Notice that $\phi$ is surjective by definition. So $\phi$ is a [[Bijection|bijection]]. Then [[Lagrange's theorem (on Finite Groups)]] yields that the size of the left coset space of $G$ modulo $S_{x}$ is $\frac{|G|}{|S_{x}|}.$ $\blacksquare$ ###### Proof For all $y\in O_{x},$ let $S_{xy}=\{ g\in G: gx=y \}.$ Define $\phi: S_{x}\to S_{xy}$ by $\phi: g \mapsto hg.$ Then the map $\psi: S_{xy}\to S_{y}$ defined by $\psi: u \mapsto h^{-1}u$ inverts $\phi$ so by [[Bijection iff Invertible]], $|S_{x}|=|S_{xy}|$ for all $y\in O_{x}.$ But the sets $S_{xy}$ with $y\in O_{x}$ partition $G.$ It follows that $|G|=|S_{x}||O_{x}|.$ $\blacksquare$ Note that $S_{xy}$ is a left coset of $S_{x}.$ ###### Proof The subset of elements in $G$ that fix $x$ is called the stabiliser of $x$, which we denote $G_{x}$. We use $Gx$ to denote the orbit of $x$. The group action axiom $g*(h*x)=(gh)*x$, for $g,h\in G$, gives a natural bijection between the stabilisers of $x$ and $g*x$ - take $h\in G_x$ and conjugate with $g$ to get an element of $G_{g*x}$ - meaning that all elements of this multiset have the same multiplicity. Now this multiset has cardinality $\# G$. Since this multiset has $\# Gx$ elements each of multiplicity $\# G_{x}$, its cardinality also equals $\#Gx\cdot \# G_{x}$ which gives $\#Gx\cdot \# G_{x} = \#G$. This is the statement of the \textit{orbit-stabiliser theorem}. # Applications **Consequences**: [[Orbit Counting Theorem]]. # References 1. https://gowers.wordpress.com/2011/11/09/group-actions-ii-the-orbit-stabilizer-theorem/