Orbit-Stabiliser-Theorem

Description

The Orbit-Stabiliser-Theoem connects the size of the Orbit (Group action) to the size of the Stabiliser

Definition

Let $G$ be a Group acting on $X$ and let $x\in X$. Then there exists a bijective map ${\varphi }_x:G/G_x\to G(x)$ with ${\varphi }_x(g{G}_x)=g\cdot x$ for all $g\in G$. If $X$ is finite thenthe index $(G:G_x)$ is finite too and the following holds $(G:G_x)=|G(x)|$

A simple corollary of the above is the orbit equation. This separates the orbits into distinct classes.

The orbit equation

Let $G$ be a Group acting on a finite set $X$. Denote $F\subseteq X$ the set of fixed points under $X$. Let $R\subset X$ be a Repräsentantensystem of the Orbit $G(x)$ (assuming it has at least 2 elements). Then the following holds: $|X|=|F|+{\sum }_{x\in R}(G:G_x)$ Where $(G:G_x)>1$ for all $x\in R$

This equation is probably most well known from its use in the Class equation