counting theorem

Given a group action of a finite group $G$ on a finite set $X$, the following expression gives the number of distinct orbits

$$\frac{1}{|G|}\sum _{g\in G}{\mathrm{stab}}_g(X)$$

Where ${\mathrm{stab}}_g(X)$ is the number of elements fixed by the action of $g$.

Title	counting theorem
Canonical name	CountingTheorem
Date of creation	2013-03-22 12:22:23
Last modified on	2013-03-22 12:22:23
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	12
Author	mathcam (2727)
Entry type	Theorem
Classification	msc 20M30
Synonym	Cauchy-Frobenius-Burnside formula