class equation theorem

Let $G$ be a group acting on a finite set $X$. Define the set of invariants in $X$ by the action of $G$ as $G_X=\{x\in X\mathit{\hspace{1em}}|\mathit{\hspace{1em}}gx=x\mathit{\hspace{1em}}\forall g\in G\}$. Then there are $H_1,\mathrm{\dots },H_r$ subgroups of $G$ with $H_i\ne G\mathit{\hspace{1em}}\forall i$ such that $\mathrm{\#}X=\mathrm{\#}{G}_X+{\sum }_{i=1}^r[G:H_i]$

Title	class equation theorem
Canonical name	ClassEquationTheorem
Date of creation	2013-03-22 14:20:50
Last modified on	2013-03-22 14:20:50
Owner	gumau (3545)
Last modified by	gumau (3545)
Numerical id	5
Author	gumau (3545)
Entry type	Theorem
Classification	msc 20D20
Synonym	class equation
Related topic	Centralizer