proof of class equation theorem

$X$ is a finite disjoint union of finite orbits: $X={\cup }_{i}G{x}_i$. We can separate this union by considerating first only the orbits of 1 element and then the rest: $X={\cup }_{j=1}^l\{x_{i_j}\}\cup {\cup }_{k=1}^{s}G{x}_{i_k}=G_X{\cup }_{k=1}^{s}G{x}_{i_k}$ Then using the orbit-stabilizer theorem, we have $\mathrm{\#}X=\mathrm{\#}{G}_X+{\sum }_{k=1}^s[G:G_{x_{i_k}}]$ where for every $k$, $[G:G_{x_{i_k}}]\ge 2$, because if one of them were 1, then it would be associated to an orbit of 1 element, but we counted those orbits first. Then this stabilizers are not $G$. This finishes the proof.

Title	proof of class equation theorem
Canonical name	ProofOfClassEquationTheorem
Date of creation	2013-03-22 14:20:52
Last modified on	2013-03-22 14:20:52
Owner	gumau (3545)
Last modified by	gumau (3545)
Numerical id	4
Author	gumau (3545)
Entry type	Proof
Classification	msc 20D20