# proof of class equation theorem

$X$ is a finite disjoint union of finite orbits: $X=\cup_{i}Gx_{i}$. We can separate this union by considerating first only the orbits of 1 element and then the rest: $X=\cup_{j=1}^{l}\{x_{i_{j}}\}\cup\cup_{k=1}^{s}Gx_{i_{k}}=G_{X}\cup_{k=1}^{s}% Gx_{i_{k}}$ Then using the orbit-stabilizer theorem, we have $\#X=\#G_{X}+\sum_{k=1}^{s}[G:G_{x_{i_{k}}}]$ where for every $k$, $[G:G_{x_{i_{k}}}]\geq 2$, 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 ProofOfClassEquationTheorem 2013-03-22 14:20:52 2013-03-22 14:20:52 gumau (3545) gumau (3545) 4 gumau (3545) Proof msc 20D20