# 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 |