# 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 |