# 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 |
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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^{} |