# proof of counting theorem

Let $N$ be the cardinality of the set of all the couples $(g,x)$ such that $g\cdot x=x$. For each $g\in G$, there exist ${\mathrm{stab}}_{g}(X)$ couples with $g$ as the first element, while for each $x$, there are $|{G}_{x}|$ couples with $x$ as the second element. Hence the following equality holds:

$$N=\sum _{g\in G}{\mathrm{stab}}_{g}(X)=\sum _{x\in X}|{G}_{x}|.$$ |

From the orbit-stabilizer theorem it follows that:

$$N=|G|\sum _{x\in X}\frac{1}{|G(x)|}.$$ |

Since all the $x$ belonging to the same orbit $G(x)$ contribute with

$$|G(x)|\frac{1}{|G(x)|}=1$$ |

in the sum, then ${\sum}_{x\in X}1/|G(x)|$ precisely equals the number of distinct orbits $s$. We have therefore

$$\sum _{g\in G}{\mathrm{stab}}_{g}(X)=|G|s,$$ |

which proves the theorem.

Title | proof of counting theorem |
---|---|

Canonical name | ProofOfCountingTheorem |

Date of creation | 2013-03-22 12:47:07 |

Last modified on | 2013-03-22 12:47:07 |

Owner | n3o (216) |

Last modified by | n3o (216) |

Numerical id | 5 |

Author | n3o (216) |

Entry type | Proof |

Classification | msc 20M30 |