# orbit-stabilizer theorem

Suppose that $G$ is a group acting (http://planetmath.org/GroupAction) on a set $X$.
For each $x\in X$, let $Gx$ be the orbit of $x$,
let ${G}_{x}$ be the stabilizer^{} of $x$,
and let ${\mathcal{L}}_{x}$ be the set of left cosets^{} of ${G}_{x}$.
Then for each $x\in X$ the function $f:Gx\to {\mathcal{L}}_{x}$
defined by $gx\mapsto g{G}_{x}$ is a bijection.
In particular,

$$|Gx|=[G:{G}_{x}]$$ |

and

$$|Gx|\cdot |{G}_{x}|=|G|$$ |

for all $x\in X$.

Proof:

If $y\in Gx$ is such that $y={g}_{1}x={g}_{2}x$ for some ${g}_{1},{g}_{2}\in G$,
then we have ${g}_{2}^{-1}{g}_{1}x={g}_{2}^{-1}{g}_{2}x=1x=x$, and so ${g}_{2}^{-1}{g}_{1}\in {G}_{x}$,
and therefore ${g}_{1}{G}_{x}={g}_{2}{G}_{x}$.
This shows that $f$ is well-defined.

It is clear that $f$ is surjective^{}.
If $g{G}_{x}={g}^{\prime}{G}_{x}$, then $g={g}^{\prime}h$ for some $h\in {G}_{x}$,
and so $gx=({g}^{\prime}h)x={g}^{\prime}(hx)={g}^{\prime}x$.
Thus $f$ is also injective^{}.

Title | orbit-stabilizer theorem |
---|---|

Canonical name | OrbitstabilizerTheorem |

Date of creation | 2013-03-22 12:23:10 |

Last modified on | 2013-03-22 12:23:10 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 22 |

Author | yark (2760) |

Entry type | Theorem^{} |

Classification | msc 20M30 |