# proof of Lagrange’s theorem

We know that the cosets $Hg$ form a partition^{} of $G$ (see the coset entry for proof of this.) Since $G$ is finite, we know it can be completely decomposed into a finite number of cosets. Call this number $n$. We denote the $i$th coset by $H{a}_{i}$ and write $G$ as

$$G=H{a}_{1}\cup H{a}_{2}\cup \mathrm{\cdots}\cup H{a}_{n}$$ |

since each coset has $|H|$ elements, we have

$$|G|=|H|\cdot n$$ |

and so $|H|$ divides $|G|$, which proves Lagrange’s theorem. $\mathrm{\square}$

Title | proof of Lagrange’s theorem |
---|---|

Canonical name | ProofOfLagrangesTheorem |

Date of creation | 2013-03-22 12:15:47 |

Last modified on | 2013-03-22 12:15:47 |

Owner | akrowne (2) |

Last modified by | akrowne (2) |

Numerical id | 6 |

Author | akrowne (2) |

Entry type | Proof |

Classification | msc 20D99 |