# Jordan-Hölder decomposition theorem

Every finite group^{} $G$ has a filtration^{}

$$G\supset {G}_{0}\supset \mathrm{\cdots}\supset {G}_{n}=\{1\},$$ |

where each ${G}_{i+1}$ is normal in ${G}_{i}$ and each quotient group^{} ${G}_{i}/{G}_{i+1}$ is a simple group^{}. Any two such decompositions of $G$ have the same multiset of simple groups ${G}_{i}/{G}_{i+1}$ up to ordering.

A filtration of $G$ satisfying the properties above is called a Jordan–Hölder decomposition of $G$.

Title | Jordan-Hölder decomposition theorem |
---|---|

Canonical name | JordanHolderDecompositionTheorem |

Date of creation | 2013-03-22 12:08:44 |

Last modified on | 2013-03-22 12:08:44 |

Owner | djao (24) |

Last modified by | djao (24) |

Numerical id | 9 |

Author | djao (24) |

Entry type | Theorem |

Classification | msc 20E32 |

Related topic | SubnormalSeries |

Defines | Jordan-Hölder decomposition |