# proof of the Jordan Hölder decomposition theorem

Let $|G|=N$. We first prove existence, using induction^{} on $N$. If $N=1$ (or, more generally, if $G$ is simple) the result is clear. Now suppose $G$ is not simple. Choose a maximal proper normal subgroup ${G}_{1}$ of $G$. Then ${G}_{1}$ has a Jordan–Hölder decomposition by induction, which produces a Jordan–Hölder decomposition for $G$.

To prove uniqueness, we use induction on the length $n$ of the decomposition series. If $n=1$ then $G$ is simple and we are done. For $n>1$, suppose that

$$G\supset {G}_{1}\supset {G}_{2}\supset \mathrm{\cdots}\supset {G}_{n}=\{1\}$$ |

and

$$G\supset {G}_{1}^{\prime}\supset {G}_{2}^{\prime}\supset \mathrm{\cdots}\supset {G}_{m}^{\prime}=\{1\}$$ |

are two decompositions of $G$. If ${G}_{1}={G}_{1}^{\prime}$ then we’re done (apply the induction hypothesis to ${G}_{1}$), so assume ${G}_{1}\ne {G}_{1}^{\prime}$. Set $H:={G}_{1}\cap {G}_{1}^{\prime}$ and choose a decomposition series

$$H\supset {H}_{1}\supset \mathrm{\cdots}\supset {H}_{k}=\{1\}$$ |

for $H$. By the second isomorphism theorem, ${G}_{1}/H={G}_{1}{G}_{1}^{\prime}/{G}_{1}^{\prime}=G/{G}_{1}^{\prime}$ (the last equality is because ${G}_{1}{G}_{1}^{\prime}$ is a normal subgroup^{} of $G$ properly containing ${G}_{1}$). In particular, $H$ is a normal subgroup of ${G}_{1}$ with simple quotient^{}. But then

$${G}_{1}\supset {G}_{2}\supset \mathrm{\cdots}\supset {G}_{n}$$ |

and

$${G}_{1}\supset H\supset \mathrm{\cdots}\supset {H}_{k}$$ |

are two decomposition series for ${G}_{1}$, and hence have the same simple quotients by the induction hypothesis; likewise for the ${G}_{1}^{\prime}$ series. Therefore $n=m$. Moreover, since $G/{G}_{1}={G}_{1}^{\prime}/H$ and $G/{G}_{1}^{\prime}={G}_{1}/H$ (by the second isomorphism theorem), we have now accounted for all of the simple quotients, and shown that they are the same.

Title | proof of the Jordan Hölder decomposition theorem^{} |
---|---|

Canonical name | ProofOfTheJordanHolderDecompositionTheorem |

Date of creation | 2013-03-22 12:08:49 |

Last modified on | 2013-03-22 12:08:49 |

Owner | djao (24) |

Last modified by | djao (24) |

Numerical id | 9 |

Author | djao (24) |

Entry type | Proof |

Classification | msc 20E22 |