# Wielandt-Kegel theorem

###### Theorem.

If a finite group^{} is the product^{} of two nilpotent subgroups^{},
then it is solvable.

That is, if $H$ and $K$ are nilpotent subgroups of a finite group $G$, and $G=HK$, then $G$ is solvable.

This result can be considered as
a generalization^{} of Burnside’s $p$-$q$ Theorem (http://planetmath.org/BurnsidePQTheorem),
because if a group $G$ is of order ${p}^{m}{q}^{n}$, where $p$ and $q$ are distinct primes, then $G$ is the product of a Sylow $p$-subgroup (http://planetmath.org/SylowPSubgroup) and Sylow $q$-subgroup, both of which are nilpotent.

Title | Wielandt-Kegel theorem |
---|---|

Canonical name | WielandtKegelTheorem |

Date of creation | 2013-03-22 16:17:37 |

Last modified on | 2013-03-22 16:17:37 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 11 |

Author | yark (2760) |

Entry type | Theorem |

Classification | msc 20D10 |

Synonym | Kegel-Wielandt theorem |