# Fitting’s theorem

*Fitting’s Theorem* states that if $G$ is a group
and $M$ and $N$ are normal nilpotent subgroups^{} (http://planetmath.org/Subgroup) of $G$,
then $MN$ is also a normal nilpotent subgroup
(of nilpotency class less than or equal to
the sum of the nilpotency classes of $M$ and $N$).

Thus, any finite group^{} has a unique largest normal nilpotent subgroup, called its *Fitting subgroup ^{}*.
More generally, the Fitting subgroup of a group $G$ is defined to be the subgroup of $G$ generated by the normal nilpotent subgroups of $G$;
Fitting’s Theorem shows that the Fitting subgroup is always locally nilpotent.
A group that is equal to its own Fitting subgroup is sometimes called a

*Fitting group*.

Title | Fitting’s theorem |
---|---|

Canonical name | FittingsTheorem |

Date of creation | 2013-03-22 13:51:39 |

Last modified on | 2013-03-22 13:51:39 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 12 |

Author | yark (2760) |

Entry type | Theorem |

Classification | msc 20D25 |

Defines | Fitting subgroup |

Defines | Fitting group |