# subnormal subgroup

Let $G$ be a group, and $H$ a subgroup^{} of $G$.
Then $H$ is a *subnormal subgroup ^{}* of $G$
if there is a natural number

^{}$n$ and subgroups ${H}_{0},\mathrm{\dots},{H}_{n}$ of $G$ such that

$$H={H}_{0}\u25c1{H}_{1}\u25c1\mathrm{\cdots}\u25c1{H}_{n}=G,$$ |

where ${H}_{i}$ is a normal subgroup^{} of ${H}_{i+1}$ for $i=0,\mathrm{\dots},n-1$.

Subnormality is a , as normality of subgroups is not transitive.

We may write $H\mathrm{sn}G$ or $H\u25c1\u25c1G$ or $H\mathrm{\u22b4}\mathrm{\u22b4}G$ to indicate that $H$ is a subnormal subgroup of $G$.

In a nilpotent group^{}, all subgroups are subnormal.

Subnormal subgroups are ascendant and descendant^{}.

Title | subnormal subgroup |

Canonical name | SubnormalSubgroup |

Date of creation | 2013-03-22 13:16:27 |

Last modified on | 2013-03-22 13:16:27 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 21 |

Author | yark (2760) |

Entry type | Definition |

Classification | msc 20D35 |

Classification | msc 20E15 |

Synonym | subinvariant subgroup |

Synonym | attainable subgroup |

Related topic | SubnormalSeries |

Related topic | ClassificationOfFiniteNilpotentGroups |

Related topic | NormalSubgroup |

Related topic | CharacteristicSubgroup |

Related topic | FullyInvariantSubgroup |

Defines | subnormal |

Defines | subnormality |