# normal closure

Let $S$ be a subset of a group $G$.
The *normal closure ^{}* of $S$ in $G$ is the intersection of all normal subgroups

^{}of $G$ that contain $S$, that is

$$\bigcap _{S\subseteq N\mathrm{\u22b4}G}N.$$ |

The normal closure of $S$ is the smallest normal subgroup of $G$ that contains $S$, and so is also called the *normal subgroup generated by* $S$.

It is not difficult to show that the normal closure of $S$ is the subgroup^{} generated by all the conjugates of elements of $S$.

The normal closure of $S$ in $G$ is variously denoted by $\u27e8{S}^{G}\u27e9$ or ${\u27e8S\u27e9}^{G}$ or ${S}^{G}$.

If $H$ is a subgroup of $G$,
and $H$ is of finite index in its normal closure,
then $H$ is said to be *nearly normal*.

Title | normal closure |

Canonical name | NormalClosure1 |

Date of creation | 2013-03-22 14:41:50 |

Last modified on | 2013-03-22 14:41:50 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 9 |

Author | yark (2760) |

Entry type | Definition |

Classification | msc 20A05 |

Synonym | normal subgroup generated by |

Synonym | conjugate closure |

Synonym | smallest normal subgroup containing |

Related topic | Normalizer^{} |

Related topic | CoreOfASubgroup |

Defines | nearly normal |