# normal subgroup

A subgroup^{} $H$ of a group $G$ is normal if $aH=Ha$ for all $a\in G$. Equivalently, $H\subset G$ is normal if and only if $aH{a}^{-1}=H$ for all $a\in G$, i.e., if and only if each conjugacy class^{} of $G$ is either entirely inside $H$ or entirely outside $H$.

The notation $H\mathrm{\u22b4}G$ or $H\u25c1G$ is often used to denote that $H$ is a normal subgroup^{} of $G$.

The kernel $\mathrm{ker}(f)$ of any group homomorphism^{} $f:G\u27f6{G}^{\prime}$ is a normal subgroup of $G$. More surprisingly, the converse^{} is also true: any normal subgroup $H\subset G$ is the kernel of some homomorphism^{} (one of these being the projection map $\rho :G\u27f6G/H$, where $G/H$ is the quotient group^{}).

Title | normal subgroup |
---|---|

Canonical name | NormalSubgroup |

Date of creation | 2013-03-22 12:08:07 |

Last modified on | 2013-03-22 12:08:07 |

Owner | djao (24) |

Last modified by | djao (24) |

Numerical id | 11 |

Author | djao (24) |

Entry type | Definition |

Classification | msc 20A05 |

Synonym | normal |

Related topic | QuotientGroup |

Related topic | Normalizer^{} |

Defines | normality |