# kernel

Let $\rho :G\to K$ be a group homomorphism^{}. The preimage^{} of the
codomain identity element^{} ${e}_{K}\in K$ forms a subgroup^{} of the domain
$G$, called the *kernel* of the homomorphism^{};

$$\mathrm{ker}(\rho )=\{s\in G\mid \rho (s)={e}_{K}\}$$ |

The kernel is a normal subgroup^{}. It is the trivial subgroup if and
only if $\rho $ is a monomorphism^{}.

Title | kernel |
---|---|

Canonical name | Kernel |

Date of creation | 2013-03-22 11:58:24 |

Last modified on | 2013-03-22 11:58:24 |

Owner | rmilson (146) |

Last modified by | rmilson (146) |

Numerical id | 14 |

Author | rmilson (146) |

Entry type | Definition |

Classification | msc 20A05 |

Synonym | kernel of a group homomorphism |

Related topic | GroupHomomorphism |

Related topic | Kernel |

Related topic | AHomomorphismIsInjectiveIffTheKernelIsTrivial |