# natural projection

Proposition^{}. If $H$ is a normal subgroup^{} of a group $G$, then the mapping

$$\phi :G\to G/H\mathit{\hspace{1em}}\text{where}\mathit{\hspace{1em}}\phi (x)=xH\forall x\in G$$ |

is a surjective^{} homomorphism^{} whose kernel is $H$.

*Proof.* Because every coset appears as image, the mapping $\phi $ is surjective. It is also homomorphic, since for all elements $x,y$ of $G$, one has

$$\phi (xy)=(xy)H=xH\cdot yH=\phi (x)\phi (y).$$ |

The identity element^{} of the factor group $G/H$ is the coset $eH=H$, whence

$$\mathrm{ker}(\phi )=\{x\in G\mathrm{\vdots}\phi (x)=H\}=\{x\in G\mathrm{\vdots}xH=H\}=H.$$ |

The mapping $\phi $ in the proposition is called *natural projection ^{}* or

*canonical homomorphism*.

Title | natural projection |
---|---|

Canonical name | NaturalProjection |

Date of creation | 2013-03-22 19:10:16 |

Last modified on | 2013-03-22 19:10:16 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 4 |

Author | pahio (2872) |

Entry type | Definition |

Classification | msc 20A05 |

Synonym | canonical homomorphism |

Synonym | natural homomorphism^{} |

Related topic | QuotientGroup |

Related topic | KernelOfAGroupHomomorphismIsANormalSubgroup |