# example of fibre product

Let $G$, ${G}^{\prime}$, and $H$ be groups, and suppose we have homomorphisms^{} $f:G\to H$ and ${f}^{\prime}:{G}^{\prime}\to H$. Then we can construct the fibre product $G{\times}_{H}{G}^{\prime}$. It is the following group:

$$\left\{(g,{g}^{\prime})\in G\times {G}^{\prime}\text{such that}f(g)={f}^{\prime}({g}^{\prime})\right\}.$$ |

Observe that since $f$ and ${f}^{\prime}$ are homomorphisms, it is closed under^{} the group operations^{}.

Note also that the fibre product depends on the maps $f$ and ${F}^{\prime}$, although the notation does not reflect this.

Title | example of fibre product |
---|---|

Canonical name | ExampleOfFibreProduct |

Date of creation | 2013-03-22 14:08:38 |

Last modified on | 2013-03-22 14:08:38 |

Owner | archibal (4430) |

Last modified by | archibal (4430) |

Numerical id | 4 |

Author | archibal (4430) |

Entry type | Example |

Classification | msc 14A15 |

Related topic | Group |

Related topic | Homomorphism |

Related topic | CartesianProduct |