# direct products of homomorphisms

Assume that ${\{{f}_{i}:{G}_{i}\to {H}_{i}\}}_{i\in I}$ is a family of homomorphisms^{} between groups. Then we can define the Cartesian product^{} (or unrestricted direct product) of this family as a homomorphism

$$\prod _{i\in I}{f}_{i}:\prod _{i\in I}{G}_{i}\to \prod _{i\in I}{H}_{i}$$ |

such that

$$\left(\prod _{i\in I}{f}_{i}\right)\left(g\right)(j)={f}_{j}(g(j))$$ |

for each $g\in {\prod}_{i\in I}{G}_{i}$ and $j\in I$.

One can easily show that ${\prod}_{i\in I}{f}_{i}$ is a group homomorphism. Moreover it is clear that

$$\left(\prod _{i\in I}{f}_{i}\right)\left(\underset{i\in I}{\oplus}{G}_{i}\right)\subseteq \underset{i\in I}{\oplus}{H}_{i},$$ |

so ${\prod}_{i\in I}{f}_{i}$ induces a homomorphism

$$\underset{i\in I}{\oplus}{f}_{i}:\underset{i\in I}{\oplus}{G}_{i}\to \underset{i\in I}{\oplus}{H}_{i},$$ |

which is a restriction^{} of ${\prod}_{i\in I}{f}_{i}$ to ${\oplus}_{i\in I}{G}_{i}$. This homomorphism is called the direct product^{} (or restricted direct product) of ${\{{f}_{i}:{G}_{i}\to {H}_{i}\}}_{i\in I}$.

Title | direct products of homomorphisms |
---|---|

Canonical name | DirectProductsOfHomomorphisms |

Date of creation | 2013-03-22 18:36:00 |

Last modified on | 2013-03-22 18:36:00 |

Owner | joking (16130) |

Last modified by | joking (16130) |

Numerical id | 4 |

Author | joking (16130) |

Entry type | Definition |

Classification | msc 20A99 |