# external direct product of groups

The *external direct product* $G\times H$ of two groups $G$ and $H$ is defined to be the set of ordered pairs $(g,h)$, with $g\in G$ and $h\in H$. The group operation^{} is defined by

$(g,h)({g}^{\prime},{h}^{\prime})=(g{g}^{\prime},h{h}^{\prime})$

It can be shown that $G\times H$ obeys the group axioms. More generally, we can define the external direct product of $n$ groups, in the obvious way. Let $G={G}_{1}\times \mathrm{\dots}\times {G}_{n}$ be the set of all ordered n-tuples $\{({g}_{1},{g}_{2}\mathrm{\dots},{g}_{n})\mid {g}_{i}\in {G}_{i}\}$ and define the group operation by componentwise multiplication as before.

Title | external direct product of groups |
---|---|

Canonical name | ExternalDirectProductOfGroups |

Date of creation | 2013-03-22 12:23:17 |

Last modified on | 2013-03-22 12:23:17 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 8 |

Author | mathcam (2727) |

Entry type | Definition |

Classification | msc 20K25 |

Synonym | direct product^{} |

Related topic | CategoricalDirectProduct |

Related topic | DirectProductAndRestrictedDirectProductOfGroups |