# Direct product

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Direct product | |
---|---|

Symbol | , |

Size formula | |

Algebraic properties | |

Algebraic structure | Commutative monoid |

Associative | Yes |

Commutative | Yes |

Identity | Trivial group |

Uniquely factorizable | Yes^{[note 1]} |

The **direct product** of two groups and is a group whose elements are given by the ordered pairs , with and . Its group operation is defined as

When both groups are abelian (commutative), so is the direct product. The order of the direct product of two groups is the product of their orders.

The symmetry group of the prism product of two polytopes is the direct product of their symmetry groups.

## Notes[edit | edit source]

- ↑ For finite groups. See the Krull-Remak-Schmidt theorem for a more granular classification.

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