# order (of a group)

The order of a group $G$ is the number of elements of $G$, denoted $|G|$; if $|G|$ is finite, then $G$ is said to be a finite group^{}.

The order of an element $g\in G$ is the smallest positive integer $n$ such that ${g}^{n}=e$, where $e$ is the identity element^{}; if there is no such $n$, then $g$ is said to be of infinite order. By Lagrange’s theorem, the order of any element in a finite group divides the order of the group.

Title | order (of a group) |

Canonical name | OrderofAGroup |

Date of creation | 2013-03-22 12:36:47 |

Last modified on | 2013-03-22 12:36:47 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 9 |

Author | mathcam (2727) |

Entry type | Definition |

Classification | msc 20A05 |

Synonym | order |

Related topic | Group |

Related topic | Cardinality |

Related topic | OrdersOfElementsInIntegralDomain |

Related topic | OrderRing |

Related topic | IdealOfElementsWithFiniteOrder |

Defines | finite group |

Defines | infinite order |

Defines | order (of a group element) |