# proof that every group of prime order is cyclic

The following is a proof that every group of prime order is cyclic.

Let $p$ be a prime and $G$ be a group such that $|G|=p$. Then $G$ contains more than one element. Let $g\in G$ such that $g\ne {e}_{G}$. Then $\u27e8g\u27e9$ contains more than one element. Since $\u27e8g\u27e9\le G$, by Lagrange’s theorem, $|\u27e8g\u27e9|$ divides $p$. Since $|\u27e8g\u27e9|>1$ and $|\u27e8g\u27e9|$ divides a prime, $|\u27e8g\u27e9|=p=|G|$. Hence, $\u27e8g\u27e9=G$. It follows that $G$ is cyclic.

Title | proof that every group of prime order is cyclic |
---|---|

Canonical name | ProofThatEveryGroupOfPrimeOrderIsCyclic |

Date of creation | 2013-03-22 13:30:55 |

Last modified on | 2013-03-22 13:30:55 |

Owner | Wkbj79 (1863) |

Last modified by | Wkbj79 (1863) |

Numerical id | 7 |

Author | Wkbj79 (1863) |

Entry type | Proof |

Classification | msc 20D99 |

Related topic | ProofThatGInGImpliesThatLangleGRangleLeG |