# proof of the converse of Lagrange’s theorem for finite cyclic groups

The following is a proof that, if $G$ is a finite cyclic group^{} and $n$ is a nonnegative integer that is a divisor^{} of $|G|$, then $G$ has a subgroup^{} of order $n$.

###### Proof.

Let $g$ be a generator^{} of $G$. Then $|g|=|\u27e8g\u27e9|=|G|$. Let $z\in \mathbb{Z}$ such that $nz=|G|=|g|$. Consider $\u27e8{g}^{z}\u27e9$. Since $g\in G$, then ${g}^{z}\in G$. Thus, $\u27e8{g}^{z}\u27e9\le G$. Since $|\u27e8{g}^{z}\u27e9|=|{g}^{z}|={\displaystyle \frac{|g|}{\mathrm{gcd}(z,|g|)}}={\displaystyle \frac{nz}{\mathrm{gcd}(z,nz)}}={\displaystyle \frac{nz}{z}}=n$, it follows that $\u27e8{g}^{z}\u27e9$ is a subgroup of $G$ of order $n$.
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Title | proof of the converse of Lagrange’s theorem for finite cyclic groups |
---|---|

Canonical name | ProofOfTheConverseOfLagrangesTheoremForFiniteCyclicGroups |

Date of creation | 2013-03-22 13:30:27 |

Last modified on | 2013-03-22 13:30:27 |

Owner | Wkbj79 (1863) |

Last modified by | Wkbj79 (1863) |

Numerical id | 10 |

Author | Wkbj79 (1863) |

Entry type | Proof |

Classification | msc 20D99 |

Related topic | CyclicRing3 |

Related topic | ProofThatGInGImpliesThatLangleGRangleLeG |