# proof that $G$ is cyclic if and only if $\text{delimiter}69640972G\text{delimiter}86418188=exp(G)$

###### Theorem 1

A finite abelian group $G$ is cyclic if and only if $\mathrm{|}G\mathrm{|}\mathrm{=}\mathrm{exp}\mathit{}\mathrm{(}G\mathrm{)}$.

Proof. $G$ is cyclic if and only if it has an element of order $|G|$. But $\mathrm{exp}(G)$ is the maximum order of any element of $G$. Thus $G$ is cyclic only if these two are equal.

Title | proof that $G$ is cyclic if and only if $\text{delimiter}69640972G\text{delimiter}86418188=exp(G)$ |
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Canonical name | ProofThatGIsCyclicIfAndOnlyIflvertGrvertexpG |

Date of creation | 2013-03-22 16:34:14 |

Last modified on | 2013-03-22 16:34:14 |

Owner | rm50 (10146) |

Last modified by | rm50 (10146) |

Numerical id | 6 |

Author | rm50 (10146) |

Entry type | Proof |

Classification | msc 20A99 |