# fundamental theorem of finitely generated abelian groups

###### Theorem 1 (Fundamental Theorem of Finitely Generated Abelian Groups).

Let $G$ be a finitely generated^{} abelian group^{}. Then there is a
unique expression of the form:

$$G\cong {\mathbb{Z}}^{r}\oplus \mathbb{Z}/{n}_{1}\mathbb{Z}\oplus \mathbb{Z}/{n}_{2}\mathbb{Z}\oplus \mathrm{\dots}\oplus \mathbb{Z}/{n}_{s}\mathbb{Z}$$ |

for some integers $r\mathrm{,}{n}_{i}$ satisfying:

$$r\ge 0;\forall i,{n}_{i}\ge 2;{n}_{i+1}\mid {n}_{i}\mathit{\text{for}}1\le i\le s-1.$$ |

Title | fundamental theorem of finitely generated abelian groups |
---|---|

Canonical name | FundamentalTheoremOfFinitelyGeneratedAbelianGroups |

Date of creation | 2013-03-22 13:54:12 |

Last modified on | 2013-03-22 13:54:12 |

Owner | alozano (2414) |

Last modified by | alozano (2414) |

Numerical id | 6 |

Author | alozano (2414) |

Entry type | Theorem |

Classification | msc 20E34 |

Synonym | classification of finitely generated abelian groups |

Related topic | AbelianGroupsOfOrder120 |

Related topic | FinitelyGenerated |

Related topic | AbelianGroup2 |

Defines | fundamental theorem of finitely generated abelian groups |