# abelian groups of order $120$

Here we present an application of the fundamental theorem of finitely generated abelian groups.

Example (Abelian groups^{} of order $\mathrm{120}$):

Let $G$ be an abelian group of order $n=120$. Since the group is
finite it is obviously finitely generated^{}, so we can apply the
theorem. There exist ${n}_{1},{n}_{2},\mathrm{\dots},{n}_{s}$ with

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

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

Notice that in the case of a finite group^{}, $r$,
as in the statement of the theorem, must be equal to $0$. We have

$$n=120={2}^{3}\cdot 3\cdot 5=\prod _{i=1}^{s}{n}_{i}={n}_{1}\cdot {n}_{2}\cdot \mathrm{\dots}\cdot {n}_{s}$$ |

and by the divisibility properties of ${n}_{i}$ we must have that
every prime divisor^{} of $n$ must divide ${n}_{1}$. Thus the
possibilities for ${n}_{1}$ are the following

$$2\cdot 3\cdot 5,{2}^{2}\cdot 3\cdot 5,{2}^{3}\cdot 3\cdot 5$$ |

If ${n}_{1}={2}^{3}\cdot 3\cdot 5=120$ then $s=1$. In the case that ${n}_{1}={2}^{2}\cdot 3\cdot 5$ then ${n}_{2}=2$ and $s=2$. It remains to analyze the case ${n}_{1}=2\cdot 3\cdot 5$. Now the only possibility for ${n}_{2}$ is $2$ and ${n}_{3}=2$ as well.

Hence if $G$ is an abelian group of order $120$ it must be (up to isomorphism^{}) one of the following:

$$\mathbb{Z}/120\mathbb{Z},\mathbb{Z}/60\mathbb{Z}\oplus \mathbb{Z}/2\mathbb{Z},\mathbb{Z}/30\mathbb{Z}\oplus \mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}/2\mathbb{Z}$$ |

Also notice that they are all non-isomorphic. This is because

$$\mathbb{Z}/(n\cdot m)\mathbb{Z}\cong \mathbb{Z}/n\mathbb{Z}\oplus \mathbb{Z}/m\mathbb{Z}\iff \mathrm{gcd}(n,m)=1$$ |

which is due to the
Chinese Remainder theorem^{}.

Title | abelian groups of order $120$ |
---|---|

Canonical name | AbelianGroupsOfOrder120 |

Date of creation | 2013-03-22 13:54:17 |

Last modified on | 2013-03-22 13:54:17 |

Owner | alozano (2414) |

Last modified by | alozano (2414) |

Numerical id | 5 |

Author | alozano (2414) |

Entry type | Example |

Classification | msc 20E34 |

Related topic | FundamentalTheoremOfFinitelyGeneratedAbelianGroups |

Related topic | AbelianGroup2 |