# proof of fundamental theorem of finitely generated abelian groups

Every finitely generated^{} abelian group^{} $A$ is a direct sum^{} of its cyclic subgroups, i.e.

$$A={C}_{{m}_{1}}\oplus {C}_{{m}_{2}}\oplus \mathrm{\dots}\oplus {C}_{{m}_{k}}\oplus \mathbb{Z}\oplus \mathrm{\dots}\oplus \mathbb{Z},$$ |

where $$. The numbers ${m}_{i}$ are uniquely determined as well as the number of $\mathbb{Z}$’s, which is the rank of an abelian group.

Proof. Let $G$ be an abelian group with $n$ generators^{}. Then for a free group^{} ${F}_{n}$, $G$ is isomorphic^{} to the quotient group^{} ${F}_{n}/A$. Now ${F}_{n}$ and $A$ contain a basis ${f}_{1},\mathrm{\dots},{f}_{n}$ and ${a}_{1},\mathrm{\dots},{a}_{k}$ satisfying ${a}_{i}={m}_{i}{f}_{i}$ for all $1\le i\le k$. As $G\cong {F}_{n}/A$, it suffices to show that ${F}_{n}/A$ is a direct sum of its cyclic subgroups $\u27e8{f}_{1}+A\u27e9$.

It is clear that ${F}_{n}/A$ is generated by its subgroups^{} $\u27e8{f}_{i}+A\u27e9$. Assume that the zero element^{} of ${F}_{n}/A$ can be written as a form $A={l}_{1}{f}_{1}+\mathrm{\dots}+{l}_{n}{f}_{n}+A$. It follows that ${l}_{1}{f}_{1}+\mathrm{\dots}+{l}_{n}{f}_{n}=a\in A$. As we write $a$ as a linear combination^{} of that basis and using ${a}_{i}={m}_{i}{f}_{i}$ we get the equations

$${l}_{1}{f}_{1}+\mathrm{\dots}+{l}_{n}{f}_{n}={s}_{1}{a}_{1}+\mathrm{\dots}{s}_{k}{a}_{k}={s}_{1}{m}_{1}{f}_{1}+\mathrm{\dots}+{s}_{k}{m}_{k}{f}_{k}.$$ |

As every element can be represented uniquely as a linear combination of its free generators ${f}_{1}$, we have ${l}_{i}={s}_{i}{m}_{i}$ for every $1\le i\le k$ and ${l}_{j}=0$ for every $$.

This means that every element ${l}_{i}{f}_{i}$ belongs to $A$, so ${l}_{i}{f}_{i}+A=A$. Therefore the zero element has a unique representation as a sum of the elements of the subgroup $\u27e8{f}_{i}+A\u27e9$.

## References

- 1 P. Paajanen: Ryhmäteoria. Lecture notes, Helsinki university, Finland (fall 2008)

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

Canonical name | ProofOfFundamentalTheoremOfFinitelyGeneratedAbelianGroups |

Date of creation | 2013-03-22 18:24:58 |

Last modified on | 2013-03-22 18:24:58 |

Owner | puuhikki (9774) |

Last modified by | puuhikki (9774) |

Numerical id | 7 |

Author | puuhikki (9774) |

Entry type | Proof |

Classification | msc 20K25 |