# proof that a finite abelian group has element with $\text{delimiter}69640972g\text{delimiter}86418188=exp(G)$

###### Theorem 1

If $G$ is a finite abelian group, then $G$ has an element of order $\mathrm{exp}\mathit{}\mathrm{(}G\mathrm{)}$.

Proof. Write $\mathrm{exp}(G)=\prod {p}_{i}^{{k}_{i}}$. Since $\mathrm{exp}(G)$ is the least common multiple of the orders of each group element, it follows that for each $i$, there is an element whose order is a multiple of ${p}_{i}^{{k}_{i}}$, say $|{c}_{i}|={a}_{i}{p}_{i}^{{k}_{i}}$. Let ${d}_{i}={c}_{i}^{{a}_{i}}$. Then $|{d}_{i}|={p}_{i}^{{k}_{i}}$. The ${d}_{i}$ thus have pairwise relatively prime orders, and thus

$$\left|\prod {d}_{i}\right|=\prod \left|{d}_{i}\right|=\mathrm{exp}(G)$$ |

so that $\prod {d}_{i}$ is the desired element.

Title | proof that a finite abelian group has element with $\text{delimiter}69640972g\text{delimiter}86418188=exp(G)$ |
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Canonical name | ProofThatAFiniteAbelianGroupHasElementWithlvertGrvertexpG |

Date of creation | 2013-03-22 16:34:05 |

Last modified on | 2013-03-22 16:34:05 |

Owner | rm50 (10146) |

Last modified by | rm50 (10146) |

Numerical id | 4 |

Author | rm50 (10146) |

Entry type | Proof |

Classification | msc 20A99 |