# Sequence of Integers defining Abelian Group

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## Theorem

Let $n \in \Z_{>0}$ be a strictly positive integer.

Let $C_n$ be a finite abelian group.

Then $C_n$ is of the form:

- $C_{n_1} \times C_{n_2} \times \cdots \times C_{n_r}$

such that:

- $n = \ds \prod_{k \mathop = 1}^r n_k$

- $\forall k \in \set {2, 3, \ldots, r}: n_k \divides n_{k - 1}$

where $\divides$ denotes divisibility.

## Proof

This theorem requires a proof.In particular: This is probably just a statement of Fundamental Theorem of Finite Abelian Groups, which needs to be studied to see what it actually meansYou can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{ProofWanted}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## Examples

### Abelian Groups of Order $100$

Let $G$ be an abelian group of order $100$.

From Sequence of Integers defining Abelian Group, $G$ can be expressed in the form:

- $G = C_{n_1} C_{n_2} \cdots C_{n_r}$

The possible sequences $\tuple {n_1, n_2, \ldots n_r}$ of positive integers which can define $G$ are:

\(\ds r = 1:\) | \(\) | \(\ds \tuple {100}\) | ||||||||||||

\(\ds r = 2:\) | \(\) | \(\ds \tuple {50, 2}\) | ||||||||||||

\(\ds \) | \(\) | \(\ds \tuple {20, 5}\) | ||||||||||||

\(\ds \) | \(\) | \(\ds \tuple {10, 10}\) |

## Sources

- 1996: John F. Humphreys:
*A Course in Group Theory*... (previous) ... (next): Chapter $14$: The classification of finite abelian groups