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 means
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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

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