Sequence of Integers defining Abelian Group

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.