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**Statement:**

**Fundamental Theorem of Finitely Generated Abelian Groups**

Every finitely generated abelian group G is isomorphic to a direct product of cyclic groups in the form

$\mathbb{Z}_{p_{1}^{r_{1}}} \times \mathbb{Z}_{p_{2}^{r_{2}}} \times \dots \times \mathbb{Z}_{p_{n}^{r_{n}}} \times \mathbb{Z} \times \mathbb{Z} \times \dots \times \mathbb{Z}$

where $p_{i}$ are primes, not necessarily distinct, and the $r_{i}$ are positive integers. The direct product is unique except for possible rearrangement of the factors; that is, the number (**Betti number** of G) of factors $\mathbb{Z}$ is unique and the prime powers $(p_{i})^{r_{i}}$ are unique.