Fundamental Theorem of Finitely Generated Abelian Groups

Primary decomposition

Every finitely generated abelian group $G$ is isomorphic to a group of the form $\displaystyle \mathbb{Z}^n\oplus\mathbb{Z}_{q_1}\oplus\dots\oplus\mathbb{Z}_{q_t}$ where $n\geq 0$ and $q_1,\dots,q_t$ are powers of (not necessarily distinct) prime numbers. The values of $n, q_1, \dots, q_t$ are (up to rearrangement) uniquely determined by $G$ .

Invariant factor decomposition

We can also write $G$ as a direct sum of the form $\displaystyle \mathbb{Z}^n\oplus\mathbb{Z}_{k_1}\oplus\dots\oplus\mathbb{Z}_{k_u},$ where $k_1\mid k_2\mid k_3\mid\dots\mid k_u$ . Again the rank $n$ and the invariant factors $k_1,\dots,k_u$ are uniquely determined by $G$ .