# The Fundamental Theorem of Finite Abelian Groups

Theorem 1 (The Fundamental Theorem of Finite Abelian Groups): Let $G$ be a finite abelian group. Then $G$ is isomorphic to a direct product of cyclic groups of prime power order. |

*If $G$ is a finite abelian group of order $n$ then in general, there are many ways in which $n$ can be represented as a product of prime powers. For example, if $|G| = 12$ then $12 = 2 \cdot 2 \cdot 3$ and $12 = 4 \cdot 3$. The Fundamental Theorem of Finite Abelian groups tells us that $G$ is isomorphic to either $\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_3$ OR $\mathbb{Z}_4 \times \mathbb{Z}_3$. Unfortunately, it does NOT tell us which of these two groups $G$ will be isomorphic to. To determine that, one needs to look at specific properties of the given group $G$.*

## Example 1

Let $G$ be an abelian group of order $360$. Observe that $360$ can be factorized as:

(1)Now observe that $360$ can be factored as a product of **prime powers** in many ways, namely:

By the Fundamental Theorem of Finite Abelian groups, we must have that $G$ isomorphic to ONE of the six following groups:

(3)