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)

\begin{align} \quad 360 = 2^3 \cdot 3^2 \cdot 5 \end{align}

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

(2)

\begin{align} \quad 360 &= 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 5 \\ &= 2 \cdot 4 \cdot 3 \cdot 3 \cdot 5 \\ &= 8 \cdot 3 \cdot 3 \cdot 5 \\ &= 2 \cdot 2 \cdot 2 \cdot 9 \cdot 5 \\ &= 2 \cdot 4 \cdot 9 \cdot 5 \\ &= 8 \cdot 9 \cdot 5 \end{align}

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

(3)

\begin{align} \quad 1. & \mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z} \times \mathbb{Z}_3 \times \mathbb{Z}_5 \\ 2. & \mathbb{Z}_2 \times \mathbb{Z}_4 \times \mathbb{Z}_3 \times \mathbb{Z}_3 \times \mathbb{Z}_5 \\ 3. & \mathbb{Z}_8 \times \mathbb{Z}_3 \times \mathbb{Z}_3 \times \mathbb{Z}_5 \\ 4. & \mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_9 \times \mathbb{Z}_5 \\ 5. & \mathbb{Z}_2 \times \mathbb{Z}_4 \times \mathbb{Z}_9 \times \mathbb{Z}_5 \\ 6. & \mathbb{Z}_8 \times \mathbb{Z}_9 \times \mathbb{Z}_5 \end{align}