Fundamental Theorem of Finite Abelian Groups

Theorem
Every finite abelian group is an internal group direct product of cyclic groups of prime-power order.

The number of terms in the product and the orders of the cyclic groups are uniquely determined by the group.

Proof of the Fundamental Theorem of Finite Abelian Groups
From Lemma 2, it follows that $G$ is the product of several groups of prime-power order.

From Lemma 4, it follows that each such factor is the internal group direct product of cyclic groups.

Thus, $G$ is an internal group direct product of cyclic groups of prime-power order.

Lemma 5 demonstrates the uniqueness of those factors.