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
Let $G$ be a finite abelian group.

By means of Abelian Group is Product of Prime-power Order Groups, we factor it uniquely into groups of prime-power order.

Then, Abelian Group of Prime-power Order is Product of Cyclic Groups applies to each of these factors.

Hence we conclude $G$ factors into prime-power order cyclic groups.

The factorisation of $G$ into prime-power order factors is already unique.

Therefore, a demonstration of the uniqueness of the secondary factorisation suffices.

Lemma 5 demonstrates the uniqueness of those factors.