Fundamental Theorem of Finite Abelian Groups

Theorem
Every finite Abelian group is a 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
First Lemma: Let G be a finite Abelian group of order $$mp^n$$ where p is a prime that does not divide m. Then $$G=H \times K$$ where $$H=  \left\{ x \in G : x^{p^n}=e \right\}  $$ and $$K= \left\{ x \in G : x^m=e \right\}  $$. $$\left|{H}\right| = p^n$$.

Proof:

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