Cauchy's Group Theorem/Proof 2

Theorem
Let $G$ be a finite abelian group whose identity is $e$.

Let $p$ be a prime number which divides order of $G$.

Then $G$ has an element whose order is $p$.

Consequently, $G$ has a subgroup of order $p$.

Proof
This result obtains as a special case of Subgroups of All Prime Power Factors.