Cauchy's theorem for abelian groups

Statement
Suppose $$G$$ is a fact about::finite abelian group and $$p$$ is a prime number that divides the order of $$G$$. Then, $$G$$ contains an element of order $$p$$, or equivalently (by looking at the subgroup generated), a subgroup of order $$p$$. This subgroup is a cyclic group of prime order.