State the structure theorem for finitely generated abelian groups. Prove that a finitely generated abelian group $A$ is finite if and only if there exists a prime $p$ such that $A / p A=0$.

Show that there exist abelian groups $A \neq 0$ such that $A / p A=0$ for all primes $p$. Prove directly that your example of such an $A$ is not finitely generated.