Prove that no finitely generated abelian group is divisible.
Let be a nontrivial divisible finitely generated abelian group. By FTFGAG, we have , where is a finite abelian group. By this previous exercise, is divisible if and only if and are divisible. By this previous exercise, if is nontrivial then it is not divisible; thus . Again by this result, since , is divisible if and only if is divisible.
Note that there does not exist an element such that . Thus is not divisible, hence is not divisible, a contradiction. Thus no such group exists.