My question might be ridiculously easy. But I want prove
No finitely generated abelian group is divisible.
Let $G$ be a finitely generated abelian group. By definition, group $G$ is divisible if for any $g\in G$ and natural number $n$ there is $h\in G$ such that $g=h^n$. There 2 case:
In the first case I tried to use the fundamental theorem of finetely generated abelian group. By this theorem, finitely generated abelian group is finite group iff its free rank is zero. At this point I am now stuck.
In the second case I can use the following statement "No finite abelian group is divisible." But I cannot prove this state.