I have been thinking about the following question:
Let $G$ be an infinite group with the property that every proper subgroup of $G$ is finitely generated. Can we say that $G$ is always finitely generated?
Of course, I do not expect a positive answer to this, but I cannot think about any examples regarding this claim. Any counterexample would be appreciated.
Best Answer
Consider the group $$C(p^\infty)=\cup_{n\in N} C(p^n).$$ Each subgroup of this group is a finite group, but it is not finitely generated.
This group has many different names: the Prüfer p-group or the p-quasicyclic.