A divisible abelian group is not finitely generated: group theoretical proof

Question

Do you a group theoretical proof of the following result?:

Theorem: A (non trivial) divisible abelian group is not finitely generated.

The only proof I know uses the fundamental theorem of finitely generated abelian groups, mainly based on ring theory.

Answer 1

Hints:

1) An abelian divisible group cannot have maximal groups (or, more generally, finite index subgroups);

2) A f.g. abelian group always has maximal subgroups.