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.

The only ring theory involved in the proof of the fundamental theorem is the Euclidean algorithm in the integers, and you won't get very far in group theory without using that at some point!
@DerekHolt: It is not the problem, I think it is interesting to know whether the property holds thanks to the group structure or to the ring structure.

DonAntonio · Accepted Answer · 2013-02-20 14:36:16Z

up vote 6 down vote accepted

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.

answered Feb 20 at 14:36

DonAntonio
46.7k42371

@Seirios: It can be proved that an abelian group $G$ is f.g. iff it is a quotient of a free abelian group of finite rank. – Babak S. Feb 20 at 16:18

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