TITLE: Is an abelian subgroup of a finitely generated group finitely generated?
QUESTION [5 upvotes]: Let $H <G $ be groups, G finitely generated and H abelian. Is H then finitely generated?

REPLY [6 votes]: The Higman-Neumann-Neumann Embedding Theorem states that any countable group G can be embedded into a group generated by two elements. Now let H=$\mathbb{Q}$.