I was wondering this today, and my algebra professor didn't know the answer.
Are subgroups of finitely generated groups finitely generated?
I suppose it is necessarily true for finitely generated abelian groups, but is it true in general?
And if not, is there a simple example of a finitely generated group with a non-finitely generated subgroup?
NOTE: This question has been merged with another question, asked by an undergraduate. For an example not involving free groups, please see Andreas Caranti's answer, which was the accepted answer on the merged question.