G is a finitely generated group and F is its subgroup.


Q: Under what known sufficient conditions can we guarantee that F is finitely generated?
(e.g. G is Abelian)



Finitely generated subgroup
A consequence of the Nielsen-Schreier theorem is the following: If a group generated by $n$ elements, then every subgroup of *finite* index $k$ is generated by $kn−k+1$ elements. See also [this](http://www.artofproblemsolving.com/Forum/viewtopic.php?t=62075) aops discussion; there jmerry gives a direct algebraic proof.


