My first time posting here! Wish all of you a merry Christmas! I am alone in my apartment, reviewing the homework for the semester and found some problems…
So we have an interesting fact: a subgroup of a finitely generated group is not necessarily finitely generated (a nice exercise by considering the commutator subgroup of the free group of two generators).
So my questions are:
- Show that if a finitely generated abelian group is generated by elements, then any subgroup of is generated by elements. (This is a mid-term problem, and I forgot how to do it…)
- Is the same statement true for finite but non-abelian group? (Open to me.)