I should give a better answer. In the case where H is a subgroup that contains elements who are a product of <n+1 elements in the generator, then we can invoke the induction hypothesis. So I guess I sort of do see how the IH plays in. However, I have no idea how the quotient group helps me in...
Ah yes, I forgot to mention (I had been up for 7 hours past my bedtime trying the problem) that I had done the base case. And by done I mean I cited a theorem I never proved correctly, but assume we did in class at one point: every subgroup of a cyclic subgroup is itself cyclic. Unfortunately I...
I have a homework problem which asks to prove that the subgroups of a finitely generated abelian group are finitely generated.
The hint in the book says to prove it by induction on the size of X where the group G = <X>. It also says to consider the quotient group G/an+1 (with an+1 in X) in the...