Subgroups of a finitely generated abelian group
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 induction step.
I've spent a lot of time on this problem, but I've pretty much made no progress. I know that every element of G is a product (I'm thinking of it as a multiplicative group but I don't think it matters) of the elements in X. For the quotient group, I know that it is finitely generated. That's... pretty much it. Also, I can't use rings which seem to be the easiest way to prove this problem because we haven't covered it. However, I have covered three fundamental Isomorphism Theorems as well as a Correspondence Theorem. I'll be honest though, my understanding of these important theorems is weak, so if the inductive proof relies on an understanding of them then it's no surprise I haven't gotten anywhere. I don't really see why isomorphisms would be applicable which is why I haven't considered it very thoroughly. I have a hard time pursuing a path that doesn't seem fruitful at first glance (bad habit), so I've mostly been going over the same ideas in my head which haven't led me anywhere. For example, the Correspondence Theorem says that there is a bijection from a subgroup S to its quotient group S/K with K <= S. Any help would be great!