A question says: Using the isomorphism theorems or otherwise, prove that a subgroup of a finitely generated abelian group is finitely generated.
I would say that for a finitely generated abelian group $G$, there exists elements $g_1,\dots, g_n$ such that a linear combination of them generates the whole group. Therefore as every element of a subgroup has an element in $G$ and so can be made by a linear combination of $g_1,\dots, g_n$. This means that $g_1,\dots, g_n$ span the whole subgroup and so there exists a subset of $g_1,\dots, g_n$ which generates the subgroup.
This answer seems far too 'linear algebra-ish' rather than 'group theory-ish' and I can't seem to see how one would use the isomorphism theorems? Help would be appreciated!