Two finitely generated abelian groups are isomorphic iff they are embedded in each other

Question

Suppose $A$ and $B$ are finitely generated abelian groups, $A$ has a subgroup that's isomorphic to $B$ and vice versa. Then A is isomorphic to $B$.

My guess (since $A,B$ are finitely generated) is to use the fundamental theorem of finitely generated abelian groups and somehow trace images of generators.