Let $A,B,C$ be finitely generated abelian groups. Assume that there is an exact sequence $$0 \to C \to A^n \to B^n \to 0,$$where $A^n = A \oplus \dotsc \oplus A$ as usual. It is not assumed that $A^n \to B^n$ is induced by some $A \to B$.

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> Assume that $C$ can be generated by $<n$ elements, does it follow $C=0$?
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I have proven this for $n=2$ [here](https://arxiv.org/abs/1205.2884) and numerical experiments suggest that it also holds for $n=3$. Of course the same question can be asked for finitely generated modules over a PID.