Finitely-generated abelian groups in a long exact sequence I learned here that if the outer groups in a short exact sequence are finitely-generated, then the middle group is, too.

Question. Is there a generalization of this to long-exact sequences?

Julian Kuelshammer mentioned the horseshoe lemma but I'm not really sure how to apply it.

One always has an exact sequence
$$0\to A\to A\oplus C\to B\oplus C\to B\to 0.$$
So given $A$ and $B$ there is a four-term exact sequences
where the middle terms can be arbitrarily "large".
