I already asked this question on MSE here <https://math.stackexchange.com/questions/3254184/cute-striking-applications-of-snake-lemma-outside-homological-algebra>, but still received no answer. I hope I will be more lucky here. 


When you teach algebra to students, it's often easy to find cute/direct applications of "big" theorems to motivate how useful these results can be.


For example, group actions, Sylow theorems, or the first isomorphism theorem have nice applications and can provide non trivial results in few lines.


However, I'm struggling to find such applications concerning the Snake Lemma. As the title suggests, I would like applications outside homological algebra, so an answer like "we can use it to prove the $n$-lemma" (pick for $n$ your favorite integer), is not the kind of answer I'm looking for.


Precisely, I would like to know applications of the Snake Lemma, even direct and/or not very sophisticated, but which would have been difficult or lengthy to prove without it.


I found such an example here, which is rather sophisticated: <https://math.stackexchange.com/questions/682777/is-an-abelian-group-characterized-by-its-localizations/712351#712351>


Maybe you will have other examples , even shorter or simpler?


Thanks for your help !


Greg



Cute/striking application(s) of snake lemma outside homological algebra
The Snake Lemma is used in the Iwasawa theory of elliptic curves. This is a branch of number theory.


Control theorems in Iwasawa theory following Mazur's control theorem all rely on repeated use of the Snake Lemma. See Theorem 1.2 of Greenberg's article <https://arxiv.org/pdf/math/9809206.pdf> for a statement of this theorem. In proving the control theorem the Snake Lemma is used in undertaking an analysis of the restriction maps of the local cohomology groups (on the right) and that of the global cohomology group (in the middle). One uses the Snake Lemma to get information about the restriction map on the Selmer group (which is the map on the left).


Another classic and somewhat related example can be found in the short paper of Hachmori and Matsuno "On Finite $\Lambda$ -Submodules of Selmer Groups of Elliptic Curves". In this paper, an alternate proof of one of Greenberg's theorems is given. More precisely it is shown that the Selmer group (attached to an elliptic curve over a number field) has no nontrivial finite $\Lambda$-submodules. The argument makes use of the Cassels Tate pairing and relies on a short Snake Lemma argument (cf. equation 3). The proof is about a page long.


