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I would like to know whether there are examples where finite group theory can be directly applied to solve real world problems outside of mathematics. (Sufficiently applied mathematics such as cryptography, coding theory, or statistics still count.)

Let me clarify: I am not interested in applications of elementary group theory which happen to involve finite groups (e.g. cyclic/dihedral/easy groups as molecular symmetries). I am interested in applications of topics specifically coming from finite group theory as a discipline, like one might see in Isaacs, Huppert, or Robinson.

"The Schur multiplier has order $2640,$ so we should point the laser that way."

"Is this computer system secure?" "No - Frobenius kernels are nilpotent."

I'm aware of this MO post, but many of the applications listed there are inside mathematics or fall in the "applications of easy groups" category. It is entirely possible that what I'm looking for doesn't exist, and that finite group theory is still an untouchable, pure subject, like number theory in the days of G. H. Hardy. But perhaps not. Does anyone know of any applications of the higher level stuff?

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    $\begingroup$ Have you ever seen the Futurama episode The Prisoner of Benda or the Stargate SG-1 episode Holiday? I know that doesn't answer your question (which is why it's a comment, instead of an answer), but that sort of "fake real-world scenario" had its solution in finite group theory. $\endgroup$
    – Cameron Buie
    Commented Mar 8, 2013 at 2:12
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    $\begingroup$ Cryptography, cryptography, cryptography. $\endgroup$
    – Cheerful Parsnip
    Commented Mar 8, 2013 at 2:39
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    $\begingroup$ @JimConant Could you elaborate on that, please? Everything I've seen in cryptography has been finite fields and quadratic forms. I haven't run into finite group theory yet... I'd love an example. $\endgroup$
    – Alexander Gruber ♦
    Commented Mar 8, 2013 at 5:40
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    $\begingroup$ Isn't a Rubik's Cube a physical embodiment of a finite group? $\endgroup$
    – Jim Garrison
    Commented Mar 8, 2013 at 7:09
  • $\begingroup$ @AlexanderGruber: actually I'm not an expert, and maybe I should have said cryptanalysis, which is what I was thinking. I read a nice article a while back by one of the Polish mathematicians who worked on cracking the Enigma machine code in World War II, and their methods relied a lot on an analysis of the symmetric group. $\endgroup$
    – Cheerful Parsnip
    Commented Mar 8, 2013 at 10:30

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I think I see what Alexander means. There is no shortage of group theoretic (or number theoretic) thinking in telecommunications applications, but even though the apps are hi-tech, the group theory in use does not quite have the same sheen. Let me list a few examples:

  1. The theory of cyclic (error-correcting) codes is really all about the combinatoric properties of the summands of the left regular representation of a cyclic group over the field $\mathbb{F}_2$ (or some other finite field). Harmonic analysis of the cyclic group (or of an elementary abelian 2-group) is all over the place here.
  2. Many a coding theoretical proof or performance estimation calculation simply could not be carried out without a symmetry argument: "The points of the signal constellation form an orbit of a finite group of unitary matrices meaning that the constellation `looks similar' around any one of its points. Therefore we can w.l.o.g. assume that...", "The automorphism group of this code is doubly transitive, therefore we can w.l.o.g. assume that $0$ and $1$ are contained in the support of this codeword, and reduce the number of variables from six to four."
  3. Single shot properties of specific groups occasionally come to the fore: "This 2-group of units of this Clifford algebra only have such and such representations. This allows us to prove the non-existence of certain types of desirable multiantenna signal constellations" or "The structure of Coxeter groups allows us to solve the problem of finding optimal spherical codes with that group as a group of symmetries as well as design a very efficient algorithm locating the signal point closest to the received vector." (the last one jointly due to yours truly). Edit: As another example belonging to this set I just recalled a paper, where people working at Bell Labs proposed using finite groups with fixed-point-free representations to do differential modulation in multiantenna setting. Such groups were classified by Zassenhaus (he was studying nearfields). I don't think that the idea took off. Also most of those groups are metacyclic with a few non-solvable groups completing the list. So it doesn't really change my impression very much.
  4. Suitable Molien series are use in classifying the possible self-dual codes.

I will add more items to the list, if/when I think of them. My point is that group theoretic thinking is ubiquitous in coding theory/telcomm, but in most cases we don't really need what could be called deep group theoretic results. There are rare tailor-made exceptions like the connections between Mathieu-24 and the extended binary Golay code, but I'm not sure that that qualifies either, because that code, while grand, is too short for practical applications.

One of the reasons for this is that the really interesting groups are few and far between, but the engineers want a scalable system with a lot of flexibility in the parameters. I once described the above mentioned use of Coxeter groups to a group of engineers. I was very excited myself about my speed-up tweaks to the length reduction algorithm (generalizing the Shell sorting algorithm), and it showed. So after my presentation one of them asked: "This looks really good, but 8 is kinda low dimensional. Does it scale?"

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    $\begingroup$ How would you define group theoretic thinking? $\endgroup$
    – Pineapple Fish
    Commented Dec 11, 2020 at 21:28
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    $\begingroup$ A good question @BenjaminThoburn. Wish I knew. The examples I listed are relatively low level. Basically using symmetries present in the situation to make some deductions and/or analyze some constructions. Nothing deep, really. $\endgroup$
    – Jyrki Lahtonen
    Commented Dec 11, 2020 at 21:34
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    $\begingroup$ I guess any argument where properties of groups play a role is close to what I had in mind :-) $\endgroup$
    – Jyrki Lahtonen
    Commented Dec 12, 2020 at 5:02
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Did you mean things like:

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As a small member here, I did some papers in Chemical areas using Group theory. I think, one of the best application of Group theory can be seen in Chemistry. Shinsaku Fujita has been one of the first one who opens this door.

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    $\begingroup$ I just remember a fact I read somewhere on this site, that Polya used Polya Enumeration lots to count molecules. $\endgroup$
    – user58512
    Commented Mar 16, 2013 at 19:09
  • $\begingroup$ @caveman: Yes exactly! and the GAP helps us practically. $\endgroup$
    – Mikasa
    Commented Mar 17, 2013 at 5:01
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Perhaps a most prominent example of an application of group theory (a la symmetry study) in real life is for the study of crystals. A whole book on the subject as well as many links upon searching for "group theory crystals" will provide you with lots of concrete examples.

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This may not be what you had in mind, but a paper by C. Mochon, "Anyons from non-solvable finite groups are sufficient for universal quantum computation" seems to use some more advanced group theory in the field of quantum computation, and cites a theorem of Feit and Thompson. I know little about this field however, and cannot comment on the paper itself, but it might be a good starting point.

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    $\begingroup$ "a theorem of Feit and Thompson"?! Wouldn't that be the huge F&T theorem that states that any finite group of odd order is solvable? $\endgroup$
    – DonAntonio
    Commented Mar 8, 2013 at 3:44
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You mentioned security. There are several papers devoted to the use of group theory in cryptography.

Please allow me to mention a tiny contribution of which I am one of the coauthors, in which the O'Nan-Scott theorem is used.

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I would look in a few places.

Expander graphs, which may have originated in telephone network design and are now tied to many other areas. A construction might use Cayley graphs of generating sets of large finite groups over a finite field.

Combinatorics of Weyl groups in types B, C and D. Applications often need constructions that come in families and can scale up beyond particular sizes and dimensions. This field is probably too recent to have seen many applications.

Experimental designs in statistics that use highly symmetric block designs. This is similar in flavor to coding theory and sphere packings. Amounts of symmetry used are less than in those classical application areas, but there is a large literature and who knows what might be there. As nonparametric Big Data statistics creates new needs the math will undoubtedly get more complicated.

Closer to the continuous case, John Baez has posted in his weekly series on the Frobenius-Schur indicator as a sign of real, complex and quaternionic aspects of physics.

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Does the representation theory of finite groups still count? I'd be surprised if there aren't any applications at all in physics for the representation theory of the cyclic or symmetric groups.

(After Yuans correction), the Fock space of $n$ free (non-interacting) particles is constructed is the trivial representation of the symmetric group $S_n$ for Bosons and for the sign representation for Fermions.

Perhaps theoretical Physics is still not real world physics enough for you :).

Also, Levi-Strauss required (non-trivial) finite group theory (supplied by Andre Weil) in his analysis of kinship relations in certain Amazonian tribes. Jack Morava in On the Canonical formula of Claude levi-Strauss has this to say in the abstract:

The anthropologist Claude LeviStrauss has formulated a theory of the structure of myths using a formalism borrowed from mathematics, which has been difficult to interpret, and is somewhat controversial...in this note I propose an interpretation of LeviStrauss's `canonical formula' in terms of an anti-automorphism of the quaternion group of order eight.

and he ends with:

Perhaps it will be useful to mention that modern mathematical logic is very sophisticated, and is willing to study systems with ‘truth-values’ in quite general commutative groups,... but to my knowledge, logic with values in non-commutative groups has been studied only in contexts motivated by higher mathematic. This may explain, to some extent, the difficulty people have had, in finding a mathematical interpretation of Levi-Strauss’s ideas; but it seems clear to me that such an interpretation does exist, and that as far as I can see, it fits integrally with Levi-Strauss’s earlier work on the subject.

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    $\begingroup$ Group representations are ubiquitous in physics. $\endgroup$
    – James S. Cook
    Commented Mar 8, 2013 at 2:54
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    $\begingroup$ @Cook: I know Lie group representations are but I was struggling for examples of finite group representations. $\endgroup$
    – Mozibur Ullah
    Commented Mar 8, 2013 at 3:07
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    $\begingroup$ There are plenty of applications of $C_n$ and $S_n$ to physics, but this is precisely what I mean by "applications of elementary group theory in which the groups happen to be finite." One becomes very familiar with these groups in first year group theory courses. From a finite group theory perspective they are not very interesting. (Cyclic groups are easily understood, and symmetric groups are hard, but really they are more combinatorics than finite groups.) $\endgroup$
    – Alexander Gruber ♦
    Commented Mar 8, 2013 at 3:27
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    $\begingroup$ @Alexander: symmetric groups are not very interesting?! $\endgroup$
    – Qiaochu Yuan
    Commented Mar 8, 2013 at 7:53
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    $\begingroup$ @QiaochuYuan Symmetric groups are super interesting, I just wouldn't say they are studied much in finite group theory. They're really more combinatorial objects. (I mean, yes, there's Cayley's theorem, but that is a technicality.) Finite group theory as a discipline is (in my view) more about structural properties of finite groups in general, rather than of specific finite groups, which is what I'd like to see applied. As I said, if that's too picky and there isn't anything, that's okay. $\endgroup$
    – Alexander Gruber ♦
    Commented Mar 8, 2013 at 8:22

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