# Are there real world applications of finite group theory?

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?

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?”