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?
Best Answer
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:
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?"