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The classification of finite simple groups has been called one of the great intellectual achievements of humanity, but I don't even know one single application of it. Even worse, I know a lot of applications of simple modules over some ring/algebra $A$, but I can barely know an application of them for finite simple groups. When studying modules, one has, for example,

  1. If $S$ and $T$ are distinct simple modules, then $\operatorname{Hom}(S,T) = 0$, and one can enhance this using Jordan-Holder to prove that, if $M$ and $N$ are modules whose Jordan-Holder decomposition don't have common factors, then $\operatorname{Hom}(M,N)=0$. We may use this, for example, to try to compute some cohomology, also;
  2. The simple modules form a basis of the $K_0$ group, and therefore if we're interested in, for example, the multiplicative structure of $K_0$ it's enough to compute the (tensor) product of simple modules;
  3. If the algebra $A$ is basic (i.e. every simple representation is $1$-dimensional), which happens for path algebras, then simple modules have a group structure with respect to the tensor product (so they are an analogue for the Picard group).

For finite simple groups, the only application I know is for the (non)-solubility of polynomials, and it's a quite particular example which uses only $S_n$ and $A_n$. So I have two questions:

  1. What are some (concrete) applications of (finite simple groups + Jordan-Holder) for general finite groups?
  2. What are some (concrete) applications of the classifications of finite simple groups?
Thiago Landim
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    What do you call a "concrete" application? Would you consider an application to another mathematical area (or even another part of group theory) "concrete"?. – Geoff Robinson Jul 26 '21 at 18:54
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    “I don’t know a single application of CFSG”. Then I don’t think you’ve googled particularly well! – Carl-Fredrik Nyberg Brodda Jul 26 '21 at 18:57
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    @Carl-FredrikNybergBrodda or perhaps his own research is far from finite groups, and he just wants a global point of view from someone who understands this better? – Gabriel Jul 26 '21 at 19:02
  • Dear @GeoffRobinson, the "concrete" should mean preferably in another mathematical area or some computation with an example, but I would also be glad with applications inside group theory. – Thiago Landim Jul 26 '21 at 19:02
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    Search MathSciNet with "classification of finite simple groups" in the Review Text box (include the quotiation marks) and put a subject number in the MSC Primary Box to avoid papers on group theory (since you are more interested in applications outside of group theory), e.g., use "11" for papers in number theory. This won't give you all possible results, since not all papers using CFSG have this fact mentioned in the review. An example of that is on page 29 of the paper https://www.math.leidenuniv.nl/~hwl/PUBLICATIONS/1991d/art.pdf. – KConrad Jul 26 '21 at 19:05
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    A similar question is https://mathoverflow.net/questions/34290. – Richard Stanley Jul 26 '21 at 19:07
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    @Gabriel It’s expected that people asking here do their homework first. Someone saying they don’t know a single application of CFSG is clear evidence that they didn’t do their homework (including simply looking at MO for similar questions, as in Richard Stanley’s comment!). – Carl-Fredrik Nyberg Brodda Jul 26 '21 at 19:14
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    Babai's original proof of a quasi-polynomial time algorithm for graph isomorphism used the classification although maybe there is an alternative proof that avoids it by now. – Benjamin Steinberg Jul 26 '21 at 19:14
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    Of the "useful properties" you list for simple modules, (1) has a direct analogue for simple groups, and (2) and (3) are false. – Jeremy Rickard Jul 26 '21 at 19:16
  • A few applications are mentioned at the wikipedia page. – Noah Schweber Jul 26 '21 at 19:20
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    @Carl-FredrikNybergBrodda the comment bar says "Thiago Landim is a new contributor. Be nice [...]" and I don't think we're being very nice to someone who perhaps doesn't yet knows all the rules. – Gabriel Jul 26 '21 at 19:34
  • Thanks @KConrad – Thiago Landim Jul 26 '21 at 19:38
  • Thanks @RichardStanley, I tried to look for similar question, but wasn't able to find it – Thiago Landim Jul 26 '21 at 19:39
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    I think this is for all intents and purposes a duplicate of the question Richard Stanley linked and has already one duplicate answer so I am voting to close as a duplicate – Benjamin Steinberg Jul 26 '21 at 19:40
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    @JeremyRickard, what is the direct analogue of (1) for simple groups? C_p and A_p are simple groups, but there are a lot of morphisms from C_p to A_p – Thiago Landim Jul 26 '21 at 19:40
  • Dear @BenjaminSteinberg, yes, certainly! – Thiago Landim Jul 26 '21 at 19:42
  • @ThiagoLandim Yes, OK, fair enough. – Jeremy Rickard Jul 26 '21 at 20:29
  • Would the discovery of "monstrous moonshine" count as an application? https://en.wikipedia.org/wiki/Monstrous_moonshine The monster sporadic group was discovered in the process of proving the classification and the connection to modular functions was discovered after that... – Nick Gill Jul 27 '21 at 08:46
  • Nikolov and Segal's work on profinite groups, generalizing work of J.-P. Serre, depends on CFSG. See, for instance, this: https://annals.math.princeton.edu/wp-content/uploads/annals-v165-n1-p05.pdf Specifically they show that "every finitely generated profinite group is strongly complete". – Nick Gill Jul 27 '21 at 08:50
  • Finally, there are lots of statements about simple groups themselves which are only known to be true thanks to CFSG. The Schreier conjecture, mentioned in the answer of @arsmath, is one such. Another would be "all finite simple groups are 2-generated" (there are lots of variants of this). Another would be the "family of all non-abelian finite simple groups can be made into expanders in a uniform fashion" -- https://arxiv.org/abs/1005.0782 And so on and so on. – Nick Gill Jul 27 '21 at 08:53
  • Finally FINALLY, it is also worth mentioning the Aschbacher-O'Nan-Scott theorem describing finite primitive groups. This question -- https://mathoverflow.net/questions/155476/does-onan-scott-depend-on-cfsg -- details how CFSG can be used to strengthen this theorem. – Nick Gill Jul 27 '21 at 09:09

2 Answers2

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There's an entire book on this subject, "Applying the Classification of Finite Simple Groups: A User’s Guide" by Stephen D. Smith, published through the AMS, though you can find a draft version here.

The applications are not as simple as they are for modules, but many questions can be settled by invoking the classification of finite simple groups. For example, you can invoke the classification to in turn classifying 2-transitive groups. The only known proof of the Schreier conjecture relies on the classification. The book has many more applications.

arsmath
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    There are also many examples outside group theory due to R. Guralnick, P.H. Tiep and various co-authors to problems in algebraic geometry and number theory. Some, but not all of these, are covered in Smith's book. – Geoff Robinson Jul 26 '21 at 20:07
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The best case bound for the Jordan-Schur theorem uses heavily the classification, and that theorem shows up in a lot of different contexts.

JoshuaZ
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  • The sometimes-called "Jordan-Schur theorem" is entirely due to Jordan. – YCor Jul 26 '21 at 19:25
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    @YCor I don't think that's accurate. Jordan proved it for finite groups. Schur generalized it to periodic groups. – JoshuaZ Jul 26 '21 at 19:28
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    The idea of using CFSG to obtain better bounds for this theorem originates with B. Weisfeiler, who unfortunately disappeared in Chile before his work was published ( though the manuscripts he left were not complete). – Geoff Robinson Jul 26 '21 at 20:25
  • @JoshuaZ indeed, but here (and in most places) it's quoted/used for finite groups. – YCor Jul 26 '21 at 21:09