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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?
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    $\begingroup$ What do you call a "concrete" application? Would you consider an application to another mathematical area (or even another part of group theory) "concrete"?. $\endgroup$ Jul 26, 2021 at 18:54
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    $\begingroup$ “I don’t know a single application of CFSG”. Then I don’t think you’ve googled particularly well! $\endgroup$ Jul 26, 2021 at 18:57
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    $\begingroup$ @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? $\endgroup$
    – Gabriel
    Jul 26, 2021 at 19:02
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    $\begingroup$ 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 math.leidenuniv.nl/~hwl/PUBLICATIONS/1991d/art.pdf. $\endgroup$
    – KConrad
    Jul 26, 2021 at 19:05
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    $\begingroup$ A similar question is mathoverflow.net/questions/34290. $\endgroup$ Jul 26, 2021 at 19:07

2 Answers 2

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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.

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    $\begingroup$ 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. $\endgroup$ Jul 26, 2021 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.

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  • $\begingroup$ The sometimes-called "Jordan-Schur theorem" is entirely due to Jordan. $\endgroup$
    – YCor
    Jul 26, 2021 at 19:25
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    $\begingroup$ @YCor I don't think that's accurate. Jordan proved it for finite groups. Schur generalized it to periodic groups. $\endgroup$
    – JoshuaZ
    Jul 26, 2021 at 19:28
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    $\begingroup$ 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). $\endgroup$ Jul 26, 2021 at 20:25
  • $\begingroup$ @JoshuaZ indeed, but here (and in most places) it's quoted/used for finite groups. $\endgroup$
    – YCor
    Jul 26, 2021 at 21:09

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