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The classification of finite simple groups -- whether it be viewed as finished, or as a work in progress -- is (or will be) without doubt an enormous achievement. It clearly sheds a great deal of light on the structure of finite groups. However, as with the classification of simple Lie algebras, one might expect this to have a significant impact outside of the immediate subject. So what are some of the known, or expected, applications to the classification outside of finite group theory?

NB: This major edit is an attempt to extract what I'm guessing is the essence of the question. -D.A.

Extra edit by Will Jagy. First, quoting a comment below: $$ $$ To anybody who is considering casting the 5th and final vote to close: Please wait at least a day or so before doing so! Whether or not this question should be closed, there's not a great reason to get too hasty here. Kevin Lin.

Next, some rather negative comments you might be able to view below concern the initial version of the question, which one might be able to see by viewing the edit history.

To anyone who has doubts about the proof, please see http://ams.org/notices/200407/fea-aschbacher.pdf and do not debate that particular matter here.

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There are already many theorems whose only known proofs use the classification in an essential way. –  S. Carnahan Aug 2 '10 at 18:14
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Anweshi -- what is subjective and argumentative about asking why something is useful? Scott -- that's exactly the point: the proof of classification theorem is notoriously long and technical, and there is some controversy as to whether or not it is complete at all, so it would be nice to know which theorems depend on it. –  algori Aug 2 '10 at 18:20
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Perhaps someone might want to edit the question so that it asks the questions you guys are mentioning without the need of "personal interpretation"? :) –  Gjergji Zaimi Aug 2 '10 at 18:29
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To anybody who is considering casting the 5th and final vote to close: Please wait at least a day or so before doing so! Whether or not this question should be closed, there's not a great reason to get too hasty here. –  Kevin H. Lin Aug 2 '10 at 20:26
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@Joseph: It is better now. Your questions generally suffer from a lack of detail. You were repeatedly posting one-line questions and it was not good etiquette. The people who write answers would probably have to write at least one paragraph. It is reasonable to expect you to provide more detail to your question. –  Anweshi Aug 2 '10 at 20:38
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6 Answers 6

Nikolov and Segal proved that finite-index subgroups of finitely generated profinite groups are open. This implies that the topology in such a group is uniquely determined by the group structure. They use the classification in a crucial way.

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My look of Nikolov/Segal tells they use CFSG in various places. The 1st is various bounds (Section 5), from Saxl and Wilson. There is a long history of various results on the "class 9" of Aschbacher, depending on the characteristics involved, and I think (not sure) this is the same loop of ideas, for instance mathematik.uni-bielefeld.de/sfb343/preprints/pr98118.ps.gz $$ $$ The 2nd is their: "The theorem follows, since according to the Classification all but finitely many of the finite simple groups are either alternating or of Lie type." Only need "finitely many sporadics" here? –  Junkie Aug 4 '10 at 0:57
    
Another way of saying this is, for their application N/S need some sort of uniformity. For sporadic groups, it is trivial as (by CFSG) they are finite in number, and the statement is asymptotic. For Lie groups they divide into small/large fields, and use various results that might invoke the classification. However, many of these various results similarly use the CFSG to get a uniform statement, with the bulk working as for Lie groups. For instance, I don't think (Thm 6.1) using Liebeck/Pyber relies on CFSG, as the Theorem D is totally Lie theory, and thus (to me) seems independent of CFSG. –  Junkie Aug 4 '10 at 1:05
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Citing Graham's answers 1 and 2 to two other questions:

Definition: A polynomial $f(x)\in \mathbb C[x]$ is indecomposable if whenever $f(x)=g(h(x))$ for polynomials $g$, $h$, one of $g$ or $h$ is linear.

Theorem. Let $f, g$, be nonconstant indecomposable polynomials over $\mathbb C$. Suppose that $f(x)−g(y)$ factors in $\mathbb C[x,y]$. Then either $g(x)=f(ax+b)$ for some $a, b \in \mathbb C$, or

$$\deg f=\deg g=7,11,13,15,21, \mbox{or } 31,$$

and each of these possibilities does occur.

The proof uses the classification of the finite simple groups and is due to Fried [1980, in the proceedings of the 1979 Santa Cruz conference on finite groups], following a the reduction of the problem to a group/Galois-theoretic statement by Cassels [1970]. [W. Feit, "Some consequences of the classification of finite simple groups," 1980.]

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"The proof uses the classification of the finite simple groups and is due to Fried [1980]" $$ $$ So he proved this when CFSG was still in essence a conjecture. A better question is what part of the Classification he actually needs. Section 4 of Feit (and is careful to say the CFSG is not yet proven) states CFSG plus work of Curtis/Kantor/Seitz yields a list of all faithful doubly transitive permutation reps (Thm 4.1), and this gives all nonsolvable doubly transitives on $p$ symbols, and Fried follows as you need two inequivalent doubly transitive actions with the same character. –  Junkie Aug 5 '10 at 3:07
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I figured this deserves at least one answer before getting shut down. As many of you know, Jordan proved that a finite subgroup of $GL_n(\mathbb{C})$ contains an abelian normal subgroup of index, say $C(n)$ depending only on $n$. One can find a proof in Curtis and Reiner for instance. In the paper by B. Weisfeiler called "Post-classification version of Jordan’s theorem on finite linear groups" he uses classification to sharpen the existing bound on $C(n)$. There are some extensions to positive characteristic fields as well.

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The current version of the question asks for applications outside finite group theory, so I guess this qualifies. But probably the more interesting question is where the classification has impact on mathematics outside group theory. This is a tricky question, since some of the sporadic groups (notably the Monster) seem to have life beyond conventional group theory as do some of the groups of Lie type. But the great achievement of the classification beyond extending the list of known groups is (or will be) the proof that no others can exist. Does that have wider application? –  Jim Humphreys Aug 2 '10 at 22:21
    
Donu, I recommend putting (in the question ) the quote from Kevin Lin about not hastily closing, plus the explicit link to the Aschbacher pdf, I can do that if you are not sure how. –  Will Jagy Aug 2 '10 at 22:31
    
Jim: Of course, you are asking a better version of the question which I hope someone (who knows more than me) can answer. Will: Feel free to add/modify whatever you see fit. –  Donu Arapura Aug 2 '10 at 22:48
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Thanks, Donu. I fiddled with it. As to Jim's question, it would appear Scott C knows something. As Anweshi mentioned Moonshine, perhaps Richard Borcherds can think of something fitting to say. Mostly, I would be disappointed that we put in this much effort to rehabilitate the question if we then got few substantial responses. –  Will Jagy Aug 2 '10 at 23:13
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There are all kinds of applications of CFSG within the rest of algebra. I am afraid they are too numerous to list here. Let me mention this book which I personally find very interesting, and which has a number of such results. On the other hand, if you read it carefully, you will see that many applications of CFSG also follow from a (weaker but readable) result by Larsen & Pink (1998).

Let me turn and give another answer. If you study currently best bound due to Babai & Luks, on the complexity of graph isomorphism, you will also see that it is based on CFSG, although again on a relatively easy looking consequence of it. Although most experts would say that this problem is strongly connected to group theory, as stated, it lies outside of algebra. Hope you find this example convincing enough.

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I have a few comments, but will make this an answer due to length, and I just got a brainstorm of a real answer anyhow. I've now rewritten this to more answer the question: how does the Classification attach itself to other mathematical areas.

  • Group actions are quite common in mathematics. Showing that only finitely many types of group actions occur in a problem is a typical idea. The theorem of Fried involving indecomposable polynomials fits into this, as there is an action on branched covers.

  • There are a number of corollaries of CFSG, which are essentially classifications in their own right. The above Fried result depends on a type of doubly transitive action being classified. Another example, rooted in Dunfield/Thurston (page 45), they note that for the orbit in question in their application, a result of Gilman suffices (they use CFSG to assert that a finite 6-transitive group action contains $A_n$). For some these, asking whether all of CFSG is needed could be apropos.

  • Another type of corollary is that some bounds are lowered, due to the fact that we now know (for instance) that all groups have (say) a representation satisfying a certain bound. The existence of a qualitatively different bound under CFSG (say polynomial opposed to exponential) has more interest than just making numbers smaller. Looking at the Babai paper for graph isomorphism, they even note (Theorem 3.1) that a easier weaker result suffices. http://people.cs.uchicago.edu/~laci/papers/hypergraphiso.pdf

  • The work of Aschbacher, followed by the book of Kleidman and Liebeck, on maximal subgroups of finite classical groups is another source. Here $SL, SO, Sp$ and $SU$ are all involved. Aschbacher's theorem (here is a survey by King) says that there are 8 types of subgroups (stablizers of: subspaces, direct sums, spreads, forms, extension fields, tensor products, subfields; extraspecial normalizers, plus the exotic ninth class). Another survey (precursor to their book) is by Kleidman and Liebeck. Once the degree is above 14, I think, the 8 classes become uniform in description (though the exotics persist). This relates directly to group theory of course, but many math branches use these constructs.

My specific example was a paper of Bachoc and Nebe that showed that an 80-dimensional lattice with a large minimal norm (of 8) had a known automorphism group (related to $M_{22}$) that was maximal finite in $GL_{80}(Z)$. They then used to show that their lattice was not isometric to a different one they constructed. More generally, if there is no possible common finite supergroup of the known automorphisms of two lattices, they are not isometric. To prove this can require CFSG in one form or another.

I agree with what was stated in a comment above: "But probably the more interesting question is where the classification has impact on mathematics outside group theory."

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I guess that's another question that could be asked: how is the Classification typically used? –  Junkie Aug 4 '10 at 0:37
    
Monster group is certainly a byproduct of classification and the whole Monstrous Moonshine transcends group theory. This "one-time" impact opened a fertile area of research (generalized Kac-Moody Lie algebras, vertex algebras, meromorphic modular forms, etc). –  Victor Protsak Aug 4 '10 at 1:11
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I cannot agree. Applications of the Monster often are almost totally independent of the classification and solely rely on its construction which is logically independent, though historically intermixed. Constructing the Monster is nontrivial but is much less than the 10K pages of the CFSG proof. –  Junkie Aug 4 '10 at 1:50
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Maybe it is difficult to say what is strictly "outside" of finite group theory. However, to add to Igor's answers, I can suggest that you look at the work of Bob Guralnick and various collaborators. Among the interesting results which use the classification is the proof by Fried, Guralnick and Saxl of a 1966 conjecture of Carlitz. Let f(x) be a polynomial with coefficients in the finite field F_q. Then f is called "exceptional" if, for infinitely many finite extensions K of F_q, the induced function f:K-->K is a permutation. The conjecture of Carlitz was that if q is odd and f is exceptional then f has odd degree.

There is also a large body of work saying that the structure, both as an abstract group and as a permutation group, of the monodromy group of a finite branched covering of connected Riemann surfaces is controlled by the genus of the covering surface. For example, the Guralnick-Thompson Conjecture (now a theorem, with the final piece of the proof due to Frohardt and Magaard) says that if we bound the genus of the covering surface, we can obtain only finitely many nonalternating, noncyclic groups as composition factors of the monodromy group. I don't think anyone knows how to prove results of this nature without the classification.

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As far as I know, and MR supports this, ams.org/mathscinet-getitem?mr=1341953, Carlitz-Wan conjecture now admits an independent and quite elementary proof. So the classification was the big hammer which first cracked open a nut, but later ways to gently open an even tougher one were found. –  Victor Protsak Aug 3 '10 at 8:12
    
Thanks for pointing this out. –  John Shareshian Aug 3 '10 at 17:54
    
Just a clarification: while there is an elementary proof of the Carlitz-Wan conjecture -- which constrains the degrees of exceptional polynomials -- the Fried-Guralnick-Saxl paper achieves vastly more, making significant progress towards a classification of all exceptional polynomials. For this they crucially used the classification of finite simple groups, and no elementary proof of their result is known. –  Michael Zieve Sep 28 '13 at 1:58
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