Question

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.

Answer 1

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.

Answer 2

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.]

Answer 3

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.

Answer 4

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.

Answer 5

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."

Answer 6

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.

Answer 7

I must apologize because i do not really have the details at all, but i thought you might enjoy the application. At http://www.math.wayne.edu/~rrb/MTS/archive/0710.html Nathan Dunfield used the CFSG at some point to identify a specific group. I really can not recall why, and unfortunately there does not seem to be any paper listed on his CV around the time of the talk that seems like it could shed some light. This is not my area really, but i think it was something about having a group that acted in a particular way on a set, ie 2-transitively (something of that nature), and he used the CFSG to identify the group or rule out possible cases. I will try to see if i can find a copy of the notes so i can give more details. Or perhaps someone who is more familiar with his work will edit this post. I got the impression that using CFSG was a little extreme but that he did not have another way to do it... i think.

EDIT: In a little blurb on his main page http://www.math.uiuc.edu/~nmd/ he states that he uses this result but not in which projects, I will dig more tomorrow. or someone else can.