What are some interesting corollaries of the classification of finite simple groups?

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?

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NB:

• this question was significantly edited by other users soon after being posted (Aug 2 '10). The few critical comments below were actually addressed before the question was edited and improved.

• To anyone who has doubts about the proof, please see Aschbacher's Notices paper "The status of the classification of the finite simple groups" and please rather don't debate that particular matter here.

Answer 1

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 ["Exposition on an arithmetic-group theoretic connection via Riemann's existence theorem", 1980, in the proceedings of the 1979 Santa Cruz conference on finite groups], following 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 2

Nikolov and Segal proved in On finitely generated profinite groups II, products in quasisimple groups 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 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 $\operatorname{GL}_n(\mathbb{C})$ 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 B. Weisfeiler’s paper 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 Lubotsky and Segal - Subgroup growth 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 (Finite subgroups of algebraic groups, 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 (Finite covers of random 3-manifolds, page 45), they note that for the orbit in question in their application, a result of Gilman (Finite quotients of the automorphism group of a free group) suffices (they use CFSG to assert that a finite 6-transitive group action contains $A_n$). For some of 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 and Codenotti paper Isomorphism of hypergraphs of low rank in moderately exponential time for graph isomorphism, they even note (Theorem 3.1) that a easier weaker result suffices.

• The work of Aschbacher, followed by the book of Kleidman and Liebeck (The subgroup structure of the finite classical groups), on maximal subgroups of finite classical groups is another source. Here $\operatorname{SL}$, $\operatorname{SO}$, $\operatorname{Sp}$ and $\operatorname{SU}$ are all involved. Aschbacher's theorem (King has a survey The subgroup structure of finite classical groups in terms of geometric configurations) 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 (A survey of the maximal subgroups of the finite simple groups. Geometries and groups). 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 (Extremal lattices of minimum $8$ related to the Mathieu group $M_{22}$) 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 $\operatorname{GL}_{80}(Z)$. 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 by Jim Humphries 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 in Schur covers and Carlitz's conjecture of a 1966 conjecture of Carlitz. Let $f(x)$ be a polynomial with coefficients in the finite field $\mathbb F_q$. Then $f$ is called "exceptional" if, for infinitely many finite extensions $K$ of $\mathbb F_q$, the induced function $f:K\to K$ 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, in Composition factors and monodromy groups) 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.