TITLE: What are some interesting corollaries of the classification of finite simple groups?
QUESTION [39 upvotes]: 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 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.

REPLY [5 votes]: 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."