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This is an oft-quoted excerpt from John Thompson's article "Finite Non-Solvable Groups":

“... the classification of finite simple groups is an exercise in taxonomy. This is obvious to the expert and to the uninitiated alike. To be sure, the exercise is of colossal length, but length is a concomitant of taxonomy. Those of us who have been engaged in this work are the intellectual confreres of Linnaeus. Not surprisingly, I wonder if a future Darwin will conceptualize and unify our hard won theorems. The great sticking point, though there are several, concerns the sporadic groups. I find it aesthetically repugnant to accept that these groups are mere anomalies...Possibly... The Origin of Groups remains to be written, along lines foreign to those of Linnean outlook.”

I would very much like to know about ongoing efforts to find cleaner methods to classify the finite simple groups. Could we create a list of contemporary methods to study finite simple groups? They could be radically different from existing methods, or they could be a natural outgrowth of methods which have been successfully applied to the CFSG.

Some possibilities:

  1. I've heard about fusion systems, which generalize the properties of Sylow subgroups and are relevant to algebraic topology.

  2. I've heard (only in passing) that some people have tried to generalize Tits' geometric methods (buildings, chambers, etc.), and that these geometric methods are somehow analogous to local analysis (the study of normalizers of subgroups of prime-power order). I read in Alperin's article "Finite groups viewed locally" that applying local analysis is "an art and not a science" and that it requires a "lot of patience and ingenuity". Are there any recent developments in local analysis that make it easier to apply?

  3. I've heard that the alternating groups can be meaningfully considered as "groups of Lie type over the field with one element".

  4. I know that in many situations, questions about finite groups can be "reduced to" problems in linear algebra (I have in mind the seminal Hall-Higman paper, which contained tricks that were later used by Thompson to prove his normal $p$-complement theorem and also in the Feit-Thompson proof). Is there an enlightening general theory that allows us to reduce group-theoretic problems to linear-algebraic problems?

  5. I've heard that the Burnside ring of a finite group $G$ has nontrivial idempotents if and only if $G$ is solvable, and that people used to consider this a promising approach to proving Burnside's odd-order conjecture back when it was still a conjecture. Is this approach still promising?

  6. I know that group cohomology has been applied to obtain results about finite groups which are of a "classical" flavour (i.e. theorems that Burnside, Frobenius, etc. would have found interesting), like Tate's transfer theorem. Has anyone tried to apply homological methods to simplify the CFSG? I've also heard that homotopical algebra can be considered a nonabelian generalization of homological algebra, so perhaps it is of greater relevance to the study of nonabelian finite groups?

  7. I've heard that modular representation theory à la Brauer was useful, but not as much as Brauer had hoped it would be. I've also heard that most of the ordinary character theory in the proof of CFSG is contained in the proof of the Feit-Thompson theorem, though I suspect that's probably an exaggeration. I read in Alperin's article "Finite groups viewed locally" that people suspect that Brauer's block theory can be applied to prove the so-called B-conjecture. Has anyone tried to use modular/ordinary character theory to simplify the CFSG?

Edit (24/7):

  1. I've heard that the theory of finite solvable groups is very mature, and that some attempts have been made to generalise some of the features of this theory (see Classes of Finite Groups)

Edit (25/7)

  1. The amalgam method (see comments by Timothy Chow and j.p. below)
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    $\begingroup$ Well, the second-generation "GLS" proof is still in the process of being written up. The first volume in the GLS series still gives a largely accurate summary of the whole proof, and you can find answers to some of your questions by reading the introductions to the individual volumes. I don't think we can reasonably expect a radically different strategy to be undertaken until the second-generation proof is complete. $\endgroup$
    – Timothy Chow
    Commented Jul 23, 2023 at 9:19
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    $\begingroup$ Regarding 7, several important proofs during the classification which used Brauer's block theory were concerned with "low-rank" situations, where the Sylow $2$-subgroup of a supposed simple group $G$ did not have elementary Abelian subgroups of rank $3$ or more. Once these were disposed of, group-theoretic methods, such as signalizers, became prevalent. An exception is Glauberman's Z*-theorem, which used block theory, and still admits no proof without it. This theorem was key to CFSG, and was one of the most cited papers in algebra in the 70s for that reason. $\endgroup$
    – Geoff Robinson
    Commented Jul 23, 2023 at 10:04
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    $\begingroup$ D. Goldchmidt proved his strongly closed Abelian (2)-subgroup theorem taking the Z*-theorem as a staring point, but he did not use much more block theory ( I think he did use some more character-theory). Such a theorem for odd primes would be useful to simplify some revisionist proofs, if it were available. Also, some contemporary "local-global" conjectures in block theory might at least offer a path to further applications of block theory to CFSG ( if they themselves could be proved without CFSG). $\endgroup$
    – Geoff Robinson
    Commented Jul 23, 2023 at 10:48
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    $\begingroup$ I recall reading somewhere that one reason Mason lost motivation to keep working on the quasithin case was that he believed his results would soon be superseded by work that others were doing. Of course that didn't happen, but I'm wondering what that other line of research was. Unfortunately, I can't remember where I read this. The closest thing I've found is Aschbacher's 2004 Notices article; at the end, he mentions "new methods" including the "amalgam method" which seemed promising at one point in time. I wonder to what extent that promise has been fulfilled in the last 20 years. $\endgroup$
    – Timothy Chow
    Commented Jul 23, 2023 at 17:49
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    $\begingroup$ Related: mathoverflow.net/questions/38161/… $\endgroup$
    – Terry Tao
    Commented Jul 23, 2023 at 18:08

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