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For reference, the Feit-Thompson Theorem states that every finite group of odd order is necessarily solvable. Equivalently, the theorem states that there exist no non-abelian finite simple groups of odd order.

I am well aware of the complexity and length of the proof. However, would it be possible to provide a rough outline of the ideas and techniques in the proof? More specifically, the sub-questions of this question are:

  • Are the techniques in this proof purely group-theoretic or are techniques from other areas of mathematics borrowed? (Such as, for example, other branches of algebra.) In the same vein, how great an influence do the techniques (if any) from number theory and combinatorics have on the proof? (Here "combinatorics" is of course not very specific. I should emphasize that I mean "tools from combinatorics that are pure and solely derived from techniques within the area of combinatorics and that do not require "deep" group theory to derive". Similarly for "number theory".)

  • What sorts of "character-free" techniques and ideas exist in the proof? Does a character-free proof of this result exist? (Since I suspect the answer to the latter is in the negative, I am primarily interested in an answer to the former.)

  • What are the underlying "intuitions" behind the proof? That is, how does one come up with such a proof, or at least, certain parts of it? This is a rough question of course; "coming up" with things in mathematics is very difficult to describe. However, since the argument is so long, I suspect some sort of inspiration must have driven the proof.

  • I have observed in group theory that many arguments naturally divide into "cases" and often the individual cases are easy to tackle and the arguments naturally "flow". Of course, here I speak of arguments whose lengths are no more than a few pages. Does the proof of the Feit-Thompson theorem share the same "structure" as smaller proofs, or is the proof structurally unique?

  • How often do explicit "elementwise computations" arise in the proof?

  • Is there any hope that one day someone might discover a considerably shorter proof of the Feit-Thompson Theorem? For example, would the existence of a proof of this theorem less than 50 or so pages be likely? (A proof making strong use of the classification of finite simple groups, or any other non-trivial consequence of the Feit-Thompson Theorem, does not count.) If not, why is it so difficult in group theory to provide more concise arguments?

While I have Gorenstein's excellent book entitled Finite Groups at hand, I did not go far enough (when I was reading it) to actually get into the "real meat" of the discussion of the Feit-Thompson theorem; that is, to actually get a sense of the mathematics used to prove the theorem. Nor do I intend to do so in the near future. (Don't get me wrong, I would be really interested to see this proof, but it seems too much unless you intend to research finite group theory or a related area.)

Thank you very much for any answers. I am aware that some aspects of this question are imprecise; I have tried my best to be as clear as possible in some cases, but there might still be possible sources of ambiguity and I apologize if they are. (If there are, I would appreciate it if you could try to look for the "obvious interpretation".) Also, I have a relatively strong background in finite group theory (but not a "research-level" background in the area) so feel free to use more complex group-theoretic terminology and ideas if necessary, but if possible, try to give an exposition of the proof that is as elementary as possible. Thanks again!

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The original proof contained lots of character theory and lots of case-by-case analysis. With the general material in Gorenstein presupposed, a shorter and more relaxed proof can be found in the two books "Local Analysis for the Odd Order Theorem" and "Character Theory for the Odd Order Theorem". –  Steve D Jul 14 '10 at 9:05
    
The algebra textbook by Dummit and Foote has a series of exercises somewhere in it that are supposed to be designed to give one a feeling for the proof. However, I don't think there's any character theory in these exercises, and I can't say how much they really convey about the proof. Still, you might try to find them... –  Dan Ramras Jul 14 '10 at 16:27
    
The two books cited by Steve D are of course still technical, well beyond Glauberman's short survey: MR1311244 (96h:20036), Bender, Helmut (D-KIEL); Glauberman, George (1-CHI), Local analysis for the odd order theorem. London Mathematical Society Lecture Note Series, 188. Cambridge University Press, Cambridge, 1994. MR1747393 Peterfalvi, Thomas (F-PARIS7), Character theory for the odd order theorem. Translated from the 1986 French original by Robert Sandling and revised by the author. London Mathematical Society Lecture Note Series, 272. Cambridge University Press, Cambridge, 2000. –  Jim Humphreys Jul 15 '10 at 13:03
    
During my stay at the mathematics department at Kiel some professor claimed that H. Bender considered quite a lot of the theory in the book he coauthored with G. Glauberman unnecessary for the proof of the odd order theorem. But I didn't ask H. Bender directly to confirm this statement and don't know any details. –  Someone Jul 23 '10 at 14:05
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The proof could be simplified if number theorist were able to proof the Feit-Thompson conjecture (en.wikipedia.org/wiki/Feit%E2%80%93Thompson_conjecture). –  Someone Jul 23 '10 at 14:06

3 Answers 3

up vote 18 down vote accepted

During a discussion at the n-category theory cafe Stephen Harris sent me this excellent expository article by Glauberman which goes into a bit more depth than wikipedia.

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It's good to have this online, since the publication occurred in an out-of-the-way conference volume: MR1756828 (2001b:20027) 20D10, Glauberman, George (1-CHI), Anew look at the Feit-Thompson odd order theorem. 15th School of Algebra (Portuguese) (Canela, 1998). Mat. Contemp. 16 (1999), 73–92. Glauberman is a reliable source for this area of finite group theory. Another of his distinctions is having as his first Ph.D. student a prominent player in recent Iraqi politics, Ahmad Chalabi (less prominent in mathematics). –  Jim Humphreys Jul 14 '10 at 18:55
    
This is an extremely helpful comment. Thank you very much! I have printed out the expository article by Glauberman and really like the way it is written based on my reading of the first few pages. I look forward to reading it in its entirety in the near future! –  Amitesh Datta Jul 15 '10 at 14:53

I won't presume to attempt a precis of the Feit-Thompson proof. But I would suggest that your question about the hope of finding a much shorter proof is impossible to answer meaningfully. The current answer, backed up by almost 50 years of recent history, is probably ``with currently available techniques, there appears to be little prospect of any dramatic shortening of the length of the proof of the odd order theorem." It should also be remembered that many of the currently available accepted techniques of finite group theory were developed to attack this problem, and proved later to be very powerful in a wider context. Many of the techniques are such an integral part of the weaponry of many modern group theorists that they implicitly impose an inevitability and naturality to the structure of the proof of the odd order theorem, complex and forbidding though the details are. But had the question been asked, say in 1955, "Is there any prospect of proving the solvability of finite groups of odd order in the near future?", the answer likely to be given at the time can only be a matter of speculation (for most of us at any rate), but with the benefit of hindsight we can see at present that to make the prospect of such a proof a reality, many new and innovative techniques had to be developed, and profound new insights brought to bear.

However, it would be a rash mathematician (and one who took little account of the history of the subject) who would pronounce it impossible to find a significantly shorter proof at some point in the future. It might be a safer bet to suggest that a significantly shorter proof would require some genuinely new insights and ideas, but even a statement such as that might eventually be proved to be presumptuous.

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http://en.wikipedia.org/wiki/Odd_order_theorem is worth reading.

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Thanks for your answer! I did scan through that Wikipedia article prior to asking this question and should probably read it more thoroughly now. It is indeed a very detailed article and does address some aspects of my question. However, there are certain aspects of my question that the article does not address, and certain aspects that the article does not discuss in complete detail. But thanks anyway! –  Amitesh Datta Jul 14 '10 at 12:13
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Its validity aside, I pity the student of math sociology whose views would be inordinately coloured by "Perhaps the most revolutionary aspect of the proof was its length: before the Feit-Thompson paper, few arguments in group theory were more than a few pages long and most could be read in a day. Once group theorists realized that such long arguments could work, a series of papers that were several hundred pages long started to appear." –  Junkie May 3 '11 at 16:38

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