Feit-Thompson theorem: the Odd order paper

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.

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.

The Wikipedia article Odd order theorem is worth reading.