Feit–Thompson theorem

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In mathematics, the Feit–Thompson theorem, or odd order theorem, states that every finite group of odd order is solvable. It was proved by Walter Feit and John Griggs Thompson (1962, 1963)

1 History
2 Significance of the proof
3 Revision of the proof
4 An outline of the proof
5 See also
6 References

[edit] History

William Burnside (1911) conjectured that every nonabelian finite simple group has even order. Richard Brauer suggested using this as the basis for the classification of finite simple groups, by showing that a finite simple group can usually be determined if the centralizer of an involution is known. A group of odd order has no involutions, so to carry out Brauer's program it is first necessary to show that non-cyclic finite simple groups never have odd order. This is equivalent to showing that odd order groups are solvable, which is what Feit and Thompson proved.

The attack on Burnside's conjecture was started by Michio Suzuki (1957), who studied CA groups; these are groups such that the Centralizer of every non-trivial element is Abelian. In a pioneering paper he showed that all CA groups of odd order are solvable. (He later classified all the simple CA groups, and more generally all simple groups such that the centralizer of any involution has a normal 2-Sylow subgroup, finding an overlooked family of simple groups of Lie type in the process, that are now called Suzuki groups.)

Feit, Hall, and Thompson (1960) extended Suzuki's work to the family of CN groups; these are groups such that the Centralizer of every non-trivial element is Nilpotent. They showed that every CN group of odd order is solvable. Their proof is similar to Suzuki's proof. It was about 17 pages long, which at the time was thought to be very long for a proof in group theory.

The Feit–Thompson theorem can be thought of as the next step in this process: they show that there is no non-cyclic simple group of odd order such that every proper subgroup is solvable. This proves that every finite group of odd order is solvable, as a minimal counterexample must be a simple group such that every proper subgroup is solvable. Although the proof follows the same general outline as the CA theorem and the CN theorem, the details are vastly more complicated. The final paper is 255 pages long.

[edit] Significance of the proof

The Feit–Thompson theorem is often considered to be the start of the classification of finite simple groups. Many of the techniques they introduced in their proof, especially the idea of local analysis, were developed further into tools used in the classification. Perhaps the most revolutionary and important new idea was that of the very long paper: before their 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 started to appear. Some of these dwarfed even the Feit–Thompson paper; Aschbacher and Smith's paper on quasithin groups was well over 1000 pages long.

[edit] Revision of the proof

Many mathematicians have simplified parts of the original Feit–Thompson proof. However all of these improvements are in some sense local; the global structure of the argument is still the same, but some of the details of the arguments have been simplified.

The simplified proof has been published in two books: (Bender & Glauberman 1995) and (Peterfalvi 2000). This simplified proof is still very hard, and is about the same length as the original proof (but is written in a more leisurely style).

(Gonthier et al. 2006) have begun a long-term project to produce a computer verified formal proof of the theorem.

[edit] An outline of the proof

It takes a professional group theorist about a year of hard work to understand the proof completely, so the following summary should not be taken too seriously. Instead of describing the Feit–Thompson theorem directly, it is easier to describe Suzuki's CA theorem and then comment on some of the extensions needed for the CN-theorem and the odd order theorem. The proof can be broken up into three steps. We let G be a non-abelian (minimal) simple group of odd order satisfying the CA condition. For a more detailed description of the odd order paper see the book (Gorenstein 1980).

Step 1. Local analysis of the structure of the group G. This is easy in the CA case because the relation "a commutes with b" is an equivalence relation on the non-identity elements. So the elements break up into equivalence classes, such that each equivalence class is the set of non-identity elements of a maximal abelian subgroup. The normalizers of these maximal abelian subgroups turn out to be exactly the maximal proper subgroups of G. These normalizers are Frobenius groups whose character theory is reasonably transparent, and well-suited to manipulations involving character induction. Also, the set of prime divisors of |G| is partitioned according to the primes which divide the orders of the distinct conjugacy classes of maximal abelian subgroups of |G|. This pattern of partitioning the prime divisors of |G| according to conjugacy classes of nilpotent Hall subgroups ( a Hall subgroup is one whose order and index are relatively prime) whose normalizers give all the maximal subgroups of G (up to conjugacy) is repeated in both the proof of the Feit-Hall-Thompson CN-theorem and in the proof of the Feit-Thompson odd-order theorem. The proof of the CN-case is already considerably more difficult than the CA-case, while this part of the proof of the odd-order theorem takes over 100 journal pages. Whereas in the CN-case, the resulting maximal subgroups are still Frobenius groups, the maximal subgroups which occur in the proof of the odd-order theorem need no longer have this structure, and the analysis of their structure and interplay produces 5 very complicated possible configurations.

Step 2. Character theory of G. If X is an irreducible character of the normalizer H of the maximal abelian subgroup A of the CA group G, not containing A in its kernel, we can induce X to a character Y of G, which is not necessarily irreducible. Because of the known structure of G, it is easy to find the character values of Y on all but the identity element of G. This implies that if X₁ and X₂ are two such irreducible characters of H and Y₁ and Y₂ are the corresponding induced characters, then Y₁ − Y₂ is completely determined, and calculating its norm shows that it is the difference of two irreducible characters of G (these are sometimes known as exceptional characters of G with respect to H). A counting argument shows that each non-trivial irreducible character of G arises exactly once as an exceptional character associated to the normalizer of some maximal abelian subgroup of G. A similar argument (but replacing abelian Hall subgroups by nilpotent Hall subgroups) works in the proof of the CN-theorem. However, in the proof of the odd-order theorem, the arguments for constructing characters of G from characters of subgroups are far more delicate, and involve more subtle maps between character rings than character induction, since the maximal subgroups have a more complicated structure and are embedded in a less transparent way.

Step 3. By step 2, we have a complete and precise description of the character table of the CA group G. From this, and using the fact that G has odd order, sufficient information is available to obtain estimates for |G| and arrive at a contradiction to the assumption that G is simple. This part of the argument works similarly in the CN-group case.

In the proof of the Feit–Thompson theorem, however, this step is (as usual) vastly more complicated. The character theory only eliminates four of the possible five configurations left after step 1. To eliminate the final case, fearsomely complicated manipulations with generators and relations are necessary. This part is considered by many to be the hardest and most mysterious part of the proof.