Feit–Thompson theorem

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)

History

The contrast that these results shew between groups of odd and even order suggests inevitably that simple groups of odd order do not exist.

William Burnside (1911, p. 503 note M)

William Burnside (1911, p. 503 note M) conjectured that every nonabelian finite simple group has even order. Richard Brauer (1957) suggested using the centralizers of involutions of simple groups as the basis for the classification of finite simple groups, as the Brauer-Fowler theorem shows that there are only a finite number of finite simple groups with given centralizer of an involution. 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.

Significance of the proof

The Feit–Thompson theorem showed that the classification of finite simple groups using centralizers of involutions might be possible, as every nonabelian simple group has an involution. 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 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. Some of these dwarfed even the Feit–Thompson paper; Aschbacher and Smith's paper on quasithin groups was 1221 pages long.

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), which covers everything except the character theory, and (Peterfalvi 2000, part I) which covers the character theory. This revised proof is still very hard, and is longer than the original proof, but is written in a more leisurely style.

An outline of the proof

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 exposition of the odd order paper see Thompson (1963) or (Gorenstein 1980) or Glauberman (1999).

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. (Bender later simplified this part of the proof using Bender's method.) 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. Peterfalvi (2000) used the Dade isometry to simplify the character theory.

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, Thompson used some fearsomely complicated manipulations with generators and relations (which were later simplified by Peterfalvi (1984), whose argument is reproduced in (Bender & Glauberman 1994). The Feit-Thompson conjecture would simplify this step if it were proven.

References

• Bender, Helmut; Glauberman, George (1994), Local analysis for the odd order theorem, London Mathematical Society Lecture Note Series, 188, Cambridge University Press, ISBN 978-0-521-45716-3, MR1311244 
• Brauer, R. (1957), "On the structure of groups of finite order", Proceedings of the International Congress of Mathematicians, Amsterdam, 1954, Vol. 1, Erven P. Noordhoff N.V., Groningen, pp. 209–217, MR0095203, http://mathunion.org/ICM/ICM1954.1/ 
• Burnside, William (1911), Theory of groups of finite order, New York: Dover Publications, ISBN 978-0-486-49575-0, MR0069818, http://books.google.com/books?isbn=1440035458 
• Feit, Walter; Thompson, John G.; Hall, Marshall, Jr. (1960), "Finite groups in which the centralizer of any non-identity element is nilpotent", Math. Z. 74: 1–17, doi:10.1007/BF01180468, MR0114856 
• Feit, Walter; Thompson, John G. (1962), "A solvability criterion for finite groups and some consequences", Proc. Nat. Acad. Sci. 48 (6): 968–970, doi:10.1073/pnas.48.6.968, JSTOR 71265, MR0143802 
• Feit, Walter; Thompson, John G. (1963), "Solvability of groups of odd order", Pacific Journal of Mathematics 13: 775–1029, ISSN 0030-8730, MR0166261, http://projecteuclid.org/Dienst/UI/1.0/Journal?authority=euclid.pjm&issue=1103053941 
• Glauberman, George (1999), "A new look at the Feit-Thompson odd order theorem", Matemática Contemporânea 16: 73–92, ISSN 0103-9059, MR1756828, http://www.mat.unb.br/~matcont/16_5.ps 
• Gorenstein, D. (1980), Finite groups (2nd ed.), New York: Chelsea Publishing Co., ISBN 978-0-8284-0301-6, MR569209, http://www.ams.org/bookstore-getitem/item=CHEL-301-H 
• Peterfalvi, Thomas (1984), "Simplification du chapitre VI de l'article de Feit et Thompson sur les groupes d'ordre impair", Comptes Rendus des Séances de l'Académie des Sciences. Série I. Mathématique 299 (12): 531–534, ISSN 0249-6291, MR770439 
• Peterfalvi, Thomas (2000), Character theory for the odd order theorem, London Mathematical Society Lecture Note Series, 272, Cambridge University Press, ISBN 978-0-521-64660-4, MR1747393, http://books.google.com/books?isbn=052164660X 
• Suzuki, Michio (1957), "The nonexistence of a certain type of simple groups of odd order", Proceedings of the American Mathematical Society (Proceedings of the American Mathematical Society, Vol. 8, No. 4) 8 (4): 686–695, doi:10.2307/2033280, JSTOR 2033280, MR0086818 
• Thompson, John G. (1963), "Two results about finite groups", Proc. Internat. Congr. Mathematicians (Stockholm, 1962), Djursholm: Inst. Mittag-Leffler, pp. 296–300, MR0175972, http://mathunion.org/ICM/ICM1962.1/