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.

History

William Burnside conjectured that every 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 , 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 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 odd CN group 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 odd non-cyclic simple group such that every subgroup is solvable. (This proves that every odd group is solvable, as a minimal counterexample must be a simple group such that every 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 about 250 pages long.

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 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; one was over 1000 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:

• Local Analysis for the Odd Order Theorem (London Mathematical Society Lecture Note Series)

by Helmut Bender, George Glauberman. ISBN 0521457165

• Character Theory for the Odd Order Theorem (London Mathematical Society Lecture Note Series)

by T. Peterfalvi, ISBN 052164660X

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).

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. The proof can be broken up into three steps. We let G be a simple group of odd order satisfying the CA condition.

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 subgroups of G. In the odd order paper the analysis of the maximal subgroups takes about 100 pages instead of a few lines, and produces 5 very complicated possible configurations.

Step 2. Character theory of G. If X is an irreducible character of the maximal abelian subgroup A of the CA group G, 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 characters of A and Y₁ and Y₂ are the corresponding induced characters, then

Y₁ − Y₂

is known completely, and calculating its norm shows that it is the difference of two irreducible characters of G. (These are sometimes called exceptional characters of G.) In this way, it is possible to associate an irreducible character of G with an irreducible character of A. A counting argument shows that we get all irreducible characters of G (other than the trivial character) from irreducible characters of maximal abelian subgroups of G.

In the Feit-Thompson theorem, the arguments for constructing characters of G from characters of subgroups are far more delicate, because the structure of the subgroups is more complicated.

Step 3. By step 2, we have a complete and precise description of the character table of the CA group G. From this is is easy to get a contradiction if G is odd and simple.

In the Feit-Thompson theorem, things are (as usual) vastly more complicated. The character theory only eliminates four of the possible five configurations from step 1. To eliminate the final case, they had to use some fearsomely complicated manipulations with generators and relations. This part is considered the hardest and most mysterious part of the proof.

For a more detailed description of the odd order paper see the book Finite groups by Daniel Gorenstein, ISBN 0828403015.

References

• Feit, Walter; Thompson, John G., "Solvability of groups of odd order", Pacific Journal of Mathematics, 13 1963 775--1029.
• Feit, Walter; Hall, Marshall, Jr.; Thompson, John G., "Finite groups in which the centralizer of any non-identity element is nilpotent", Math. Z. 74 1960 1--17.
• Suzuki, Michio, "The nonexistence of a certain type of simple groups of odd order", Proceedings of the American Mathematical Society, volume 8 (1957), 686--695.