In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Feit–Thompson theorem, or odd order theorem, states that every finite
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
of odd
order is
solvable. It was proved by .
History
conjectured that every nonabelian
finite simple group
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marked ...
has even order. suggested using the
centralizer
In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group ''G'' is the set of elements \mathrm_G(S) of ''G'' such that each member g \in \mathrm_G(S) commutes with each element of ''S'', o ...
s of involutions of simple groups as the basis for the
classification of finite simple groups
In mathematics, the classification of the finite simple groups is a result of group theory stating that every finite simple group is either cyclic, or alternating, or it belongs to a broad infinite class called the groups of Lie type, or else it ...
, as the
Brauer–Fowler theorem shows that there are only a finite number of finite simple groups with given
centralizer
In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group ''G'' is the set of elements \mathrm_G(S) of ''G'' such that each member g \in \mathrm_G(S) commutes with each element of ''S'', o ...
of an
involution
Involution may refer to:
* Involute, a construction in the differential geometry of curves
* '' Agricultural Involution: The Processes of Ecological Change in Indonesia'', a 1963 study of intensification of production through increased labour inpu ...
. 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 , 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
In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow that give detailed information about the number of subgroups of fixe ...
, finding an overlooked family of simple
groups of Lie type
In mathematics, specifically in group theory, the phrase ''group of Lie type'' usually refers to finite groups that are closely related to the group of rational points of a reductive linear algebraic group with values in a finite field. The phras ...
in the process, that are now called
Suzuki groups
In the area of modern algebra known as group theory, the Suzuki groups, denoted by Sz(22''n''+1), 2''B''2(22''n''+1), Suz(22''n''+1), or ''G''(22''n''+1), form an infinite family of groups of Lie type found by , that are simple for ''n'' ≥ 1. ...
.)
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
In mathematics, a minimal counterexample is the smallest example which falsifies a claim, and a proof by minimal counterexample is a method of proof which combines the use of a minimal counterexample with the ideas of proof by induction and proof ...
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
In mathematics, the term local analysis has at least two meanings, both derived from the idea of looking at a problem relative to each prime number ''p'' first, and then later trying to integrate the information gained at each prime into a 'global' ...
, 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; the paper by
Michael Aschbacher
Michael George Aschbacher (born April 8, 1944) is an American mathematician best known for his work on finite groups. He was a leading figure in the completion of the classification of finite simple groups in the 1970s and 1980s. It later turne ...
and Stephen D. Smith on
quasithin group In mathematics, a quasithin group is a finite simple group that resembles a group of Lie type of rank at most 2 over a field of characteristic 2. More precisely it is a finite simple group of characteristic 2 type and width 2. Here characteristic ...
s was 1,221 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: , which covers everything except the
character theory
In mathematics, more specifically in group theory, the character of a group representation is a function on the group that associates to each group element the trace of the corresponding matrix. The character carries the essential information about ...
, and , 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.
A fully formal proof, checked with the
Coq
Coq is an interactive theorem prover first released in 1989. It allows for expressing mathematical assertions, mechanically checks proofs of these assertions, helps find formal proofs, and extracts a certified program from the constructive proof ...
proof assistant
In computer science and mathematical logic, a proof assistant or interactive theorem prover is a software tool to assist with the development of formal proofs by human-machine collaboration. This involves some sort of interactive proof editor ...
, was announced in September 2012 by
Georges Gonthier
Georges Gonthier is a Canadian computer scientist and one of the leading practitioners in formal mathematics. He led the formalization of the four color theorem and Feit–Thompson proof of the odd-order theorem. (Both were written using the ...
and fellow researchers at
Microsoft Research
Microsoft Research (MSR) is the research subsidiary of Microsoft. It was created in 1991 by Richard Rashid, Bill Gates and Nathan Myhrvold with the intent to advance state-of-the-art computing and solve difficult world problems through technologi ...
and
INRIA
The National Institute for Research in Digital Science and Technology (Inria) () is a French national research institution focusing on computer science and applied mathematics.
It was created under the name ''Institut de recherche en informatiq ...
.
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 or or .
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
In mathematics, a Frobenius group is a transitive permutation group on a finite set, such that no non-trivial element
fixes more than one point and some non-trivial element fixes a point.
They are named after F. G. Frobenius.
Structure
Suppos ...
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 certain
Hall subgroup
In mathematics, specifically group theory, a Hall subgroup of a finite group ''G'' is a subgroup whose order is coprime to its index. They were introduced by the group theorist .
Definitions
A Hall divisor (also called a unitary divisor) o ...
s (a Hall subgroup is one whose order and
index
Index (or its plural form indices) may refer to:
Arts, entertainment, and media Fictional entities
* Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index''
* The Index, an item on a Halo megastru ...
are relatively prime) which correspond to 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. Each maximal subgroup ''M'' has a certain nilpotent Hall subgroup ''M''
σ with normalizer contained in ''M'', whose order is divisible by certain primes forming a set σ(''M''). Two maximal subgroups are conjugate if and only if the sets σ(''M'') are the same, and if they are not conjugate then the sets σ(''M'') are disjoint. Every prime dividing the order of ''G'' occurs in some set σ(''M''). So the primes dividing the order of ''G'' are partitioned into equivalence classes corresponding to the conjugacy classes of maximal subgroups. The proof of the CN-case is already considerably more difficult than the CA-case: the main extra problem is to prove that two different Sylow subgroups intersect in the identity. This part of the proof of the odd-order theorem takes over 100 journal pages. A key step is the proof of the
Thompson uniqueness theorem
In mathematical finite group theory, Thompson's original uniqueness theorem states that in a minimal simple finite group of odd order there is a unique maximal subgroup containing a given elementary abelian subgroup of rank
Rank is the relative ...
, stating that abelian subgroups of normal rank at least 3 are contained in a unique maximal subgroup, which means that the primes ''p'' for which the Sylow ''p''-subgroups have normal rank at most 2 need to be considered separately. Bender later simplified the proof of the uniqueness theorem using
Bender's method In group theory, Bender's method is a method introduced by for simplifying the local group theoretic analysis of the odd order theorem. Shortly afterwards he used it to simplify the Walter theorem on groups with abelian Sylow 2-subgroups , and Go ...
. Whereas in the CN-case, the resulting maximal subgroups ''M'' are still Frobenius groups, the maximal subgroups that 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 possible types of maximal subgroups, called types I, II, III, IV, V. Type I subgroups are of "Frobenius type", a slight generalization of Frobenius group, and in fact later on in the proof are shown to be Frobenius groups. They have the structure ''M''
''F''⋊''U'' where ''M''
''F'' is the largest normal nilpotent Hall subgroup, and ''U'' has a subgroup ''U''
0 with the same exponent such that ''M''
''F''⋊''U''
0 is a Frobenius group with kernel ''M''
''F''. Types II, III, IV, V are all
3-step groups with structure ''M''
''F''⋊''U''⋊''W''
1, where ''M''
''F''⋊''U'' is the derived subgroup of ''M''. The subdivision into types II, III, IV and V depends on the structure and embedding of the subgroup ''U'' as follows:
*Type II: ''U'' is nontrivial abelian and its normalizer is not contained in ''M''.
*Type III: ''U'' is nontrivial abelian and its normalizer is contained in ''M''.
*Type IV: ''U'' is nonabelian.
*Type V: ''U'' is trivial.
All but two classes of maximal subgroups are of type I, but there may also be two extra classes of maximal subgroups, one of type II, and one of type II, III, IV, or V.
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 X
1 and X
2 are two such irreducible characters of ''H'' and Y
1 and Y
2 are the corresponding induced characters, then Y
1 − Y
2 is completely determined, and calculating its
norm
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envi ...
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 use the
Dade isometry In mathematical finite group theory, the Dade isometry is an isometry from class function on a subgroup ''H'' with support on a subset ''K'' of ''H'' to class functions on a group ''G'' . It was introduced by as a generalization and simplificatio ...
between character rings rather than character induction, since the maximal subgroups have a more complicated structure and are embedded in a less transparent way. The theory of exceptional characters is replaced by the theory of a
coherent set of characters to extend the Dade isometry. Roughly speaking, this theory says that the Dade isometry can be extended unless the groups involved have a certain precise structure. described a simplified version the character theory due to Dade, Sibley, and Peterfalvi.
Step 3. The final contradiction
By step 2, we have a complete and precise description of the
character table In group theory, a branch of abstract algebra, a character table is a two-dimensional table whose rows correspond to irreducible representations, and whose columns correspond to conjugacy classes of group elements. The entries consist of character ...
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 some of the possible configurations left after step 1. First they show that the maximal subgroups of type I are all Frobenius groups. If all maximal subgroups are type I then an argument similar to the CN case shows that the group ''G'' cannot be an odd-order minimal simple group, so there are exactly two classes of maximal subgroups of types II, III, IV or V. Most of the rest of the proof now focuses on these two types of maximal subgroup ''S'' and ''T'' and the relation between them. More character-theoretic arguments show that they cannot be of types IV or V. The two subgroups have a precise structure: the subgroup ''S'' is of order ''p''
''q''×''q''×(''p''
''q''–1)/(''p''–1) and consists of all automorphisms of the underlying set of the finite field of order ''p''
''q'' of the form ''x''→''ax''
''σ''+''b'' where ''a'' has norm 1 and ''σ'' is an automorphism of the finite field, where ''p'' and ''q'' are distinct primes. The maximal subgroup ''T'' has a similar structure with ''p'' and ''q'' reversed. The subgroups ''S'' and ''T'' are closely linked. Taking ''p''>''q'', one can show that the cyclic subgroup of ''S'' of order (''p''
''q''–1)/(''p''–1) is conjugate to a subgroup of the cyclic subgroup of ''T'' of order (''q''
''p''–1)/(''q''–1). (In particular, the first number divides the second, so if the
Feit–Thompson conjecture
In mathematics, the Feit–Thompson conjecture is a conjecture in number theory, suggested by . The conjecture states that there are no distinct prime numbers ''p'' and ''q'' such that
:\frac divides \frac.
If the conjecture were true, it would ...
is true, it would assert that this cannot happen, and this could be used to finish the proof at this point. The conjecture is still unproven, however.)
The conclusion from applying character theory to the group ''G'' is that ''G'' has the following structure: there are primes ''p''>''q'' such that (''p''
''q''–1)/(''p''–1) is coprime to ''p''–1 and ''G'' has a subgroup given by the semidirect product ''PU'' where ''P'' is the additive group of a finite field of order ''p''
''q'' and ''U'' its elements of norm 1. Moreover ''G'' has an abelian subgroup ''Q'' of order prime to ''p'' containing an element ''y'' such that ''P''
0 normalizes ''Q'' and (''P''
0)
''y'' normalizes ''U'', where ''P''
0 is the additive group of the finite field of order ''p''. (For ''p''=2 a similar configuration occurs in the group SL
2(2
''q''), with ''PU'' a Borel subgroup of upper triangular matrices and ''Q'' the subgroup of order 3 generated by
.) To eliminate this final case, Thompson used some fearsomely complicated manipulations with
generators and relations
In mathematics, a presentation is one method of specifying a group. A presentation of a group ''G'' comprises a set ''S'' of generators—so that every element of the group can be written as a product of powers of some of these generators—and ...
, which were later simplified by , whose argument is reproduced in . The proof examines the set of elements ''a'' in the finite field of order ''p''
''q'' such that ''a'' and 2–a both have norm 1. One first checks that this set has at least one element other than 1. Then a rather difficult argument using generators and relations in the group ''G'' shows that the set is closed under taking inverses. If ''a'' is in the set and not equal to 1 then the polynomial N((1–''a'')''x''+1)–1 has degree ''q'' and has at least ''p'' distinct roots given by the elements ''x'' in F
''p'', using the fact that ''x''→1/(2–''x'') maps the set to itself, so ''p''≤''q'', contradicting the assumption ''p''>''q''.
Use of oddness
The fact that the order of the group ''G'' is odd is used in several places in the proof, as follows .
*The
Hall–Higman theorem In mathematical group theory
In abstract algebra, group theory studies the algebraic structures known as groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and v ...
is sharper for groups of odd order.
*For groups of odd order, all non-principal characters occur in complex conjugate pairs.
*Several results about ''p''-groups only hold for odd primes ''p''.
*If a group of odd order has no elementary abelian subgroups of rank 3, then its derived group is nilpotent. (This fails for the symmetric group ''S''
4 of even order.)
*Several arguments involving character theory fail for small primes, especially for the prime 2.
References
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{{DEFAULTSORT:Feit-Thompson theorem
Theorems about finite groups