Feit–Thompson theorem
Encyclopedia
In mathematics
, the Feit–Thompson theorem, or odd order theorem, states that every finite group
of odd order
is solvable
. It was proved by
, 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 , 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.)
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.
, 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 group
s was 1221 pages long.
The simplified proof has been published in two books: , which covers everything except the character theory, 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.
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. 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 , whose argument is reproduced in . The Feit-Thompson conjecture would simplify this step if it were proven.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Feit–Thompson theorem, or odd order theorem, states that every finite group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
of odd order
Order (group theory)
In group theory, a branch of mathematics, the term order is used in two closely related senses:* The order of a group is its cardinality, i.e., the number of its elements....
is solvable
Solvable group
In mathematics, more specifically in the field of group theory, a solvable group is a group that can be constructed from abelian groups using extensions...
. It was proved by
History
conjectured that every nonabelian finite simple group has even order. suggested using the centralizers of involutions of simple groups as the basis for the classification of finite simple groupsClassification of finite simple groups
In mathematics, the classification of the finite simple groups is a theorem stating that every finite simple group belongs to one of four categories described below. These groups can be seen as the basic building blocks of all finite groups, in much the same way as the prime numbers are the basic...
, 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
Solvable group
In mathematics, more specifically in the field of group theory, a solvable group is a group that can be constructed from abelian groups using extensions...
, which is what Feit and Thompson proved.
The attack on Burnside's conjecture was started by , who studied CA groups
CA group
In mathematics, in the realm of group theory, a group is said to be a CA-group or centralizer abelian group if the centralizer of any nonidentity element is an abelian subgroup. Finite CA-groups are of historical importance as an early example of the type of classifications that would be used in...
; these are groups such that the Centralizer of every non-trivial element is Abelian
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
. 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.)
extended Suzuki's work to the family of CN groups
CN group
In mathematics, in the area of algebra known as group theory, a more than fifty-year effort was made to answer a conjecture of : are all groups of odd order solvable? Progress was made by showing that CA-groups, groups in which the centralizer of a non-identity element is abelian, of odd order are...
; these are groups such that the Centralizer of every non-trivial element is Nilpotent
Nilpotent group
In mathematics, more specifically in the field of group theory, a nilpotent group is a group that is "almost abelian". This idea is motivated by the fact that nilpotent groups are solvable, and for finite nilpotent groups, two elements having relatively prime orders must commute...
. 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
Solvable group
In mathematics, more specifically in the field of group theory, a solvable group is a group that can be constructed from abelian groups using extensions...
. This proves that every finite group of odd order is solvable, as a minimal counterexample
Minimal counterexample
In mathematics, the method of considering a minimal counterexample combines the ideas of inductive proof and proof by contradiction. Abstractly, in trying to prove a proposition P, one assumes that it is false, and that therefore there is at least one 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 analysisLocal 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' picture...
, 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 group
Quasithin group
In mathematics, a quasithin group is roughly a finite simple group of characteristic 2 type and width 2. Here characteristic 2 type means that its centralizers of involutions resemble those of groups of Lie type over fields of characteristic 2, and the width is roughly the maximal rank of an...
s 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: , which covers everything except the character theory, 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.
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
Ferdinand Georg Frobenius
Ferdinand Georg Frobenius was a German mathematician, best known for his contributions to the theory of differential equations and to group theory...
whose character theory is reasonably transparent, and well-suited to manipulations involving character induction
Character theory
In mathematics, more specifically in group theory, the character of a group representation is a function on the group which associates to each group element the trace of the corresponding matrix....
. 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
Philip Hall
Philip Hall FRS , was an English mathematician.His major work was on group theory, notably on finite groups and solvable groups.-Biography:...
subgroups (a Hall subgroup is one whose order and index
Index of a subgroup
In mathematics, specifically group theory, the index of a subgroup H in a group G is the "relative size" of H in G: equivalently, the number of "copies" of H that fill up G. For example, if H has index 2 in G, then intuitively "half" of the elements of G lie in H...
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
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 Walter's analysis of groups with abelian Sylow 2-subgroups , and Gorenstein and Walter's classification of groups with...
.) 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. used the Dade isometry
Dade isometry
In mathematical finite group theory, the Dade isometry is an isometry from class functions on a subgroup H with support on a subset K of H to class functions on a group G...
to simplify the character theory.
Step 2. Character theory
Character theory
In mathematics, more specifically in group theory, the character of a group representation is a function on the group which associates to each group element the trace of the corresponding matrix....
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
Norm (mathematics)
In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector...
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 , whose argument is reproduced in . The Feit-Thompson conjecture would simplify this step if it were proven.