In mathematics, a signalizer functor gives the intersections of a potential subgroup of a finite group with the centralizers of nontrivial elements of an abelian group. The signalizer functor theorem gives conditions under which a signalizer functor comes from a subgroup. The idea is to try and construct a -subgroup of a finite group , which has a good chance of being normal in , by taking as generators certain -subgroups of the centralizers of nonidentity elements in one or several given elementary abelian -subgroups of The technique has origins in the Feit–Thompson theorem

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 - History : conjectured that every nonabelian finite simple group has even order...

, and was subsequently developed by many people including who defined signalizer functors, who proved the Signalizer Functor Theorem for solvable groups, and who proved it for all groups. This theorem is needed to prove the so-called "dichotomy" stating that a given nonabelian finite simple group

Simple group

In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, a normal subgroup and the quotient group, and the process can be repeated...

 either has local characteristic two, or is of component type. It thus plays a major role in the classification of finite simple groups

Classification 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...

.

Definition

Let be an elementary abelian -subgroup of the finite group An A-signalizer functor on or simply a signalizer functor when and are clear is a mapping from the set of nonidentity elements of to the set of -invariant -subgroups of satisfying the following properties:

• For every nonidentity , the group is contained in
• For every nonidentity , we have

The second condition above is called the balance condition. If the subgroups are all 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...

, then the signalizer functor itself is said to be solvable.

Solvable signalizer functor theorem

Given certain additional, relatively mild, assumptions allow one to prove that the subgroup of generated by the subgroups is in fact a -subgroup. The Solvable Signalizer Functor Theorem proved by Glauberman and mentioned above says that this will be the case if is solvable and has at least three generators. The theorem also states that under these assumptions, itself will be solvable.

Using the classification of finite simple groups, showed that is a -group without the assumption that is solvable.

Completeness

The terminology of completeness is often used in discussions of signalizer functors. Let be a signalizer functor as above, and consider the set И of all -invariant -subgroups of satisfying the following condition:

• for all nonidentity

For example, the subgroups belong to И by the balance condition. The signalizer functor is said to be complete if И has a unique maximal element when ordered by containment. In this case, the unique maximal element can be shown to coincide with above, and is called the completion of . If is complete, and turns out to be solvable, then is said to be solvably complete.

Thus, the Solvable Signalizer Functor Theorem can be rephrased by saying that if has at least three generators, then every solvable -signalizer functor on is solvably complete.

Examples of signalizer functors

The easiest way to obtain a signalizer functor is to start with an -invariant -subgroup of and define for all nonidentity In practice, however, one begins with and uses it to construct the -invariant -group.

The simplest signalizer functor used in practice is this:


A few words of caution are needed here. First, note that as defined above is indeed an -invariant -subgroup of because is abelian. However, some additional assumptions are needed to show that this satisfies the balance condition. One sufficient criterion is that for each nonidentity the group is solvable (or -solvable or even -constrained). Verifying the balance condition for this under this assumption requires a famous lemma, known as Thompson's

John G. Thompson

John Griggs Thompson is a mathematician at the University of Florida noted for his work in the field of finite groups. He was awarded the Fields Medal in 1970, the Wolf Prize in 1992 and the 2008 Abel Prize....

 -lemma. (Note, this lemma is also called Thompson's -lemma, but the in this use must not be confused with the appearing in the definition of a signalizer functor!)

Coprime action

To obtain a better understanding of signalizer functors, it is essential to know the following general fact about finite groups:

• Let be an abelian noncyclic group acting on the finite group Assume that the orders of and are relatively prime. Then

To prove this fact, one uses the Schur–Zassenhaus theorem

Schur–Zassenhaus theorem

The Schur–Zassenhaus theorem is a theorem in group theory which states that if G is a finite group, and N is a normal subgroup whose order is coprime to the order of the quotient group G/N, then G is a semidirect product of N and G/N....

to show that for each prime dividing the order of the group has an -invariant Sylow -subgroup. This reduces to the case where is a -group. Then an argument by induction on the order of reduces the statement further to the case where is elementary abelian with acting irreducibly. This forces the group to be cyclic, and the result follows. See either of the books quoted below for details.

This is used in both the proof and applications of the Solvable Signalizer Functor Theorem. To begin, notice that it quickly implies the claim that if is complete, then its completion is the group defined above.

Normal completion

The completion of a signalizer functor has a "good chance" of being normal in according to the top of the article. Here, the coprime action fact will be used to motivate this claim. Let be a complete -signalizer functor on

Let be a noncyclic subgroup of Then the coprime action fact together with the balance condition imply that
.

To see this, observe that because is B-invariant, we have



The equality above uses the coprime action fact, and the containment uses the balance condition. Now, it is often the case that satisfies an "equivariance" condition, namely that for each and nonidentity



The superscript denotes conjugation by For example, the mapping (which is often a signalizer functor!) satisfies this condition. If satisfies equivariance, then the normalizer of will normalize It follows that if is generated by the normalizers of the noncyclic subgroups of then the completion of (i.e. W) is normal in