# Signalizer functor

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 to construct a $p'$ -subgroup of a finite group $G$ , which has a good chance of being normal in $G$ , by taking as generators certain $p'$ -subgroups of the centralizers of nonidentity elements in one or several given noncyclic elementary abelian $p$ -subgroups of $G.$ The technique has origins in the Feit–Thompson theorem, and was subsequently developed by many people including Template:Harvtxt who defined signalizer functors, Template:Harvtxt who proved the Solvable Signalizer Functor Theorem for solvable groups, and Template:Harvs who proved it for all groups. This theorem is needed to prove the so-called "dichotomy" stating that a given nonabelian finite simple group either has local characteristic two, or is of component type. It thus plays a major role in the classification of finite simple groups.

## Definition

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

The second condition above is called the balance condition. If the subgroups $\theta (a)$ are all solvable, then the signalizer functor $\theta$ itself is said to be solvable.

## Solvable signalizer functor theorem

Given $\theta ,$ certain additional, relatively mild, assumptions allow one to prove that the subgroup $W=\langle \theta (a)\mid a\in A,a\neq 1\rangle$ of $G$ generated by the subgroups $\theta (a)$ is in fact a $p'$ -subgroup. The Solvable Signalizer Functor Theorem proved by Glauberman and mentioned above says that this will be the case if $\theta$ is solvable and $A$ has at least three generators. The theorem also states that under these assumptions, $W$ itself will be solvable.

Several earlier versions of the theorem were proven: Template:Harvtxt proved this under the stronger assumption that $A$ had rank at least 5. Template:Harvs proved this under the assumption that $A$ had rank at least 4 or was a 2-group of rank at least 3. Template:Harvtxt gave a simple proof for 2-groups using the ZJ theorem, and a proof in a similar spirit has been given for all primes by Template:Harvtxt. Template:Harvtxt gave the definitive result for solvable signalizer functors. Using the classification of finite simple groups, Template:Harvs showed that $W$ is a $p'$ -group without the assumption that $\theta$ is solvable.

### Completeness

For example, the subgroups $\theta (a)$ belong to И by the balance condition. The signalizer functor $\theta$ 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 $W$ above, and $W$ is called the completion of $\theta$ . If $\theta$ is complete, and $W$ turns out to be solvable, then $\theta$ is said to be solvably complete.

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

## Examples of signalizer functors

The simplest signalizer functor used in practice is this: $\theta (a)=O_{p'}(C_{G}(a)).$ ## Coprime action

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

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 $\theta$ is complete, then its completion is the group $W$ defined above.

## Normal completion

The completion of a signalizer functor has a "good chance" of being normal in $G,$ according to the top of the article. Here, the coprime action fact will be used to motivate this claim. Let $\theta$ be a complete $A$ -signalizer functor on $G$ To see this, observe that because $\theta (a)$ 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 $\theta$ satisfies an "equivariance" condition, namely that for each $g\in G$ and nonidentity $a\in A$ 