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 formula_1-subgroup of a finite group formula_2, which has a good chance of being normal in formula_2, by taking as generators certain formula_1-subgroups of the centralizers of nonidentity elements in one or several given noncyclic elementary abelian formula_5-subgroups of formula_6 The technique has origins in the Feit–Thompson theorem, and was subsequently developed by many people including who defined signalizer functors, who proved the Solvable 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 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 formula_12 are all solvable, then the signalizer functor formula_13 itself is said to be solvable.
Solvable signalizer functor theorem.
Given formula_14 certain additional, relatively mild, assumptions allow one to prove that the subgroup formula_15 of formula_2 generated by the subgroups formula_12 is in fact a formula_1-subgroup. The Solvable Signalizer Functor Theorem proved by Glauberman and mentioned above says that this will be the case if formula_13 is solvable and formula_20 has at least three generators. The theorem also states that under these assumptions, formula_21 itself will be solvable.
Several earlier versions of the theorem were proven: proved this under the stronger assumption that formula_20 had rank at least 5. proved this under the assumption that formula_20 had rank at least 4 or was a 2-group of rank at least 3. 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 . gave the definitive result for solvable signalizer functors. Using the classification of finite simple groups, showed that formula_21 is a formula_1-group without the assumption that formula_13 is solvable.
Completeness.
The terminology of completeness is often used in discussions of signalizer functors. Let formula_13 be a signalizer functor as above, and consider the set И of all formula_20-invariant formula_1-subgroups formula_30 of formula_2 satisfying the following condition:
For example, the subgroups formula_12 belong to И by the balance condition. The signalizer functor formula_13 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 formula_21 above, and formula_21 is called the completion of formula_13. If formula_13 is complete, and formula_21 turns out to be solvable, then formula_13 is said to be solvably complete.
Thus, the Solvable Signalizer Functor Theorem can be rephrased by saying that if formula_20 has at least three generators, then every solvable formula_20-signalizer functor on formula_2 is solvably complete.
Examples of signalizer functors.
The easiest way to obtain a signalizer functor is to start with an formula_20-invariant formula_1-subgroup formula_47 of formula_48 and define formula_49 for all nonidentity formula_33 In practice, however, one begins with formula_13 and uses it to construct the formula_20-invariant formula_1-group.
The simplest signalizer functor used in practice is this:
formula_54
A few words of caution are needed here. First, note that formula_12 as defined above is indeed an formula_20-invariant formula_1-subgroup of formula_2 because formula_20 is abelian. However, some additional assumptions are needed to show that this formula_13 satisfies the balance condition. One sufficient criterion is that for each nonidentity formula_61 the group formula_62 is solvable (or formula_5-solvable or even formula_5-constrained). Verifying the balance condition for this formula_13 under this assumption requires a famous lemma, known as Thompson's formula_66-lemma. (Note, this lemma is also called Thompson's formula_67-lemma, but the formula_20 in this use must not be confused with the formula_20 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:
formula_74
To prove this fact, one uses the Schur–Zassenhaus theorem to show that for each prime formula_75 dividing the order of formula_76 the group formula_73 has an formula_70-invariant Sylow formula_75-subgroup. This reduces to the case where formula_73 is a formula_75-group. Then an argument by induction on the order of formula_73 reduces the statement further to the case where formula_73 is elementary abelian with formula_70 acting irreducibly. This forces the group formula_85 to be cyclic, and the result follows. See either of the books or 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 formula_13 is complete, then its completion is the group formula_21 defined above.
Normal completion.
The completion of a signalizer functor has a "good chance" of being normal in formula_48 according to the top of the article. Here, the coprime action fact will be used to motivate this claim. Let formula_13 be a complete formula_20-signalizer functor on formula_2
Let formula_92 be a noncyclic subgroup of formula_93 Then the coprime action fact together with the balance condition imply that
formula_94.
To see this, observe that because formula_12 is "B"-invariant, we have
formula_96
The equality above uses the coprime action fact, and the containment uses the balance condition. Now, it is often the case that formula_13 satisfies an "equivariance" condition, namely that for each formula_98 and nonidentity formula_99
formula_100
The superscript denotes conjugation by formula_101 For example, the mapping formula_102 (which is often a signalizer functor!) satisfies this condition. If formula_13 satisfies equivariance, "then the normalizer of formula_92 will normalize formula_105" It follows that if formula_2 is generated by the normalizers of the noncyclic subgroups of formula_107 then the completion of formula_13 (i.e. W) is normal in formula_6