Signalizer functor – Wikipedia

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

${displaystyle p’}$

-subgroup of a finite group

${displaystyle G}$

, which has a good chance of being normal in

${displaystyle G}$

, by taking as generators certain

${displaystyle p’}$

-subgroups of the centralizers of nonidentity elements in one or several given noncyclic elementary abelian

${displaystyle p}$

-subgroups of

${displaystyle G.}$

The technique has origins in the Feit–Thompson theorem, and was subsequently developed by many people including Gorenstein (1969) who defined signalizer functors, Glauberman (1976) who proved the Solvable Signalizer Functor Theorem for solvable groups, and McBride (1982a, 1982b) 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[edit]

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

${displaystyle theta (a)}$

are all solvable, then the signalizer functor

${displaystyle theta }$

itself is said to be solvable.

Solvable signalizer functor theorem[edit]

Given

${displaystyle theta ,}$

certain additional, relatively mild, assumptions allow one to prove that the subgroup

${displaystyle W=langle theta (a)mid ain A,aneq 1rangle }$

of

${displaystyle G}$

generated by the subgroups

${displaystyle theta (a)}$

is in fact a

${displaystyle p’}$

-subgroup. The Solvable Signalizer Functor Theorem proved by Glauberman and mentioned above says that this will be the case if

${displaystyle theta }$

is solvable and

${displaystyle A}$

has at least three generators. The theorem also states that under these assumptions,

${displaystyle W}$

itself will be solvable.

Several earlier versions of the theorem were proven: Gorenstein (1969) proved this under the stronger assumption that

${displaystyle A}$

had rank at least 5. Goldschmidt (1972a, 1972b) proved this under the assumption that

${displaystyle A}$

had rank at least 4 or was a 2-group of rank at least 3. Bender (1975) 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 Flavell (2007). Glauberman (1976) gave the definitive result for solvable signalizer functors. Using the classification of finite simple groups, McBride (1982a, 1982b) showed that

${displaystyle W}$

is a

${displaystyle p’}$

-group without the assumption that

${displaystyle theta }$

is solvable.

Completeness[edit]

The terminology of completeness is often used in discussions of signalizer functors. Let

${displaystyle theta }$

be a signalizer functor as above, and consider the set И of all

${displaystyle A}$

-invariant

${displaystyle p’}$

-subgroups

${displaystyle H}$

of

${displaystyle G}$

satisfying the following condition:

• 
  ${displaystyle Hcap C_{G}(a)subseteq theta (a)}$

  for all nonidentity

  ${displaystyle ain A.}$

  

For example, the subgroups

${displaystyle theta (a)}$

belong to И by the balance condition. The signalizer functor

${displaystyle theta }$

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

${displaystyle W}$

above, and

${displaystyle W}$

is called the completion of

${displaystyle theta }$

. If

${displaystyle theta }$

is complete, and

${displaystyle W}$

turns out to be solvable, then

${displaystyle theta }$

is said to be solvably complete.

Thus, the Solvable Signalizer Functor Theorem can be rephrased by saying that if

${displaystyle A}$

has at least three generators, then every solvable

${displaystyle A}$

-signalizer functor on

${displaystyle G}$

is solvably^[spelling?] complete.

Examples of signalizer functors[edit]

The easiest way to obtain a signalizer functor is to start with an

${displaystyle A}$

-invariant

${displaystyle p’}$

-subgroup

${displaystyle M}$

of

${displaystyle G,}$

and define

${displaystyle theta (a)=Mcap C_{G}(a)}$

for all nonidentity

${displaystyle ain A.}$

In practice, however, one begins with

${displaystyle theta }$

and uses it to construct the

${displaystyle A}$

-invariant

${displaystyle p’}$

-group.

The simplest signalizer functor used in practice is this:

${displaystyle theta (a)=O_{p’}(C_{G}(a)).}$

A few words of caution are needed here. First, note that

${displaystyle theta (a)}$

as defined above is indeed an

${displaystyle A}$

-invariant

${displaystyle p’}$

-subgroup of

${displaystyle G}$

because

${displaystyle A}$

is abelian. However, some additional assumptions are needed to show that this

${displaystyle theta }$

satisfies the balance condition. One sufficient criterion is that for each nonidentity

${displaystyle ain A,}$

the group

${displaystyle C_{G}(a)}$

is solvable (or

${displaystyle p}$

-solvable or even

${displaystyle p}$

-constrained). Verifying the balance condition for this

${displaystyle theta }$

under this assumption requires a famous lemma, known as Thompson’s

${displaystyle Ptimes Q}$

-lemma. (Note, this lemma is also called Thompson’s

${displaystyle Atimes B}$

-lemma, but the

${displaystyle A}$

in this use must not be confused with the

${displaystyle A}$

appearing in the definition of a signalizer functor!)

Coprime action[edit]

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

• Let
  ${displaystyle E}$

  be an abelian noncyclic group acting on the finite group

  ${displaystyle X.}$

  Assume that the orders of

  ${displaystyle E}$

  and

  ${displaystyle X}$

  are relatively prime. Then

${displaystyle X=langle C_{X}(E_{0})mid E_{0}subseteq E,{text{ and }}E/E_{0}{text{ cyclic }}rangle }$

To prove this fact, one uses the Schur–Zassenhaus theorem to show that for each prime

${displaystyle q}$

dividing the order of

${displaystyle X,}$

the group

${displaystyle X}$

has an

${displaystyle E}$

-invariant Sylow

${displaystyle q}$

-subgroup. This reduces to the case where

${displaystyle X}$

is a

${displaystyle q}$

-group. Then an argument by induction on the order of

${displaystyle X}$

reduces the statement further to the case where

${displaystyle X}$

is elementary abelian with

${displaystyle E}$

acting irreducibly. This forces the group

${displaystyle E/C_{E}(X)}$

to be cyclic, and the result follows. See either of the books Aschbacher (2000) or Kurzweil & Stellmacher (2004) 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

${displaystyle theta }$

is complete, then its completion is the group

${displaystyle W}$

defined above.

Normal completion[edit]

The completion of a signalizer functor has a “good chance” of being normal in

${displaystyle G,}$

according to the top of the article. Here, the coprime action fact will be used to motivate this claim. Let

${displaystyle theta }$

be a complete

${displaystyle A}$

-signalizer functor on

${displaystyle G}$

Let

${displaystyle B}$

be a noncyclic subgroup of

${displaystyle A.}$

Then the coprime action fact together with the balance condition imply that

${displaystyle W=langle theta (a)mid ain A,aneq 1rangle =langle theta (b)mid bin B,bneq 1rangle }$

.

To see this, observe that because

${displaystyle theta (a)}$

is B-invariant, we have

${displaystyle theta (a)=langle theta (a)cap C_{G}(b)mid bin B,bneq 1rangle subseteq langle theta (b)mid bin B,bneq 1rangle .}$

The equality above uses the coprime action fact, and the containment uses the balance condition. Now, it is often the case that

${displaystyle theta }$

satisfies an “equivariance” condition, namely that for each

${displaystyle gin G}$

and nonidentity

${displaystyle ain A}$

${displaystyle theta (a^{g})=theta (a)^{g}.,}$

The superscript denotes conjugation by

${displaystyle g.}$

For example, the mapping

${displaystyle amapsto O_{p’}(C_{G}(a))}$

(which is often a signalizer functor!) satisfies this condition. If

${displaystyle theta }$

satisfies equivariance, then the normalizer of

${displaystyle B}$

will normalize

${displaystyle W.}$

It follows that if

${displaystyle G}$

is generated by the normalizers of the noncyclic subgroups of

${displaystyle A,}$

then the completion of

${displaystyle theta }$

(i.e. W) is normal in

${displaystyle G.}$

References[edit]

• Aschbacher, Michael (2000), Finite Group Theory, Cambridge University Press, ISBN 978-0-521-78675-1
• Bender, Helmut (1975), “Goldschmidt’s 2-signalizer functor theorem”, Israel Journal of Mathematics, 22 (3): 208–213, doi:10.1007/BF02761590, ISSN 0021-2172, MR 0390056
• Flavell, Paul (2007), A new proof of the Solvable Signalizer Functor Theorem (PDF), archived from the original (PDF) on 2012-04-14
• Goldschmidt, David M. (1972a), “Solvable signalizer functors on finite groups”, Journal of Algebra, 21: 137–148, doi:10.1016/0021-8693(72)90040-3, ISSN 0021-8693, MR 0297861
• Goldschmidt, David M. (1972b), “2-signalizer functors on finite groups”, Journal of Algebra, 21: 321–340, doi:10.1016/0021-8693(72)90027-0, ISSN 0021-8693, MR 0323904
• Glauberman, George (1976), “On solvable signalizer functors in finite groups”, Proceedings of the London Mathematical Society, Third Series, 33 (1): 1–27, doi:10.1112/plms/s3-33.1.1, ISSN 0024-6115, MR 0417284
• Gorenstein, D. (1969), “On the centralizers of involutions in finite groups”, Journal of Algebra, 11: 243–277, doi:10.1016/0021-8693(69)90056-8, ISSN 0021-8693, MR 0240188
• Kurzweil, Hans; Stellmacher, Bernd (2004), The theory of finite groups, Universitext, Berlin, New York: Springer-Verlag, doi:10.1007/b97433, ISBN 978-0-387-40510-0, MR 2014408
• McBride, Patrick Paschal (1982a), “Near solvable signalizer functors on finite groups” (PDF), Journal of Algebra, 78 (1): 181–214, doi:10.1016/0021-8693(82)90107-7, hdl:2027.42/23875, ISSN 0021-8693, MR 0677717
• McBride, Patrick Paschal (1982b), “Nonsolvable signalizer functors on finite groups”, Journal of Algebra, 78 (1): 215–238, doi:10.1016/0021-8693(82)90108-9, hdl:2027.42/23876, ISSN 0021-8693