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In mathematics, a signalizer functor gives the intersections of a potential subgroup of a finite group with the
centralizer In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group ''G'' is the set of elements \mathrm_G(S) of ''G'' such that each member g \in \mathrm_G(S) commutes with each element of ''S'', o ...
s 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 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. suggested using th ...
, 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 SIMPLE Group Limited is a conglomeration of separately run companies that each has its core area in International Consulting. The core business areas are Legal Services, Fiduciary Activities, Banking Intermediation and Corporate Service. The d ...
either has local characteristic two, or is of component type. It thus plays a major role in the
classification of finite simple groups In mathematics, the classification of the finite simple groups is a result of group theory stating that every finite simple group is either cyclic, or alternating, or it belongs to a broad infinite class called the groups of Lie type, or else it ...
.


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: *For every nonidentity a\in A, the group \theta(a) is contained in C_G(a). *For every nonidentity a, b\in A, we have \theta(a) \cap C_G(b) \subseteq \theta(b). 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: proved this under the stronger assumption that A had rank at least 5. proved this under the assumption that A 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 In mathematics, George Glauberman's ZJ theorem states that if a finite group ''G'' is ''p''-constrained and ''p''-stable and has a normal ''p''-subgroup for some odd prime ''p'', then ''O'p''′(''G'')''Z''(''J''(''S'')) is a normal subgroup ...
, 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 W is a p'-group without the assumption that \theta is solvable.


Completeness

The terminology of completeness is often used in discussions of signalizer functors. Let \theta be a signalizer functor as above, and consider the set И of all A-invariant p'-subgroups H of G satisfying the following condition: *H\cap C_G(a) \subseteq \theta(a) for all nonidentity a \in A. 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 easiest way to obtain a signalizer functor is to start with an A-invariant p'-subgroup M of G, and define \theta(a) = M\cap C_G(a) for all nonidentity a \in A. In practice, however, one begins with \theta and uses it to construct the A-invariant p'-group. The simplest signalizer functor used in practice is this: \theta(a) = O_(C_G(a)). A few words of caution are needed here. First, note that \theta(a) as defined above is indeed an A-invariant p'-subgroup of G because A is abelian. However, some additional assumptions are needed to show that this \theta satisfies the balance condition. One sufficient criterion is that for each nonidentity a \in A, the group C_G(a) is solvable (or p-solvable or even p-constrained). Verifying the balance condition for this \theta under this assumption requires a famous lemma, known as Thompson's P\times Q-lemma. (Note, this lemma is also called Thompson's A \times B-lemma, but the A in this use must not be confused with the A 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 E be an abelian noncyclic group acting on the finite group X. Assume that the orders of E and X are relatively prime. Then X = \langle C_X(E_0) \mid E_0 \subseteq E, \text E/E_0 \text\rangle To prove this fact, one uses the
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 (or split extension ...
to show that for each prime q dividing the order of X, the group X has an E-invariant Sylow q-subgroup. This reduces to the case where X is a q-group. Then an argument by induction on the order of X reduces the statement further to the case where X is elementary abelian with E acting irreducibly. This forces the group E/C_E(X) 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 \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 Let B be a noncyclic subgroup of A. Then the coprime action fact together with the balance condition imply that W= \langle \theta(a) \mid a \in A, a \neq 1\rangle = \langle \theta(b) \mid b \in B, b \neq 1\rangle . To see this, observe that because \theta(a) is ''B''-invariant, we have \theta(a) = \langle \theta(a) \cap C_G(b) \mid b \in B, b \neq 1\rangle \subseteq \langle \theta(b) \mid b \in B, b \neq 1\rangle. 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 \theta(a^g) = \theta(a)^g. \, The superscript denotes conjugation by g. For example, the mapping a \mapsto O_(C_G(a)) (which is often a signalizer functor!) satisfies this condition. If \theta satisfies equivariance, ''then the normalizer of B will normalize W.'' It follows that if G is generated by the normalizers of the noncyclic subgroups of A, then the completion of \theta (i.e. W) is normal in G.


References

* * * * * * * * * *{{Citation , last1=McBride , first1=Patrick Paschal , title=Nonsolvable signalizer functors on finite groups , doi=10.1016/0021-8693(82)90108-9 , doi-access=free , year=1982b , journal=
Journal of Algebra ''Journal of Algebra'' (ISSN 0021-8693) is an international mathematical research journal in algebra. An imprint of Academic Press, it is published by Elsevier. ''Journal of Algebra'' was founded by Graham Higman, who was its editor from 1964 to 1 ...
, issn=0021-8693 , volume=78 , issue=1 , pages=215–238, hdl=2027.42/23876 , hdl-access=free
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 functo ...