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Signalizer functor

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In mathematics, in the area of abstract algebra, a signalizer functor is a mapping from a potential finite subgroup to the centralizers of the nontrivial elements of an abelian group. The signalizer functor theorem provides the conditions under which the source of such a functor is in fact a subgroup.

The signalizer functor was first defined by Daniel Gorenstein.[1] George Glauberman proved the Solvable Signalizer Functor Theorem for solvable groups[2] and Patrick McBride proved it for general groups.[3][4] Results concerning signalizer functors play a major role in the classification of finite simple groups.

Definition

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Let A be a non-cyclic 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 element , the group is contained in
  • For every pair of nonidentity elements , we have

The second condition above is called the balance condition. If the subgroups are all solvable, then the signalizer functor itself is said to be solvable.

Solvable signalizer functor theorem

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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 states that this will be the case if is solvable and has at least three generators.[2] The theorem also states that under these assumptions, itself will be solvable.

Several weaker versions of the theorem were proven before Glauberman's proof was published. Gorenstein proved it under the stronger assumption that had rank at least 5.[1] David Goldschmidt proved it under the assumption that had rank at least 4 or was a 2-group of rank at least 3.[5][6] Helmut Bender gave a simple proof for 2-groups using the ZJ theorem,[7] and Paul Flavell gave a proof in a similar spirit for all primes.[8] Glauberman gave the definitive result for solvable signalizer functors.[2] Using the classification of finite simple groups, McBride showed that is a -group without the assumption that is solvable.[3][4]

Completeness

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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 as a result of the balance condition of θ.

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

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The easiest way to obtain a signalizer functor is to start with an -invariant -subgroup of and define for all nonidentity However, it is generally more practical to begin with and use it to construct the -invariant -group.

The simplest signalizer functor used in practice is

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 can be done using Thompson's -lemma.

Coprime action

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

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

This fact can be proven using the Schur–Zassenhaus theorem 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. [9][10]

This fact is used in both the proof and applications of the Solvable Signalizer Functor Theorem.

For example, one useful result is that it implies that if is complete, then its completion is the group defined above.

Normal completion

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Another result that follows from the fact above is that the completion of a signalizer functor is often normal in :

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,

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 , where the superscript denotes conjugation by For example, the mapping , the example of a signalizer functor given above, 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

References

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  1. ^ a b Gorenstein, D. (1969), "On the centralizers of involutions in finite groups", Journal of Algebra, 11 (2): 243–277, doi:10.1016/0021-8693(69)90056-8, ISSN 0021-8693, MR 0240188
  2. ^ a b c 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
  3. ^ a b 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
  4. ^ a b 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
  5. ^ 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
  6. ^ Goldschmidt, David M. (1972b), "2-signalizer functors on finite groups", Journal of Algebra, 21 (2): 321–340, doi:10.1016/0021-8693(72)90027-0, ISSN 0021-8693, MR 0323904
  7. ^ 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
  8. ^ Flavell, Paul (2007), A new proof of the Solvable Signalizer Functor Theorem (PDF), archived from the original (PDF) on 2012-04-14
  9. ^ Aschbacher, Michael (2000), Finite Group Theory, Cambridge University Press, ISBN 978-0-521-78675-1
  10. ^ 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