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.
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: The second condition above is called the balance condition.
certain additional, relatively mild, assumptions allow one to prove that the subgroup
The Solvable Signalizer Functor Theorem proved by Glauberman states that this will be the case if
Several weaker versions of the theorem were proven before Glauberman's proof was published.
Gorenstein proved it under the stronger assumption that
[1] David Goldschmidt proved it under the assumption that
[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
[3][4] The terminology of completeness is often used in discussions of signalizer functors.
be a signalizer functor as above, and consider the set И of all
belong to И as a result of the balance condition of θ.
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
Thus, the Solvable Signalizer Functor Theorem can be rephrased by saying that if
The easiest way to obtain a signalizer functor is to start with an
The simplest signalizer functor used in practice is
However, some additional assumptions are needed to show that this
To obtain a better understanding of signalizer functors, it is essential to know the following general fact about finite groups: This fact can be proven using the Schur–Zassenhaus theorem to show that for each prime
[9][10] This fact is used in both the proof and applications of the Solvable Signalizer Functor Theorem.
Another result that follows from the fact above is that the completion of a signalizer functor is often normal in
Then the coprime action fact together with the balance condition imply that
The equality above uses the coprime action fact, and the containment uses the balance condition.
, the example of a signalizer functor given above, satisfies this condition.
is generated by the normalizers of the noncyclic subgroups of