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. George Glauberman proved the Solvable Signalizer Functor Theorem for solvable groups and Patrick McBride proved it for general groups. 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:
a\inA
\theta(a)
CG(a).
a,b\inA
\theta(a)\capCG(b)\subseteq\theta(b).
The second condition above is called the balance condition. If the subgroups
\theta(a)
\theta
Given
\theta,
W=\langle\theta(a)\mida\inA,a ≠ 1\rangle
G
\theta(a)
p'
The Solvable Signalizer Functor Theorem proved by Glauberman states that this will be the case if
\theta
A
W
Several weaker versions of the theorem had already been proven by the time Glauberman's proof was published. Gorenstein proved it under the stronger assumption that
A
A
W
p'
\theta
The terminology of completeness is often used in discussions of signalizer functors. Let
\theta
A
p'
H
G
H\capCG(a)\subseteq\theta(a)
a\inA.
\theta(a)
The signalizer functor
\theta
W
W
\theta
\theta
W
\theta
Thus, the Solvable Signalizer Functor Theorem can be rephrased by saying that if
A
A
G
The easiest way to obtain a signalizer functor is to start with an
A
p'
M
G,
\theta(a)=M\capCG(a)
a\inA.
\theta
A
p'
The simplest signalizer functor used in practice is
\theta(a)=Op'(CG(a)).
As defined above,
\theta(a)
A
p'
G
A
\theta
a\inA,
CG(a)
p
p
Verifying the balance condition for this
\theta
P x Q
To obtain a better understanding of signalizer functors, it is essential to know the following general fact about finite groups:
E
X.
E
X
X=\langleCX(E0)\midE0\subseteqE,andE/E0cyclic\rangle
This fact can be proven using the Schur–Zassenhaus theorem to show that for each prime
q
X,
X
E
q
X
q
X
X
E
E/CE(X)
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
\theta
W
Another result that follows from the fact above is that the completion of a signalizer functor is often normal in
G
Let
\theta
A
G
Let
B
A.
W=\langle\theta(a)\mida\inA,a ≠ 1\rangle=\langle\theta(b)\midb\inB,b ≠ 1\rangle.
To see this, observe that because
\theta(a)
\theta(a)=\langle\theta(a)\capCG(b)\midb\inB,b ≠ 1\rangle\subseteq\langle\theta(b)\midb\inB,b ≠ 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
g\inG
a\inA
\theta(ag)=\theta(a)g
g.
a\mapstoOp'(CG(a))
If
\theta
B
W.
G
A,
\theta
G.