Signalizer functor

Definition with symbols
Let $$G$$ be a group and $$A$$ an Abelian subgroup of $$G$$. Then, a signalizer functor on $$A$$ in $$G$$ is a map $$\theta$$ from the set of non-identity elements of $$A$$ to the set of $$A$$-invariant subgroups (that is, the set of subgroups of $$G$$ that commute with every element of $$A$$).

Completeness
A signalizer functor is said to be complete if there exists a subgroup $$\theta(G)$$ such that for every $$a$$ in $$A$$, $$\theta(a)$$ is the same as the centralizer of $$a$$ in $$\theta(G)$$.

Solvability
A signalizer functor is said to be solvable if all its images are solvable groups, viz $$\theta$$ is a solvable signalizer functor if for every $$a$$, $$\theta(a)$$ is a solvable group.

Signalizer functor theorem
If $$m(A)$$ is at least 3, every signalizer functor on $$A$$ is complete.

A special case of this is the solvable signalizer functor theorem.