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 functor comes from a subgroup. The idea is to try to construct a -subgroup of a finite group , which has a good chance of being normal in , by taking as generators certain -subgroups of the centralizers of nonidentity elements in one or several given noncyclic elementary abelian -subgroups of The technique has origins in the Feit–Thompson theorem, 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 either has local characteristic two, or is of component type. It thus plays a major role in the classification of finite simple groups. 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 , the group is contained in For every nonidentity , 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 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 and mentioned above says that this will be the case if is solvable and has at least three generators. The theorem also states that under these assumptions, itself will be solvable. Several earlier versions of the theorem were proven: proved this under the stronger assumption that had rank at least 5. proved this under the assumption that 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, 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