Signalizer functor

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 ${\displaystyle p'}$-subgroup of a finite group ${\displaystyle G}$, which has a good chance of being normal in ${\displaystyle G}$, by taking as generators certain ${\displaystyle p'}$-subgroups of the centralizers of nonidentity elements in one or several given noncyclic elementary abelian ${\displaystyle p}$-subgroups of ${\displaystyle G.}$ The technique has origins in the Feit–Thompson theorem, and was subsequently developed by many people including Gorenstein (1969) who defined signalizer functors, Glauberman (1976) who proved the Solvable Signalizer Functor Theorem for solvable groups, and McBride (1982a, 1982b) 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.

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