and the result follows. See either of the books or for details. This is used in both the proof and applications of the Solvable Signalizer Functor Theorem. To begin, notice that it quickly implies the claim that if is complete, then its completion is the group defined above. Normal completion The completion of a signalizer functor has a "good chance" of being normal in according to the top of the article. Here, the coprime action fact will be used to motivate this claim. Let be a complete -signalizer functor on Let be a noncyclic subgroup of Then the coprime action fact together with the balance condition imply that . To see this, observe that because is B-invariant, we have The equality above uses the coprime action fact, and the containment uses the balance condition. Now, it is often the case that satisfies an "equivariance" condition, namely that for each and nonidentity The superscript denotes conjugation by For example, the mapping (which is often a signalizer functor!) satisfies this condition. If satisfies equivariance, then the normalizer of will normalize It follows that if is generated by the normalizers of the noncyclic subgroups of then the completion of (i.e. W) is normal in References Signalizer functor