of finite simple groups, showed that is a -group without the assumption that is solvable. Completeness The terminology of completeness is often used in discussions of signalizer functors. Let be a signalizer functor as above, and consider the set И of all -invariant -subgroups of satisfying the following condition: for all nonidentity For example, the subgroups belong to И by the balance condition. The signalizer functor is said to be complete if И has a unique maximal element when ordered by containment. In this case, the unique maximal element can be shown to coincide with above, and is called the completion of . If is complete, and turns out to be solvable, then is said to be solvably complete. Thus, the Solvable Signalizer Functor Theorem can be rephrased by saying that if has at least three generators, then every solvable -signalizer functor on is solvably complete. Examples of signalizer functors The easiest way to obtain a signalizer functor is to start with an -invariant -subgroup of and define for all nonidentity In practice, however, one begins with and uses it to construct the -invariant -group. The simplest signalizer functor used in practice is this: A few words of caution are needed here. First, note that as defined above is indeed an -invariant -subgroup of because is abelian. However, some additional assumptions are needed to show that this satisfies the balance condition. One sufficient criterion is that for each nonidentity the group is solvable (or -solvable or even -constrained). Verifying the balance condition for this under this assumption requires a famous lemma, known as Thompson's -lemma. (Note, this lemma is also called Thompson's -lemma, but the in this use must not be confused with the appearing in the definition of a signalizer functor!) Coprime action To obtain a better understanding of signalizer functors, it is essential to know the following general fact about finite groups: Let be an abelian noncyclic group acting on the finite group Assume that the orders of and are relatively prime. Then To prove this fact, one uses the Schur–Zassenhaus theorem to show that for each prime dividing the order of the group has an -invariant Sylow -subgroup. This reduces to the case where is a -group. Then an argument by induction on the order of reduces the statement further to the case where is elementary abelian with acting irreducibly. This forces the group to be cyclic,