Characteristic p-functor that controls normal p-complements

This article defines a property that can be evaluated for a characteristic p-functor in the context of a finite group.|View other such properties

Definition

Suppose $p$ is a prime number and $W$ is a characteristic p-functor. We say that $W$ controls normal $p$-complements in a finite group $G$ if the following holds: if there exists a $p$-Sylow subgroup $P$, such that $N_{G}(W(P))$ possesses a normal p-complement, $G$ also possesses a normal $p$-complement.

We say that $W$ controls normal $p$-complements in general if it controls normal $p$-complements in every finite group.

Facts

• Characterization of minimal counterexamples to a characteristic p-functor controlling normal p-complements
• The generalized Glauberman-Thompson normal p-complement theorem gives sufficient conditions for a characteristic p-functor to control normal p-complements in every finite group.
• The Glauberman-Thompson normal p-complement theorem states that the ZJ-functor controls normal $p$-complements for odd primes $p$.
• The Glauberman-Solomon normal p-complement theorem states that the D*-subgroup functor controls normal $p$-complements for odd primes $p$.