# Characteristic p-functor that controls normal p-complements

From Groupprops

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 is a prime number and is a characteristic p-functor. We say that controls normal -complements in a finite group if the following holds: if there exists a -Sylow subgroup , such that possesses a normal p-complement, also possesses a normal -complement.

We say that controls normal -complements in general if it controls normal -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 -complements for odd primes .
- The Glauberman-Solomon normal p-complement theorem states that the D*-subgroup functor controls normal -complements for odd primes .