# Glauberman-Thompson normal p-complement theorem

This article gives the statement, and possibly proof, of a normal p-complement theorem: necessary and/or sufficient conditions for the existence of a Normal p-complement (?). In other words, it gives necessary and/or sufficient conditions for a given finite group to be a P-nilpotent group (?) for some prime number $p$.
View other normal p-complement theorems
This article states a fact about the behavior of a finite group relative to a prime number. This fact is true only for odd primes, i.e., it breaks down for the prime two.
View similar facts

## Statement

### Direct statement

Suppose $G$ is a finite group and $p$ is an odd prime number. Let $P$ be a $p$-Sylow subgroup. Then, if $N_G(Z(J(P))$ possesses a normal p-complement, so does $G$. In other words, $P$ is a retract of $G$.

### In terms of functors and control of complements

For an odd prime $p$, the ZJ-functor is a characteristic p-functor that controls normal p-complements in every finite group.

## Facts used

1. Generalized Glauberman-Thompson normal p-complement theorem
2. Strongly p-solvable implies Glauberman type for odd p: This states that in a strongly p-solvable group, the ZJ-subgroup functor is a characteristic p-functor whose normalizer generates whole group with p'-core. In particular, for a p'-core-free strongly p-solvable group, the ZJ-subgroup of any p-Sylow subgroup is normal (in fact, characteristic) in the whole group. See also group of Glauberman type for a prime.

## Proof

The proof follows directly from Facts (1) and (2). Fact (2) basically says that the ZJ-subgroup functor satisfies the necessary conditions for us to be able to apply Fact (1).