P-constrained and p-stable implies Glauberman type for odd p

Name
This result was proved by Glauberman, and is sometimes termed the Glauberman ZJ-theorem.

General statement
Suppose $$G$$ is a finite group and $$p$$ is an odd prime number. If $$G$$ is both p-constrained and  p-stable, then $$G$$ is a  group of Glauberman type for $$p$$.

More explicitly, if $$G$$ is both p-constrained and  p-stable, then the  ZJ-functor is a  characteristic p-functor whose normalizer generates whole group with p'-core.

Statement for p'-core-free finite groups
Suppose $$G$$ is a finite group and $$p$$ is an odd prime number. Suppose $$O_{p'}(G)$$ is trivial, i.e., $$G$$ has no nontrivial normal $$p'$$-subgroup. If $$G$$ is both p-constrained and  p-stable, then $$G$$ is a  group of Glauberman type for $$p$$. Explicitly, the following equivalent conditions are satisfied:


 * 1) For one (and hence every) $$p$$-Sylow subgroup $$P$$ of $$G$$, $$Z(J(P))$$ is a normal subgroup of $$G$$.
 * 2) For one (and hence every) $$p$$-Sylow subgroup $$P$$ of $$G$$, $$Z(J(P))$$ is a characteristic subgroup of $$G$$.

Similar facts

 * P-constrained and p-stable implies normalizer of D*-subgroup generates whole group with p'-core for odd p: Here, we replace the ZJ-subgroup with the D*-subgroup and the same statement holds.

Proof
Given: An odd prime $$p$$, a finite group $$G$$ that is both $$p$$-constrained and $$p$$-stable. In particular, $$O_p(G)$$ is nontrivial if $$G$$ is nontrivial.

To prove: $$\! G = O_{p'}(G)N_G(Z(J(P)))$$.

Proof:

The case where $$O_{p'}(G)$$ is trivial
We restate the Given and To prove.

Given: An odd prime $$p$$, a finite group $$G$$ that is both $$p$$-constrained and $$p$$-stable. In particular, $$O_p(G)$$ is nontrivial if $$G$$ is nontrivial. Further, $$O_{p'}(G)$$ is trivial.

To prove: $$Z(J(P))$$ is a normal subgroup of $$G$$.

Proof:

The general case
Given: A finite group $$G$$, an odd prime $$p$$ such that $$G$$ is both $$p$$-constrained and $$p$$-stable.

To prove: $$G$$ is of Glauberman type for $$p$$

Proof: