Equivalence of definitions of group of Glauberman type for a prime

Statement
Suppose $$G$$ is a finite group and $$p$$ is a prime number. In the discussion below, $$Z(J(P))$$ denotes the subgroup obtained by applying the ZJ-functor to $$P$$. The ZJ-functor is defined as the center of the Thompson subgroup $$J(P)$$, which in turn is defined as the join of abelian subgroups of maximum order. The following are equivalent:


 * 1) For one (and hence every) $$p$$-Sylow subgroup $$P$$ of $$G$$, $$G = O_{p'}(G)N_G(Z(J(P)))$$: Here, $$O_{p'}(G)$$ denotes the $$p'$$-core of $$G$$,
 * 2) For one (and hence every) $$p$$-Sylow subgroup $$P$$ of $$G$$, the image of $$Z(J(P))$$ in the quotient $$G/O_{p'}(G)$$ is a normal subgroup of $$G/O_{p'}(G)$$.
 * 3) For one (and hence every) $$p$$-Sylow subgroup $$Q$$ of $$K = G/O_{p'}(G)$$, $$Z(J(Q))$$ is a normal subgroup of $$K$$.
 * 4) For one (and hence every) $$p$$-Sylow subgroup $$Q$$ of $$K = G/O_{p'}(G)$$, $$Z(J(Q))$$ is a characteristic subgroup of $$K$$.

Facts used

 * 1) uses::Equivalence of definitions of characteristic p-functor whose normalizer generates whole group with p'-core

Proof
The proof follows directly from Fact (1), where the characteristic p-functor that we use is the ZJ-functor.