Equivalence of definitions of group of Glauberman type for a prime

This article gives a proof/explanation of the equivalence of multiple definitions for the term group of Glauberman type for a prime
View a complete list of pages giving proofs of equivalence of definitions

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. 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.