Group of Glauberman type for a prime

Definition for a general finite group
Suppose $$G$$ is a finite group and $$p$$ is a prime number. We say that $$G$$ is of Glauberman type with respect to $$p$$ if the defining ingredient::ZJ-functor is a defining ingredient::characteristic p-functor whose normalizer generates whole group with p'-core. In the discussion below, $$Z(J(P))$$ denotes the subgroup obtained by applying the defining ingredient::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 defining ingredient::join of abelian subgroups of maximum order.

Explicitly, the following equivalent conditions are satisfied:


 * 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 $$K$$.

Definition for p'-core-free finite group
Suppose $$G$$ is a finite group and $$p$$ is a prime number. Suppose the p'-core $$O_{p'}(G)$$ is trivial, i.e., $$G$$ has no nontrivial normal subgroup of order not divisible by $$p$$. Then, $$G$$ is termed a group of Glauberman type for $$p$$ if it satisfies the following equivalent conditions:


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

Equivalence of definitions and its significance
It turns out that, for a finite group $$G$$ and prime number $$p$$:

$$G$$ is a group of Glauberman type for $$p$$ $$\iff$$ $$G/O_{p'}(G)$$ is a group of Glauberman type for $$p$$

This can be used to provide an alternative definition of group of Glauberman type.

Stronger properties

 * Weaker than::p-nilpotent group
 * Weaker than::strongly p-solvable group (for an odd prime $$p$$):
 * Group that is both p-stable and p-constrained for an odd prime $$p$$:

Weaker properties

 * Stronger than::Group in which the ZJ-functor controls fusion for a prime:

p'-core-free examples
In these example groups, $$Z(J(P))$$ is normal in $$G$$ for one (and hence every) $$p$$-Sylow subgroup $$P$$ of $$G$$.

On the other hand, symmetric group:S4 is not a group of Glauberman type for the prime 2.