Group of Glauberman type for a prime

The article defines a property of groups, where the definition may be in terms of a particular prime that serves as parameter
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Definition

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 ZJ-functor is a 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 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.

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

Further information: Equivalence of definitions of group of Glauberman type for a prime

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.

Relation with other properties

Stronger properties

• p-nilpotent group
• strongly p-solvable group (for an odd prime $p$): For full proof, refer: strongly p-solvable implies Glauberman type for odd p
• Group that is both p-stable and p-constrained for an odd prime $p$: For full proof, refer: p-constrained and p-stable implies Glauberman type for odd p

Weaker properties

• Group in which the ZJ-functor controls fusion for a prime: For full proof, refer: Glauberman type implies ZJ-functor controls fusion

Examples

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 $G$.

Group	Order	Prime of interest	$p$-Sylow subgroup	$Z(J(P))$	Comment
symmetric group:S3	6	3	A3 in S3	A3 in S3	Here, $P = Z(J(P))$ and it itself is normal in $G$.
alternating group:A4	12	2	V4 in A4	V4 in A4	Here, $P = Z(J(P))$ and it itself is normal in $G$.

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