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

This article states a fact about the behavior of a finite group relative to a prime number. This fact is true only for odd primes, i.e., it breaks down for the prime two.
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## Name

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

## Statement

### 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 $G$.

## Definitions used

Term Definition for finite group $G$ and prime $p$ Definition for finite group $G$ and prime $p$ where $O_{p'}(G)$ is trivial
p-constrained group For one (and hence any) $p$-Sylow subgroup $P$ of $G$, $C_G(P \cap O_{p',p}(G)) \le O_{p',p}(G)$ $C_G(O_p(G)) \le O_p(G)$, i.e., the p-core $O_p(G)$ is a self-centralizing subgroup
p-stable group Either $G$ is trivial or $G$ has a nontrivial normal $p$-subgroup and $G$ satisfies the following: Suppose $P$ is a $p$-subgroup of $G$ such that $O_{p'}(G)P$ is normal in $G$. Then, if $A$ is a $p$-subgroup of $N_G(P)$ with the property that $[[P,A],A]$ is trivial, we have: $AC_G(P)/C_G(P) \le O_p(N_G(P)/C_G(P))$. Either $G$ is trivial or $G$ has a nontrivial normal $p$-subgroup and $G$ satisfies the following: Suppose $P$ is a $p$-subgroup of $G$ such that $P$ is normal in $G$. Then, if $A$ is a $p$-subgroup of $N_G(P)$ with the property that $[[P,A],A]$ is trivial, we have: $AC_G(P)/C_G(P) \le O_p(N_G(P)/C_G(P))$.
group of Glauberman type for prime $p$ For one (and hence every) $p$-Sylow subgroup $P$ of $G$, $G = O_{p'}(G)N_G(Z(J(P)))$ where $O_{p'}(G)$ is the p'-core, $Z(J(P))$ is the ZJ-subgroup of $P$, and $N_G$ denotes the normalizer operation. For one (and hence every) $p$-Sylow subgroup $P$ of $G$, $Z(J(P))$ is normal in $G$.

## Related facts

### Applications

Application name Full statement Intermediaries (if indirect application) Other facts used for the application How does it use this fact?
strongly p-solvable implies Glauberman type for odd p For any odd prime $p$, strongly p-solvable group (which is the same as a p-solvable group for $p \ge 5$, and additionally must avoid SL(2,3) as a subquotient for $p = 3$) must be of Glauberman type for $p$. strongly p-solvable implies p-stable, p-solvable implies p-constrained chain of implication
Glauberman-Thompson normal p-complement theorem Suppose $G$ is a finite group and $p$ is an odd prime number. Let $P$ be a $p$-Sylow subgroup. Then, if $N_G(Z(J(P))$ possesses a normal p-complement, so does $G$. In other words, $P$ is a retract of $G$. -- Characterization of minimal counterexamples to a characteristic p-functor controlling normal p-complements, p-solvable implies p-constrained, strongly p-solvable implies p-stable The normal p-complement theorem is true for all groups, not only for groups that are p-constrained and p-stable. The theorem is provide using the minimal counterexample approach, and it turns out that the restrictions imposed on a minimal counterexample help us reduce to the case of p-constrained and p-stable groups.

## Facts used

The table below lists key facts used directly and explicitly in the proof. Fact numbers as used in the table may be referenced in the proof. This table need not list facts used indirectly, i.e., facts that are used to prove these facts, and it need not list facts used implicitly through assumptions embedded in the choice of terminology and language.
Fact no. Statement Steps in the proof where it is used Qualitative description of how it is used What does it rely on Difficulty level Other applications
1 p-constrained and p-stable implies abelian normal subgroup of Sylow subgroup is contained in (p',p)-core Step (1) of the p'-core-free case  ?  ?
2 Glauberman's theorem on intersection with the ZJ-subgroup Step (2) of the p'-core-free case  ?  ?
3 equivalence of definitions of p-constrained group Reduction to p'-core-free case If $G$ is $p$-constrained, so is $G/O_{p'}(G)$
4 equivalence of definitions of p-stable group Reduction to p'-core-free case If $G$ is $p$-stable, so is $G/O_{p'}(G)$
5 equivalence of definitions of group of Glauberman type for a prime Reduction to p'-core-free case If $G/O_{p'}(G)$ is of Glauberman type, so is $G$

## 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:

Step no. Assertion/construction Facts used Given data used Previous steps used Explanation
1 $Z(J(P)) \le O_p(G)$ Fact (1) $p$ is an odd prime, $G$ is $p$-constrained and $p$-stable [SHOW MORE]
2 $Z(J(P))$ is normal in $G$ Fact (2) $p$ is an odd prime, $G$ is $p$-stable Step (1) [SHOW MORE]

### 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:

Step no. Assertion/construction Facts used Given data used Previous steps used Explanation
1 $G/O_{p'}(G)$ is $p$-constrained Fact (3) $G$ is $p$-constrained
2 $G/O_{p'}(G)$ is $p$-stable Fact (4) $G$ is $p$-stable
3 $G/O_{p'}(G)$ is $p'$-core-free Follows from definition of $O_{p'}$
4 $G/O_{p'}(G)$ is of Glauberman type for $p$ Steps (1)-(3), first half of proof for $p'$-core-free case step-combination direct
5 $G$ is of Glauberman type for $p$ Fact (5) Step (4) fact-step combination direct

## References

### Textbook references

• Finite Groups by Daniel Gorenstein, ISBN 0821843427, Page 279, Theorem 2.11, Chapter 8 (p-constrained and p-stable groups), More info