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.
View similar facts
Contents
Name
This result was proved by Glauberman, and is sometimes termed the Glauberman ZJ-theorem.
Statement
General statement
Suppose is a finite group and is an odd prime number. If is both p-constrained and p-stable, then is a group of Glauberman type for .
More explicitly, if 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 is a finite group and is an odd prime number. Suppose is trivial, i.e., has no nontrivial normal -subgroup. If is both p-constrained and p-stable, then is a group of Glauberman type for . Explicitly, the following equivalent conditions are satisfied:
- For one (and hence every) -Sylow subgroup of , is a normal subgroup of .
- For one (and hence every) -Sylow subgroup of , is a characteristic subgroup of .
Definitions used
Term | Definition for finite group and prime | Definition for finite group and prime where is trivial |
---|---|---|
p-constrained group | For one (and hence any) -Sylow subgroup of , | , i.e., the p-core is a self-centralizing subgroup |
p-stable group | Either is trivial or has a nontrivial normal -subgroup and satisfies the following: Suppose is a -subgroup of such that is normal in . Then, if is a -subgroup of with the property that is trivial, we have: . | Either is trivial or has a nontrivial normal -subgroup and satisfies the following: Suppose is a -subgroup of such that is normal in . Then, if is a -subgroup of with the property that is trivial, we have: . |
group of Glauberman type for prime | For one (and hence every) -Sylow subgroup of , where is the p'-core, is the ZJ-subgroup of , and denotes the normalizer operation. | For one (and hence every) -Sylow subgroup of , is normal in . |
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 , strongly p-solvable group (which is the same as a p-solvable group for , and additionally must avoid SL(2,3) as a subquotient for ) must be of Glauberman type for . | strongly p-solvable implies p-stable, p-solvable implies p-constrained | chain of implication | |
Glauberman-Thompson normal p-complement theorem | Suppose is a finite group and is an odd prime number. Let be a -Sylow subgroup. Then, if possesses a normal p-complement, so does . In other words, is a retract of . | -- | 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. |
Similar facts
- P-constrained and p-stable implies normalizer of D*-subgroup generates whole group with p'-core for odd p: Here, we replace the ZJ-subgroup with the D*-subgroup and the same statement holds.
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 is -constrained, so is | |||
4 | equivalence of definitions of p-stable group | Reduction to p'-core-free case | If is -stable, so is | |||
5 | equivalence of definitions of group of Glauberman type for a prime | Reduction to p'-core-free case | If is of Glauberman type, so is |
Proof
Given: An odd prime , a finite group that is both -constrained and -stable. In particular, is nontrivial if is nontrivial.
To prove: .
Proof:
The case where is trivial
We restate the Given and To prove.
Given: An odd prime , a finite group that is both -constrained and -stable. In particular, is nontrivial if is nontrivial. Further, is trivial.
To prove: is a normal subgroup of .
Proof:
Step no. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|
1 | Fact (1) | is an odd prime, is -constrained and -stable | [SHOW MORE] | ||
2 | is normal in | Fact (2) | is an odd prime, is -stable | Step (1) | [SHOW MORE] |
The general case
Given: A finite group , an odd prime such that is both -constrained and -stable.
To prove: is of Glauberman type for
Proof:
Step no. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|
1 | is -constrained | Fact (3) | is -constrained | ||
2 | is -stable | Fact (4) | is -stable | ||
3 | is -core-free | Follows from definition of | |||
4 | is of Glauberman type for | Steps (1)-(3), first half of proof for -core-free case | step-combination direct | ||
5 | is of Glauberman type for | 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}
Journal references
- A characteristic subgroup of a p-stable group by George Isaac Glauberman, , Volume 20, Page 1101 - 1135(Year 1968): ^{}^{More info}