Glauberman's theorem on intersection with the ZJ-subgroup
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
Statement
Suppose is an odd prime number, is a P-stable group (?), and is a -Sylow subgroup. Suppose, further, that is a nontrivial normal -subgroup of . Then, is also a nontrivial normal -subgroup of .
Related facts
- Glauberman-Thompson normal p-complement theorem
- p-constrained and p-stable implies Glauberman type for odd p
- Glauberman's replacement theorem
Facts used
- Characteristic of normal implies normal
- Normality is strongly intersection-closed
- Glauberman's replacement theorem
- Sylow satisfies permuting transfer condition
- Frattini's argument
- Any class two normal subgroup whose commutator subgroup is in the ZJ-subgroup normalizes an abelian subgroup of maximum order: This is an immediate corollary of Glauberman's replacement theorem.
- Normality is centralizer-closed
- Thompson's replacement theorem
Proof
Given: A -stable group . A nontrivial normal -subgroup of . A -Sylow subgroup of .
To prove: is a nontrivial normal -subgroup of .
Proof: We prove this by assuming a counterexample group , and a nontrivial normal -subgroup of that provides a counterexample of minimum order for that particular prime .
For convenience, denote .
Note that since is a nontrivial normal -subgroup, , and in particular, . Note also that is nontrivial, because contains the center of , and is thus normality-large.
First part of the proof: showing that satisfies the conditions for Glauberman's replacement theorem
To prove: is the normal closure of , has class two, .
Proof:
- is the normal closure of : Suppose is the normal closure of . Then, . is thus a nontrivial normal -subgroup containing . Thus, . By minimality of the order of as a counterexample, we conclude that .
- Let be the commutator subgroup. Then, is normal in : , being the commutator subgroup of , is characteristic in . By fact (1), we obtain that is normal in .
- is normal in : Since is a -group, is a proper subgroup of . Thus, by minimality of as a counterexample, and the fact that is normal in by the preceding step, is normal in .
- : and are both normal in , so by fact (2), is also normal in . Thus, . Combining this with the obvious fact that , we get .
- For any conjugate of , : Suppose . Since is normal, . Thus, . By step (3), is normal, so .
- : Since is generated by all the conjugates of (step (1)), is the normal closure in of the subgroups generated by for all conjugates of . But each of these is contained in , which is normal in and hence in . Thus, , and in particular, .
- is trivial: This is because both are subgroups of the abelian group .
- is trivial for every conjugate of in : This follows by reasoning similar to step (5), using that is normal in (step (2)).
- , so and has class two: By the previous step, contains every conjugate of . By step (1), is generated by these conjugates, so . In particular, , so has class two.
Second part of the proof: the normal core of normalizer of does not contain
Given: Let be the largest normal subgroup contained in . In other words, is the normal core of normalizer of .
To prove: does not contain .
Proof: We prove that if contains , then is normal in , a contradiction.
- is a -Sylow subgroup of : This follows from fact (4).
- : This follows from fact (5), the previous step, and the fact that is normal.
- : If , then , so by the definition of , = J(P \cap L)</math>.
- : From the previous step, , so is characteristic in . Thus, by fact (1), is normal in , so is contained in the normalizer .
- : This follows from steps (2) and (4).
- is normal in : normalizes by assumption. normalizes , and since is normal, normalizes . Thus, contains both and . The previous step thus forces .
Meat of the proof
We continue to denote by the normal core of normalizer of .
- By fact (6), there exists an abelian subgroup of maximum order in such that normalizes . In particular, : In the first part, we established that satisfies the conditions for fact (6): is class two normal and . Thus, fact (6) yields that there exists an abelian subgroup of maximum order in such that normalizes . Since normalizes , . Further, since is abelian, , so .
- Let . Then, : Both and are -subgroups. Since is normal in , so is . Further, , again since is normal, and as established in the previous step. Thus, being -stable yields . Using and yields the result.
- : Since centralizes , centralizes and thus normalizes . Also, by fact (7) and the given datum that is normal, is normal. By the fact that is the unique largest normal subgroup that normalizes , we get that is normal.
- : This is a direct consequence of the previous two steps.
- is trivial: PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
- : This is a direct consequence of the previous two steps.
- is contained in : is a subgroup of that, by the preceding step, has an abelian subgroup of maximum order in . Thus, the maximum possible order of an abelian subgroup in is the same as in . Thus, the subgroups generating form a subset of the subgroups generating , so is contained in .
- is abelian: PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
- There exists an abelian subgroup of of maximum order that is not contained in : This follows from the second part of the proof, which showed that is not contained in .
- : If were trivial, then replacing by in steps (2)-(6) of the above argument would yields contradicting the previous step.
- Among the set of possibilities for , pick such that has maximum order. Then, there exists such that normalizes and is properly contained in : This follows from fact (8) (Thompson's replacement theorem).
- : This follows from the maximality of among the subgroups of maximum order in that are not contained in .
References
Textbook references
- Finite Groups by Daniel Gorenstein, ISBN 0821843427, Page 278, Theorem 2.10, Chapter 8 (p-constrained and p-stable groups), Section 8.2 (Glauberman's theorem), ^{More info}