Thompson transitivity theorem
- 1 Statement
- 2 Related facts
- 3 Application to groups of particular kinds
- 4 Facts used
- 5 Proof
- 6 References
Suppose is a finite group and is a prime number. Suppose further that is a Group in which every p-local subgroup is p-constrained (?): the normalizer of any non-identity -subgroup of is a -constrained group.
Suppose . In other words, is maximal among abelian normal subgroups inside some -Sylow subgroup (note that normality is in , not in ) and further, has rank at least three. In other words, any generating set for must comprise at least three elements.
Then, permutes transitively under conjugation the set of all maximal -invariant -subgroups of for any prime .
Necessity of assumptions
- Analogue of Thompson transitivity theorem fails for abelian subgroups of rank two
- Analogue of Thompson transitivity theorem fails for groups in which not every p-local subgroup is p-constrained
Application to groups of particular kinds
|Group property||Meaning||Is the property stronger than being a group in which every p-local subgroup is p-constrained?||Is the conclusion of the Thompson transitivity theorem true even if we drop the rank at least three requirement? If so, why?|
|finite nilpotent group (includes the case of finite abelian group)||direct product of its Sylow subgroups, so all Sylow subgroups are normal Sylow subgroups||Yes||Yes, because a maximal -invariant -subgroup in this case is simply a -Sylow subgroup, and it is unique because the group is nilpotent).|
|p-nilpotent group||there is a normal p-complement, i.e., the -Sylow subgroup is a retract||Yes||Unclear|
|finite solvable group||no simple non-abelian composition factors||Yes||Unclear|
|p-solvable group||has a chief series where all chief factors are either -groups or -groups||Yes||Unclear|
|N-group||every local subgroup is a solvable group||Yes||Unclear|
|minimal simple group||simple non-abelian group in which every proper subgroup is solvable||Yes||Unclear|
|group in which every p-local subgroup is p-solvable||every p-local subgroup is a p-solvable group||Yes||Unclear|
In particular, we note that the Thompson transitivity theorem applies to minimal simple groups. This is an important ingredient in the classification of finite minimal simple groups and the odd-order theorem.
- Corollary of centralizer product theorem for rank at least three
- Prime power order implies nilpotent, Nilpotent implies normalizer condition
- Centralizer of coprime automorphism in homomorphic image equals image of centralizer
- Lemma on containment in p'-core for Thompson transitivity theorem
This proof uses a tabular format for presentation. Provide feedback on tabular proof formats in a survey (opens in new window/tab) | Learn more about tabular proof formats|View all pages on facts with proofs in tabular format
Given: A finite group , a prime . Every -local subgroup of is -constrained. is maximal among abelian normal subgroups in some -Sylow subgroup . Also, has rank at least three.
To prove: permutes transitively under conjugation the set of all maximal -invariant -subgroups of for any prime .
Proof: First, note that conjugation by any element in sends invariant subgroups to -invariant subgroups. In particular, it permutes the set of maximal -invariant -subgroups under conjugation.
Let be the orbits of maximal -invariant -subgroups of . We want to prove that . We break the proof into two steps.
The intersection of any two subgroups in distinct orbits is trivial
To prove: If for , then is trivial.
Proof: Suppose not. Among all possible pairs of subgroups in distinct orbits for which the intersection is nontrivial, pick a pair such that the intersection has largest possible order. Call the intersection .
|Step no.||Assertion/construction||Facts used||Given data/assumptions used||Previous steps used||Explanation|
|1||is a proper subgroup in both and||are both maximal -invariant -subgroups.||[SHOW MORE]|
|2||properly contains for||Fact (2)||are -groups, i.e., groups of prime power order||Step (1)||Fact-step-given-combination direct|
|3||Let , for . Then, are -invariant and hence acts on , both of which are -groups.||are -invariant||Steps (1), (2)||[SHOW MORE]|
|4||There exists a non-identity element of such that and are both nontrivial.||Fact (1)||is a finite abelian -group of rank at least three, and .||Step (3)||[SHOW MORE]|
|5||for||Fact (3)||, so is acting as a coprime automorphism.||Step (3)||[SHOW MORE]|
|6||is not contained in for||Steps (4), (5)||[SHOW MORE]|
|7||is -constrained||By hypothesis, normalizers of non-identity -subgroups are -constrained.||direct, noting that , so is a -subgroup.|
|8||Each is contained in .||Fact (4)||Fact-direct (need to explain).|
|9||Each is contained in .||Steps (2), (6)||[SHOW MORE]|
There is in fact only one orbit
- Solvability of groups of odd order by Walter Feit and John Griggs Thompson, Pacific Journal of Mathematics, Volume 13, Page 775 - 1029(Year 1963): This 255-page long paper gives a proof that odd-order implies solvable: any odd-order group (i.e., any finite group whose order is odd) is a solvable group.Project Euclid pageMore info