Thompson transitivity theorem
Contents
Statement
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 .
Related facts
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
Opposite facts
Corollaries/applications
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.
Facts used
- 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
Proof
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
References
Journal references
- 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 page}^{More info}
Textbook references
- Finite Groups by Daniel Gorenstein, ISBN 0821843427, Page 292, Theorem 5.4, Chapter 8 (p-constrained and p-stable groups), Section 5 (The Thompson transitivity theorem), ^{More info}