Set-stabilizers in solvable permutation groups
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David Gluck
Abstract
Let 𝐺 be a finite solvable permutation group. Then, modulo a possibly trivial normal elementary abelian 3-subgroup, some set-stabilizer in 𝐺 is a 2-group.
Dedicated to the memory of Marty Isaacs
Acknowledgements
I thank Luca Sabatini for helpful communications.
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Communicated by: Michael Giudici
References
[1] L. Babai, Asymmetric coloring of locally finite graphs and profinite permutation groups: Tucker’s conjecture confirmed, J. Algebra 607 (2022), 64–106. 10.1016/j.jalgebra.2021.10.033Suche in Google Scholar
[2] D. Gluck, Trivial set-stabilizers in finite permutation groups, Canad. J. Math. 35 (1983), no. 1, 59–67. 10.4153/CJM-1983-005-2Suche in Google Scholar
[3] D. Gluck and O. Manz, Prime factors of character degrees of solvable groups, Bull. Lond. Math. Soc. 19 (1987), no. 5, 431–437. 10.1112/blms/19.5.431Suche in Google Scholar
[4] D. Gluck and T. R. Wolf, Brauer’s height conjecture for 𝑝-solvable groups, Trans. Amer. Math. Soc. 282 (1984), no. 1, 137–152. 10.1090/S0002-9947-1984-0728707-2Suche in Google Scholar
[5] D. Gluck and T. R. Wolf, Defect groups and character heights in blocks of solvable groups. II, J. Algebra 87 (1984), no. 1, 222–246. 10.1016/0021-8693(84)90168-6Suche in Google Scholar
[6] I. M. Isaacs, Character Theory of Finite Groups, Pure Appl. Math. 69, Academic Press, New York, 1976. Suche in Google Scholar
[7] O. Manz and T. R. Wolf, Representations of Solvable Groups, London Math. Soc. Lecture Note Ser. 185, Cambridge University, Cambridge, 1993. 10.1017/CBO9780511525971Suche in Google Scholar
[8] L. Sabatini, On stabilizers in finite permutation groups, preprint (2024), https://arxiv.org/abs/2411.18534. Suche in Google Scholar
[9] A. Seress, Primitive groups with no regular orbits on the set of subsets, Bull. Lond. Math. Soc. 29 (1997), no. 6, 697–704. 10.1112/S0024609397003536Suche in Google Scholar
[10]
T. R. Wolf,
Solvable and nilpotent subgroups of
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Artikel in diesem Heft
- Frontmatter
- Bounding the degree of generic sharp transitivity
- Structure of an exotic 2-local subgroup in 𝐸7(𝑞)
- Set-stabilizers in solvable permutation groups
- Regular saturated formations of finite soluble groups
- On the 𝑝-length and 𝔘-class of a 𝑝-solvable finite group
- Finite 2-groups with exactly three automorphism orbits
- Finite class 2 nilpotent and Heisenberg groups
- Minimal degrees for faithful permutation representations of groups of order 𝑝5 where 𝑝 is an odd prime
- 𝜏-Tilting finiteness of group algebras of semidirect products of abelian 𝑝-groups and abelian 𝑝′-groups
- On the Grossman representations of the automorphism groups of free groups
- Strong indecomposability of the outer automorphism groups of nonabelian free profinite groups
- The proper geometric dimension of the mapping class group of an orientable surface with punctures
- A note on complex hyperbolic lattices and strict hyperbolization
Artikel in diesem Heft
- Frontmatter
- Bounding the degree of generic sharp transitivity
- Structure of an exotic 2-local subgroup in 𝐸7(𝑞)
- Set-stabilizers in solvable permutation groups
- Regular saturated formations of finite soluble groups
- On the 𝑝-length and 𝔘-class of a 𝑝-solvable finite group
- Finite 2-groups with exactly three automorphism orbits
- Finite class 2 nilpotent and Heisenberg groups
- Minimal degrees for faithful permutation representations of groups of order 𝑝5 where 𝑝 is an odd prime
- 𝜏-Tilting finiteness of group algebras of semidirect products of abelian 𝑝-groups and abelian 𝑝′-groups
- On the Grossman representations of the automorphism groups of free groups
- Strong indecomposability of the outer automorphism groups of nonabelian free profinite groups
- The proper geometric dimension of the mapping class group of an orientable surface with punctures
- A note on complex hyperbolic lattices and strict hyperbolization
