Extensions of a theorem of P. Hall on indexes of maximal subgroups
-
Antonio Beltrán
and
Changguo Shao
Abstract
We extend a classical theorem of P. Hall that claims that if the index of every maximal subgroup of a finite group G is a prime or the square of a prime, then G is solvable. Precisely, we prove that if one allows, in addition, the possibility that every maximal subgroup of G is nilpotent instead of having prime or squared-prime index, then G continues to be solvable. Likewise, we obtain the solvability of G when we assume that every proper non-maximal subgroup of G lies in some subgroup of index prime or squared prime.
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 12071181
Award Identifier / Grant number: 12471017
Funding statement: This work is supported by the National Nature Science Fund of China (No. 12071181 and No. 12471017). Antonio Beltrán is also supported by Generalitat Valenciana, Proyecto CIAICO/2021/193. Changguo Shao is also supported by Natural Science Research Start-up Foundation of Recruiting Talents of Nanjing University of Posts and Telecommunications (No. NY222090, No. NY222091).
References
[1] M. Bianchi, A. Gillio Berta Mauri and P. Hauck, On finite groups with nilpotent Sylow-normalizers, Arch. Math. (Basel) 47 (1986), no. 3, 193–197. 10.1007/BF01191993Search in Google Scholar
[2] J. N. Bray, D. F. Holt and C. M. Roney-Dougal, The Maximal Subgroups of the Low-Dimensional Finite Classical Groups, London Math. Soc. Lecture Note Ser. 407, Cambridge University, Cambridge, 2013. 10.1017/CBO9781139192576Search in Google Scholar
[3]
Y. Bugeaud and P. Mihăilescu,
On the Nagell–Ljunggren equation
[4]
J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson,
[5] R. M. Guralnick, Subgroups of prime power index in a simple group, J. Algebra 81 (1983), no. 2, 304–311. 10.1016/0021-8693(83)90190-4Search in Google Scholar
[6] B. Huppert, Endliche Gruppen. I, Grundlehren Math. Wiss. 134, Springer, Berlin, 1967. 10.1007/978-3-642-64981-3Search in Google Scholar
[7] B. Huppert and N. Blackburn, Finite Groups. III, Grundlehren Math. Wiss. 243, Springer, Berlin, 1982. 10.1007/978-3-642-67997-1Search in Google Scholar
[8] P. Kleidman and M. Liebeck, The Subgroup Structure of the Finite Classical Groups, London Math. Soc. Lecture Note Ser. 129, Cambridge University, Cambridge, 1990. 10.1017/CBO9780511629235Search in Google Scholar
[9] J. Lu, L. Pang and X. Zhong, Finite groups with non-nilpotent maximal subgroups, Monatsh. Math. 171 (2013), no. 3–4, 425–431. 10.1007/s00605-012-0432-7Search in Google Scholar
[10] V. S. Monakhov and V. N. Tyutyanov, On finite groups with some subgroups of prime indices, Sib. Math. J. 48 (2007), 666–668. 10.1007/s11202-007-0068-3Search in Google Scholar
[11] G. Qian, Nonsolvable groups with few primitive character degrees, J. Group Theory 21 (2018), no. 2, 295–318. 10.1515/jgth-2017-0037Search in Google Scholar
[12] D. J. S. Robinson, A Course in the Theory of Groups, 2nd ed., Grad. Texts in Math. 80, Springer, New York, 1996. 10.1007/978-1-4419-8594-1Search in Google Scholar
[13] X. Yi, S. Jiang and S. F. Kamornikov, Finite groups with given non-nilpotent maximal subgroups of prime index, J. Algebra Appl. 18 (2019), no. 5, Article ID 1950087. 10.1142/S0219498819500877Search in Google Scholar
© 2025 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Weil representations of twisted loop groups of type A n (2)
- Local Birkhoff decompositions for loop groups and a finiteness result
- Uniform boundedness of oscillatory singular integrals with rational phases
- On the stabilizer of the graph of linear functions over finite fields
- Global solution and blow-up of critical heat equation with nonlocal interaction
- Locally compact groups with all dense subgroups separable
- On the Kawaguchi–Silverman conjecture for birational automorphisms of irregular varieties
- On isometry groups of gradient Ricci solitons
- Extensions of a theorem of P. Hall on indexes of maximal subgroups
- A trace formula for Hecke operators on Fuchsian groups
- Quasi-hereditary algebras with all standard modules being isomorphic to submodules of projective modules
- Computation of endo-fixed closures in free-abelian times free groups
- Non-vanishing of Maass form 𝐿-functions of cubic level at the central point
- Traces of partition Eisenstein series
- Gabor System based on the unitary dual of the Heisenberg group
- Zeros of linear combinations of Dirichlet 𝐿-functions on the critical line
- Paley inequality for the Weyl transform and its applications
Articles in the same Issue
- Frontmatter
- Weil representations of twisted loop groups of type A n (2)
- Local Birkhoff decompositions for loop groups and a finiteness result
- Uniform boundedness of oscillatory singular integrals with rational phases
- On the stabilizer of the graph of linear functions over finite fields
- Global solution and blow-up of critical heat equation with nonlocal interaction
- Locally compact groups with all dense subgroups separable
- On the Kawaguchi–Silverman conjecture for birational automorphisms of irregular varieties
- On isometry groups of gradient Ricci solitons
- Extensions of a theorem of P. Hall on indexes of maximal subgroups
- A trace formula for Hecke operators on Fuchsian groups
- Quasi-hereditary algebras with all standard modules being isomorphic to submodules of projective modules
- Computation of endo-fixed closures in free-abelian times free groups
- Non-vanishing of Maass form 𝐿-functions of cubic level at the central point
- Traces of partition Eisenstein series
- Gabor System based on the unitary dual of the Heisenberg group
- Zeros of linear combinations of Dirichlet 𝐿-functions on the critical line
- Paley inequality for the Weyl transform and its applications
