Finite groups with permutable Hall subgroups

Abstract

Let \sigma =\{{{\displaystyle \sigma }}_i|i\in I\} be a partition of the set of all primes \mathbb{P}, and let G be a finite group. A set \mathcal{H} of subgroups of G is said to be a complete Hall\sigma-set of G if every member \ne \mathrm{1} of H is a Hall σi-subgroup of G for somei ∈ I and H contains exactly one Hall σi-subgroup of G for every i such that {{\displaystyle \sigma }}_i\cap \pi (G)\ne \phi. In this paper, we study the structure of G under the assuming that some subgroups of G permutes with all members of \mathcal{H} .

Keywords

Finite group / Hall subgroup / complete Hall σ-set / permutable subgroup / supersoluble group

References

[1] AsaadM, HelielA A. On permutable subgroups of finite groups. Arch Math, 2003, 80(2): 113–118

[2] Ballester-BolinchesA, Esteban-RomeroR, AsaadM. Products of Finite Groups. Berlin: Walter de Gruyter, 2010

[3] DoerkK, HawkesT. Finite Soluble Groups. Berlin: Walter de Gruyter, 1992

[4] GorensteinD. Finite Groups. New York: Harper & Row Publishers, 1968

[5] GriessR, SchmidP. The Frattini module. Arch Math, 1978, 30(1): 256–266

[6] GuoW. The Theory of Classes of Groups. Beijing/Dordrecht: Science Press/Kluwer Academic Publishers, 2000

[7] GuoW, SkibaA N. On Π-quasinormal subgroups of finite groups. Monatsh Math,

[8] HuppertB. Zur Sylowstruktur aufl¨osbarer gruppen. Arch Math, 1961, 12: 161–169

[9] HuppertB. Zur Sylowstruktur auflösbarer gruppen, II.Arch Math, 1964, 15: 251–257

[10] HuppertB. Endliche Gruppen I. Berlin: Springer-Verlag, 1967

[11] HuppertB, BlackburnN. Finite Groups III. Berlin: Springer-Verlag, 1982

[12] KnyaginaB N, MonakhovV S. On π'-properties of finite groups possessing a Hall π-subgroup. Sib Math J, 2011, 52(2): 297–309

[13] SkibaA N. On the F-hypercentre and the intersection of all F-maximal subgroups of a finite group. J Pure Appl Algebra, 2012, 216(4): 789–799

[14] SkibaA N. On σ-subnormal and σ-permutable subgroups of finite groups. J Algebra, 2015, 436: 1–16

[15] SkibaA N. On some results in the theory of finite partially soluble groups. Commun Math Stat,2016, 4(3): 281–309

[16] SkibaA N. A generalization of a Hall theorem. J Algebra Appl, 2016, 15(4): 1650085 (13 pp).

[17] TyutyanovV N. On the Hall conjecture. Ukra¨ın Mat Zh, 2002, 54(7): 981–990 (in Russian)

RIGHTS & PERMISSIONS

2017 Higher Education Press and Springer-Verlag GmbH Germany