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On β„³-supplemented subgroups

  • Yu Zeng EMAIL logo
Published/Copyright: March 26, 2022
Journal of Group Theory
From the journal Journal of Group Theory Volume 25 Issue 6

Abstract

Let 𝐺 be a finite group and p k a prime power dividing | G | . A subgroup 𝐻 of 𝐺 is said to be β„³-supplemented in 𝐺 if there exists a subgroup 𝐾 of 𝐺 such that G = H ⁒ K and H i ⁒ K < G for every maximal subgroup H i of 𝐻. In this paper, we complete the classification of the finite groups 𝐺 in which all subgroups of order p k are β„³-supplemented. In particular, we show that if k β‰₯ 2 , then G / O p β€² ⁒ ( G ) is supersolvable with a normal Sylow 𝑝-subgroup and a cyclic 𝑝-complement.

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 12171058

Funding statement: The project is supported by the NSF of China (No. 12171058).

Acknowledgements

The author gratefully acknowledges the support of the China Scholarship Council (CSC), and wishes to thank Professor Dolfi and Professor Qian for their valuable comments.

  1. Communicated by: Evgenii I. Khukhro

References

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Received: 2021-11-23
Revised: 2022-02-22
Published Online: 2022-03-26
Published in Print: 2022-11-01

Β© 2022 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. On a family of representations of the Higman–Thompson groups
  3. Automatic continuity for groups whose torsion subgroups are small
  4. Property π‘…βˆž for some spherical and affine Artin–Tits groups
  5. The groups 𝐺 satisfying a functional equation 𝑓(π‘₯π‘˜) = π‘₯𝑓(π‘₯) for some π‘˜ ∈ 𝐺
  6. On the 𝜎-nilpotent hypercenter of finite groups
  7. On β„³-supplemented subgroups
  8. On the proper enhanced power graphs of finite nilpotent groups
  9. Two families of unravelled abstract regular polytopes in B𝑛
  10. Asymptotics of the powers in finite reductive groups
https://doi.org/10.1515/jgth-2021-0195

Articles in the same Issue

  1. Frontmatter
  2. On a family of representations of the Higman–Thompson groups
  3. Automatic continuity for groups whose torsion subgroups are small
  4. Property π‘…βˆž for some spherical and affine Artin–Tits groups
  5. The groups 𝐺 satisfying a functional equation 𝑓(π‘₯π‘˜) = π‘₯𝑓(π‘₯) for some π‘˜ ∈ 𝐺
  6. On the 𝜎-nilpotent hypercenter of finite groups
  7. On β„³-supplemented subgroups
  8. On the proper enhanced power graphs of finite nilpotent groups
  9. Two families of unravelled abstract regular polytopes in B𝑛
  10. Asymptotics of the powers in finite reductive groups
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