Skip to main content
Log in

New criteria for hypercyclically embeddability of normal subgroups of finite groups

  • Published:
Ricerche di Matematica Aims and scope Submit manuscript

Abstract

According to Li and Zhao (Ukr Math J 64:102–109, 2012), a subgroup H of a finite group G is said to be \(S\Phi \)-supplemented in G if there exists a subnormal subgroup T of G such that \(G=HT\) and \(H\cap T\le \Phi (H)\), where \(\Phi (H)\) is the Frattini subgroup of H. In this paper, we extend the concept of \(S\Phi \)-supplemented subgroups, and give some criteria for (p-)hypercyclically embeddability of normal subgroups of finite groups by using fewer primary subgroups with given order. Our main results not only simplify, but also generalize some known theorems concerning \(S\Phi \)-supplemented subgroups.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Berkovich, Y., Isaacs, I.M.: \(p\)-Supersolvability and actions on \(p\)-groups stabilizing certain subgroups. J. Algebra 414, 82–94 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  2. Huppert, B.: Endliche Gruppen, vol. I. Springer-Verlag, Berlin (1967)

    Book  MATH  Google Scholar 

  3. Isaacs, I.M.: Finite Group Theory, Graduate Studies in Mathematics 92. American Mathematical Society, Providence, RI (2008)

    Google Scholar 

  4. Kong, Q., Guo, X.: On weakly \(s\)-semipermutable or \(ss\)-quasinormal subgroups of finite groups. Ric. Mat. 68, 571–579 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  5. Li, C.:The influence of \(\Phi \)-\(s\)-supplemented subgroups on the structure of finite groups, J. Algebra Appl. 11(2012), 1250064, 12 pp

  6. Li, C., Yi, X., Zhang, X.: A note on \(S\Phi \)-supplemented subgroups. Ukr. Math. J. 68, 1305–1307 (2017)

    Article  MathSciNet  Google Scholar 

  7. Li, X., Zhao, T.: \(S\Phi \)-supplemented subgroups of finite groups. Ukr. Math. J. 64, 102–109 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. Skiba, A.N.: On weakly \(s\)-permutable subgroups of finite groups. J. Algebra 315, 192–209 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Skiba, A.N.: A characterization of the hypercyclically embedded subgroups of finite groups. J. Pure Appl. Algebra 215, 257–261 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Tang, X., Guo, W.: On partial \(CAP^{\ast }\)-subgroups of finite groups, J. Algebra Appl. 16(2017), 1750009, 12 pp

  11. Weinstein, M. (ed.): Between nilpotent and solvable. Polygonal Publishing House, Passic (1982)

    MATH  Google Scholar 

  12. Yu, H.: On \(S\)-semipermutable or \(S\)-permutably embedded subgroups of finite groups. Acta Math. Hungar 151, 173–180 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  13. Yu, H.: On generalized \(\Pi \)-property of subgroups of finite groups. Rend. Sem. Mat. Univ. Padova 140, 237–256 (2018)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors are grateful to the referee who provided his/her valuable suggestions. Haoran Yu is supported by China Postdoctoral Science Foundation (Grant No. 2018M630317 and 2019T120231), National Natural Science Foundation of China (Grant No. 12001225 and 11871241), and The Education Department Project of Jilin Province (Grant No. JJKH20211033KJ). Xiaowei Xu is supported by National Natural Science Foundation of China (Grant No. 11971289).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Haoran Yu.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yu, H., Xu, X. New criteria for hypercyclically embeddability of normal subgroups of finite groups. Ricerche mat 72, 153–158 (2023). https://doi.org/10.1007/s11587-021-00616-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11587-021-00616-x

Keywords

Mathematics Subject Classification

Navigation