Abstract
Let H be a subgroup of a finite group G. We say that H satisfies the partial \( \Pi \)-property in G if there exists a chief series \( \varGamma _{G}: 1 =G_{0}< G_{1}< \cdots < G_{n}= G \) of G such that for every G-chief factor \( G_{i}/G_{i-1} (1\le i\le n) \) of \( \varGamma _{G} ,\) \( | G / G_{i-1}: N _{G/G_{i-1}} (HG_{i-1}/G_{i-1}\cap G_{i}/G_{i-1})| \) is a \( \pi (HG_{i-1}/G_{i-1}\cap G_{i}/G_{i-1}) \)-number. In this paper, we study the influence of some subgroups of prime power order satisfying the partial \( \Pi \)-property on the structure of a finite group.
Similar content being viewed by others
Data Availability Statement
No datasets were generated or analyzed during the current study.
References
Huppert, B.: Endliche Gruppen. I. Die Grundlehren der Mathematischen Wissenschaften, Band 134. Springer, Berlin (1967)
Chen, X., Guo, W.: On the partial \(\Pi \)-property of subgroups of finite groups. J. Group Theory 16(5), 745–766 (2013)
Qiu, Z., Liu, J., Chen, G.: On the partial \(\Pi \)-property of second minimal or second maximal subgroups of Sylow subgroups of finite groups (2023). arXiv e-prints. arXiv:2304.11451
Ballester-Bolinches, A., Pedraza-Aguilera, M.C.: On minimal subgroups of finite groups. Acta Math. Hung. 73(4), 335–342 (1996)
Gorenstein, D.: Finite Groups, 2nd edn. Chelsea Publishing Co., New York (1980)
Chen, X., Guo, W.: On \(\Pi \)-supplemented subgroups of a finite group. Commun. Algebra 44(2), 731–745 (2016)
Ward, H.N.: Automorphisms of quaternion-free \(2\)-groups. Math. Z. 112, 52–58 (1969)
Ballester-Bolinches, A., Esteban-Romero, R., Asaad, M.: Products of Finite Groups. De Gruyter Expositions in Mathematics, vol. 53. Walter de Gruyter, Berlin (2010)
Doerk, K., Hawkes, T.: Finite Soluble Groups. De Gruyter Expositions in Mathematics, vol. 4. Walter de Gruyter & Co., Berlin (1992)
Asaad, M.: On \(p\)-nilpotence of finite groups. J. Algebra 277(1), 157–164 (2004)
Guo, W.: The Theory of Classes of Groups. Mathematics and Its Applications, vol. 505, 1st edn. Springer Netherlands, Dordrecht (2000)
Guo, W.: Structure Theory for Canonical Classes of Finite Groups. Springer, Heidelberg (2015)
Ballester-Bolinches, A., Ezquerro, L.M.: Classes of Finite Groups. Mathematics and Its Applications, 1st edn. Springer, Berlin (2006)
Skiba, A.N.: On weakly \(s\)-permutable subgroups of finite groups. J. Algebra 315(1), 192–209 (2007)
Acknowledgements
This work is supported by the National Natural Science Foundation of China (Grant No. 12071376, 11971391) and the Fundamental Research Funds for the Central Universities (No. XDJK2020B052).
Author information
Authors and Affiliations
Contributions
All authors reviewed the manuscript.
Corresponding author
Ethics declarations
Conflict of Interest
The authors declare no competing interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Qiu, Z., Liu, J. & Chen, G. On the Partial \( \Pi \)-Property of Some Subgroups of Prime Power Order of Finite Groups. Mediterr. J. Math. 21, 61 (2024). https://doi.org/10.1007/s00009-024-02603-6
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00009-024-02603-6