A study on the structure of finite groups with $c$-subnormal subgroups

Document Type : Ischia Group Theory 2022

Authors

1 Department of Mathematics, Al-Hussein Bin Talal University, P.O.Box 20, Ma’an, Jordan

2 Department of Mathematics, Al-Zaytoonah University of Jordan, P.O.Box 130, Amman, Jordan

Abstract

In this paper, we use the definition of the concept ``c-Subnormal Subgroup" to study the structure of a given finite group $G$ which contains some $c-$subnormal subgroups. We prove two main theorems, which answer the question of what conditions must hold so that $G$ is an element in a formation $ \mathfrak{U}$ of supersoluble groups. Finally, we state many previous results which can be considered as special cases of these theorems.

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References

[1] Y. Wang, C-Normality of groups and its properties, J. Algebra, 180 (1996) 954–965.
[2] J. Buckley, Finite groups whose minimal subgroups are normal, Math. Z. , 116 (1970) 15–17.
[3] S. Srinivasan, Two sufficient conditions for supersolvability of finite groups, Israel J. Math., 35 210-214,
1980.
[4] M. Ramadan, Influence of normality on maximal subgroups of Sylow subgroups of a finite group, Acta Math. Hung., 59 (1992) 107–110.
[5] L. Deyu and G. Xiuyun, The influence of c-normality of subgroups on the structure of finite groups, Journal of Pure and Applied Algebra, 150 (2000) 53–60.
[6] H. Wei, On c-normal maximal and minimal subgroups of Sylow subgroups of finite groups, Comm. Algebra, 29 (2001) 2193–2200.
[7] H. Wei, Y. Wang and Y. Li, On c-normal maximal and minimal subgroups of Sylow subgroups of finite groups. II, Comm. Algebra, 31 (2003) 4807–4816.
[8] J. Jehad, Some conditions for solubility, Math. J. Okayama Univ., 42 (2000) 1–5.
[9] A. Ballester-Bollinches and L. M. Ezquerro, Classes of finite groups, Mathematics and Its Applications, 584, Springer, Dordrecht, 2006.
[10] A. Y. Alsheik Ahmad, J. J. Jaraden and Alexander N. Skiba, On Uc -normal subgroups of finite groups, Algebra Colloq., 14 (2007) 25–36.
[11] L. A. Shemetkov, Local definitions of formations of finite groups, J. Math. Sci. (N.Y.), 185 (2012) 324–334.
[12] N. S. Alexander, On weakly s-permutable subgroups of finite groups, J. Algebra, 315 (2007) 192–209.
[13] J. J. Jaraden and A. N. Skiba, On c-normal subgroups of finite groups, Comm. Algebra, 35 (2007) 3776–3788.
[14] B. Huppert, Endliche Gruppen. I, (German) Die Grundlehren der mathematischen Wissenschaften, 134, Springer-Verlag, Berlin-New York, 1967.
[15] B. Huppert and N. Blackburn, Finite groups. III, Grundlehren der Mathematischen Wissenschaften, 243, Springer-Verlag, Berlin-New York, 1982.
[16] J. B. Derr, W. E. Deskins and N. P. Mukherjee, The influence of minimal p-subgroups on the structure of finite groups, Arch. Math. (Basel), 45 (1985) 1–4.
[17] K. Doerk and Trevor O. Hawkes, Finite soluble groups, De Gruyter Expositions in Mathematics, 4, Walter de Gruyter & Co., Berlin, 1992.
[18] W. Guo, K. P. Shum and A. Skiba,G-covering subgroup systems for the classes of supersoluble and nilpotent groups, Israel J. Math., 138 (2003) 125–138.
[19] A. Alsheik-Ahmad, Finite group with given c-permutable subgroups, Algebra Discrete Math., 2004 no. 2 9–16.
[20] A. Ballester-Bolinches and Y. Wang, Finite groups with some C-normal minimal subgroups, J. Pure Appl. Algebra, 153 (2000) 121–127.
[21] A. N. Skiba, A note on c-normal subgroups of finite groups, Algebra Discrete Math., 2005 no. 3 85–95.
[22] L. Miao and W. Guo, Finite groups with some primary subgroups F-s-supplemented, Comm. Algebra, 33 (2005) 2789–2800.
International Journal of Group Theory
Volume 13, Issue 4 - Serial Number 4
Proceedings of the Ischia Group Theory 2022-Part I
December 2024
Pages 329-344

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History

  • Receive Date: 30 May 2023
  • Revise Date: 05 August 2023
  • Accept Date: 21 August 2023
  • Published Online: 13 August 2023

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