On some second maximal subgroups of non-solvable groups

Year 2023, Volume: 52 Issue: 2, 367 - 373, 31.03.2023

Yuyun Wang Long Miao Wei Liu

https://doi.org/10.15672/hujms.1106486

Abstract

We call a group $G$ belongs to the class of groups $S_{p}'$, if for every $pd$-chief factor $A/B$ of $G$, $((A/B)_{p})'=1$. In this paper, we investigate the influence of some second maximal subgroups which are related to non-$c_{p}$-normal maximal subgroups on the structure of $S_{p}'$.

Keywords

second maximal subgroups, maximal subgroups, simple groups, non-solvable groups, $c_{p}$-normal.

Supporting Institution

NSFC， NSFC-RFBR

Project Number

Grant \# 11871062，Grant \# 12011530061

References

• [1] A. Ballester-Bolinches and L. M. Ezquerro, Classes of Finite Groups, Springer, Netherlands, 2006.
• [2] M. Bianchi, A. G. B. Mauri and P. Hauck, On finite groups with nilpotent Sylownormalizers, Arch. Math. 47 (3), 193-197, 1986.
• [3] J. Cao, X. Guo and K. P. Shum, Finite non-solvable groups in which the normalizer of every nonnormal cyclic subgroup is maximal, Comm. Algebra 46 (1), 325-334, 2018.
• [4] W. Guo, The Theory of Classes of Groups, Science Press-Kluwer AcademicPublishers, Beijing-New York-Dordrecht-Boston-London, 2000.
• [5] X. Guo and K. P. Shum, Cover-avoidance properties and the structure of finite groups, J. Pure Appl. Algebra 181 (2-3), 297-308, 2003.
• [6] Z. Gao, J. Li and L. Miao, On $CAP_{S_p^*}$-subgroups of finite groups, Comm. Algebra 49 (3), 1120-1127, 2021.
• [7] B. Huppert, Endliche Gruppen I, Springer-Verlag, Berlin, New York, 1967.
• [8] B. Huppert, Normalteiler und maximale untergruppen endlichen gruppen, Math. Z. 60, 409-434, 1954.
• [9] L. Jafari, S. Kohl and M. Zarrin, A classification of the finite non-solvable minimal non-CA-groups, J. Algebra Appl. https://doi.org/10.1142/S0219498821502030, 2021.
• [10] Y. Lv and Y. Li, On $c_{p}$-normal subgroups of finite groups, Comm. Algebra 49 (4), 1405-1414, 2021.
• [11] S. Li, H. Liu and D. Liu, The solvability between finite groups and semi-subnormalcover- avoidance subgroups, J. Math. 37 (6), 1303-1308, 2017.
• [12] M. N. Konovalova, V. S. Monakhov and I. L. Sokhor, On 2-maximal subgroups of finite groups, Comm. Algebra 50 (1), 96-103, 2022.
• [13] N. Su, C. Cao and S. Qiao, A note on maximal subgroups of $\sigma$-soluble groups, Comm. Algebra 50 (4), 1580-1584, 2022.
• [14] J. Tate, Nilpotent quotient groups, Topology 3 (suppl. 1), 109-111, 1964.
• [15] Y. Wang, C-normality of groups and its properties, J. Algebra 180 (3), 954-965, 1996.
• [16] Y. Wang, L. Miao, Z. Gao and W. Liu, The influence of second maximal subgroups on the generalized p-solvability of finite groups, Comm. Algebra 50 (6), 2584-2591, 2021.