Let Z and K be subgroups of a finite group G and Z ≤ K if Z^g ≤ K for some g ∈ G implies that Z^g = Z, then Z is called weakly closed in K (respect to G). In this paper, we investigate the supersolvabilities of a finite group G by assuming that some minimal subgroups and cyclic subgroups of order 4 satisfy the weakly closed properties. Some conclusions about group formations are also obtained.

weakly closed subgroup, minimal subgroup, supersolvable, saturated formation.

Received: February 27, 2022;  Accepted: March 22, 2022; Published: May 12, 2022

How to cite this article: Tao Zhao and Ling Xue, The influence of weakly closed subgroups on the supersolvability of a finite group, JP Journal of Algebra, Number Theory and Applications 55 (2022), 1-7. http://dx.doi.org/10.17654/0972555522015

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

References:
[1] D. Gorenstein, Finite Groups, Chelsea, New York, 1968.
[2] B. Huppert, Endliche Gruppen, Vol. I, Springer, Berlin, 1967.
[3] J. Buckley, Finite groups whose minimal subgroups are normal, Math. Z. 116 (1970), 15-17.
[4] A. Shaalan, The influence of π-quasinormality of some subgroups on the structure of a finite group, Acta Math. Hungar. 56(3-4) (1990), 287-293.
[5] H. Kurzweil and B. Stellmacher, The Theory of Finite Groups, Chelsea, New York, 1968.
[6] K. Doerk, Minimal nicht uberauflosbare, endliche Gruppen, Math. Z. 91 (1966), 198-205 (in German).
[7] A. Ballester-Bolinches, H-normalizers and local definitions of saturated formations of finite groups, Israel J. Math. 67 (1989), 312-326.