The influence of $\mathscr{H}$-subgroups on $p$-nilpotency and $p$-supersolvability of finite groups

Document Type : Research Paper


1 Department of Mathematical Sciences, Kent State University, Kent, USA

2 Department of Mathematics, College of Science, China Agricultural University, Beijing, China.



Let $G$ be a finite group. A subgroup $H$ of $G$ is an $\mathscr{H}$-subgroup in $G$ if $N_G(H)\cap H^g \leq H$ for any $g \in G$. In this article, by using the concept of $\mathscr{H}$-subgroups, we study the influence of the intersection of $O^p(G_p^*)$ and the members of some fixed $\mathcal{M}_d(P)$ on the structure of the group $G$, where $P$ is a Sylow $p$-subgroup of $G.$ Some new criteria for a group to be $p$-nilpotent and $p$-supersolvable are given and some recent results are extended and generalized.


Main Subjects

[1] M. Asaad, A. A. Heliel and M. M. Al-Mosa Al-Shomrani, On weakly H -subgroups of finite groups, Comm. Algebra, 40 (2012) 3540–3550.
[2] Y. Berkovich and I. M. Isaacs, p-Supersolvability and actions on p-groups stabilizing certain subgroups, J. Algebra,
414 (2014) 82–94.
[3] M. Bianchi, A. Gillio Berta Mauri, M. Herzog and L. Verardi, On finite solvable groups in which normality is a
transitive relations, J. Group Theory, 3 (2000) 147–156.
[4] R. Chen, X. Li and X. Zhao, On weakly H -subgroups of finite groups II∗ , Comm. Algebra, 50 (2022) 4009–4015.
[5] R. J. Flores and R. M. Foote, Strongly closed subgroups of finite groups, Adv. Math, 222 (2009) 453–484.
[6] D. Goldschmidt, 2-fusion in finite groups, Ann. of Math, 99 (1974) 70–117.
[7] D. Goldschmidt, Strongly closed 2-subgroups of finite groups, Proc. Conf. on Finite Groups (Univ. Utah, Park City,
Utah), (1975) 109–110.
[8] D. Goldschmidt, Strongly closed 2-subgroups of finite groups, Ann. of Math, 103 (1975) 475–489.
[9] D. Gorenstein, Finite Groups, Harper and Row, New York-London, 1968
[10] Y. Guo and I. M. Isaacs, Conditions on p-subgroups implying p-nilpotence or p-supersolvability, Arch. Math, 105
(2015) 215–222.
[11] I. M. Isaacs, Finite Group Theory, 92, American Mathematical Society, Providence, 2008
[12] S. Li and X. He, On normally embedded subgroups of prime power order in finite groups, Comm. Algebra, 36 (2008) 2333–2340.
[13] Z. Shen and N. Du, Finite Groups with H -Subgroups, Algebra Colloq., 20 (2013) 421–426.
[14] J. G. Thompso, Normal p-complements for finite groups, J. Algebra, 1 (1964) 43–46.
[15] H. Yu, Some sufficient and necessary conditions for p-supersolvablity and p-nilpotence of a finite group, J. Algebra Appl, 16 (2017) 1750052.
[16] H. Yu, X. Xu and G. Zhang, On generalized SΦ-supplemented subgroups of finite groups, J. Algebra Appl, 18 (2019) 1950204.
[17] H. Yu, X. Xu and G. Zhang, A note on S-semipermutable and S-permutably embedded subgroups of finite groups, Ricerche mat., 105 (2022).