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

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  • Receive Date: 21 September 2022
  • Revise Date: 26 December 2022
  • Accept Date: 01 January 2023
  • Published Online: 01 March 2024