%0 Journal Article
%T The influence of $\mathscr{H}$-subgroups on $p$-nilpotency and $p$-supersolvability of finite groups
%J International Journal of Group Theory
%I University of Isfahan
%Z 2251-7650
%A Yan, Quanfu
%A Shen, Zhencai
%D 2024
%\ 03/01/2024
%V 13
%N 1
%P 55-62
%! The influence of $\mathscr{H}$-subgroups on $p$-nilpotency and $p$-supersolvability of finite groups
%K $\mathscr{H}$-subgroup
%K weakly $\mathscr{H}$-subgroup
%K $p$-supersolvablility
%K $p$-nilpotency
%R 10.22108/ijgt.2023.135208.1806
%X 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.
%U https://ijgt.ui.ac.ir/article_27183_6122c42af4f745fb4589174b1fb7e93b.pdf