Finite groups whose minimal subgroups are weakly $mathcal{H}^{ast}$-subgroups

Abstract

Let $G$ be a finite group‎. ‎A subgroup‎ ‎$H$ of $G$ is called an $mathcal{H}$-subgroup in‎ ‎$G$ if $N_G(H)cap H^{g}leq H$ for all $gin‎ ‎G$. A subgroup $H$ of $G$ is called a weakly‎ ‎$mathcal{H}^{ast}$-subgroup in $G$ if there exists a‎ ‎subgroup $K$ of $G$ such that $G=HK$ and $Hcap‎ ‎K$ is an $mathcal{H}$-subgroup in $G$. We‎ ‎investigate the structure of the finite group $G$ under the‎ ‎assumption that every cyclic subgroup of $G$ of prime order ‎$p$ or of order $4$ (if $p=2$) is a weakly ‎$mathcal{H}^{ast}$-subgroup in $G$. Our results improve‎ ‎and extend a series of recent results in the literature‎.