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

Document Type: Research Paper

Authors

1 Department of Mathematics, Faculty of Science, Beni-Suef university

2 Department of Mathematics, Faculty of Science, KAU, Saudi Arabia

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 $g\in‎ ‎G$. 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 $H\cap‎ ‎K$ 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.

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