University of IsfahanInternational Journal of Group Theory2251-76503320140901Finite groups whose minimal subgroups are weakly $mathcal{H}^{ast}$-subgroups111383710.22108/ijgt.2014.3837ENAbdelrahman AbdelhamidHelielDepartment of Mathematics, Faculty of Science, Beni-Suef universityRola AsaadHijaziDepartment of Mathematics, Faculty of Science, KAU, Saudi ArabiaReem AbdulazizAl-ObidyDepartment of Mathematics, Faculty of Science, KAU, Saudi ArabiaJournal Article20130716Let $G$ 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.http://ijgt.ui.ac.ir/article_3837_4ba7139afccee4a6543ffa5a60f76f6d.pdf