Narrow Results

We say that a subgroup H of a group G is nearly s-semipermutable in G provided G has an s-permutable subgroup T such that HT is s-permutable in G and H ∩ T ≤ H_ssG, where H_ssG is the subgroup of H generated by all those subgroups of H which are s-semipermutable in G. In this paper, we investigate the influence of near s-semipermutability of some subgroups on the p-nilpotency of finite groups.

Let $G$ be a finite group. How minimal subgroups can be embedded in $G$ is a question of particular interest in studying the structure of $G$. A subgroup $H$ of $G$ is called $s$-permutable in $G$ if $HP=PH$ for all Sylow subgroups $P$ of $G$. A subgroup $H$ of $G$ is called $n$-embedded in $G$ if there exists a normal subgroup $T$ of $G$ such that $H^G=HT$ and $H\cap T\le H_{sG}$, where $H_{sG}$ is the subgroup of $H$ generated by all those subgroups of $H$ which are $s$-permutable in $G$. In this paper, we investigate the structure of the finite group $G$ with $n$-embedded subgroups.