Narrow Results

A subgroup $H$ of a group $G$ is called an $\mathcal{𝒮}\mathcal{𝒮}\mathcal{\mathscr{H}}$-subgroup in $G$ if there exists an $s$-permutable subgroup $K$ of $G$ such that $H^{sG}=HK$ and $H^g\cap N_K(H)\le H$ for all $g\in G$, where $H^{sG}$ is the intersection of all $s$-permutable subgroup of $G$ containing $H$. In this paper, we present two new sufficient and necessary conditions on the $p$-nilpotency of finite groups by use of a small quantity of $\mathcal{𝒮}\mathcal{𝒮}\mathcal{\mathscr{H}}$-subgroups of Sylow $p$-subgroups. A number of known results are improved and extended.

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.

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.

We consider the so-called generalized center, defined by Agrawal, in the slightly wider context of periodic groups and try to find out where additional conditions are needed for refinements. In particular we consider the final terms of the corresponding ascending sequences.