Narrow Results

Let $G$ be a finite group. A subgroup $H$ of $G$ is called Hall normally embedded in $G$ if $H$ is a Hall subgroup of $H^G$, where $H^G$ is the normal closure of $H$ in $G$, that is, the smallest normal subgroup of $G$ containing $H$. A group $G$ is called a $PHN$-group if its all minimal subgroups and cyclic subgroup of order $4$ are Hall normally embedded in $G$. In this paper, we give the classification of minimal non-$PHN$-groups. Furthermore, we investigate the structure of finite group all of whose maximal subgroups of even order are $PHN$-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.

A subgroup H of G is said to be self-conjugate-permutable if HH^x = H^xH implies H^x = H. A finite group G is called PSC-group if every cyclic subgroup of group G of prime order or order 4 is self-conjugate-permutable. In the paper, first we give the structure of finite group G, all of whose maximal subgroups are PSC-groups. Then we also classified that finite group G all of whose second maximal subgroups are PSC-groups.

A subgroup H of a finite group G is said to be SS-quasinormal in G if there exists a supplement B of H in G such that H is permutable with every Sylow subgroup of B. In this paper, we get some new characterizations of supersolvability and p-nilpotency of G by assuming some subgroups of prime power order of G are SS-quasinormal.

We give a complete classification of the finite groups with a unique subgroup of order $p$ for each prime $p$ dividing its order.