Narrow Results

The subgroup structure of a finite group, under the assumption that its every non-nilpotent maximal subgroup has prime index, is studied in the paper.

The norm $N(G)$ of a group $G$ is the intersection of the normalizers of all subgroups in $G$. In this paper, the norm is generalized by studying on Sylow subgroups and ${\mathscr{H}}$-subgroups in finite groups which is denoted by $C(G)$ and $A(G)$, respectively. It is proved that the generalized norms $A(G)$ and $C(G)$ are all equal to the hypercenter of $G$.