On $p$-nilpotency of one class of finite groups

A subgroup $H$ of a group $G$ is called modular in $G$ if $H$ is a modular element (in sense of Kurosh) of the lattice $L(G)$ of all subgroups of $G$. The subgroup of $H$ generated by all modular subgroups of $G$ contained in $H$ is called the modular core of $H$ and denoted by $H_{mG}$. In the paper a new criterion of the $p$-nilpotency of a group was obtained on the basis of the concept of the $m$-supplemented subgroup which is the extension of concepts of modular and supplemented subgroups respectively.