Minimal subgroups and the structure of finite groups

A subgroup $H$ of a finite group $G$ is weakly-supplemented in $G$ if there exists a proper subgroup $K$ of $G$ such that $G=HK$ . In this paper we prove that a finite group $G$ is $p$ -nilpotent if every minimal subgroup of $P\bigcap G^{N}$ is weakly-supplemented in $G$ , and when $p=2$ either every cyclic subgroup of $P\bigcap G^{N}$ with order 4 is weakly-supplemented in $G$ or $P$ is quaternion-free, where $p$ is the smallest prime number dividing the order of $G$ , $P$ a sylow $p$ -subgroup of $G$ .