m-embedded Subgroups and p-nilpotency of Finite Groups

Abstract

Let $A$ be a subgroup of a finite group $G$ and $\sum \,=\,\{{{G}_{0}}\,\le \,{{G}_{1}}\,\le \,.\,.\,.\,\le \,{{G}_{n}}\}$ some subgroup series of $G$. Suppose that for each pair $\left( K,\,H \right)$ such that $K$ is a maximal subgroup of $H$ and ${{G}_{i-1}}\,\le \,K\,<\,H\,\le \,{{G}_{i}}$, for some i, either $A\,\cap \,H\,=\,A\,\cap \,K\,\text{or}\,\text{AH}\,\text{=}\,\text{AK}$. Then $A$ is said to be $\sum$-embedded in $G$. And $A$ is said to be $m$-embedded in $G$ if $G$ has a subnormal subgroup $T$ and $a\,\{1\,\le \,G\}$-embedded subgroup $C$ in $G$ such that $G\,=\,AT$ and $T\cap A\,\le \,C\,\le \,A$. In this article, some sufficient conditions for a finite group $G$ to be $p$-nilpotent are given whenever all subgroups with order ${{p}^{k}}$ of a Sylow $p$-subgroup of $G$ are $m$-embedded for a given positive integer $k$.