On the $ICPC$-property of finite subgroups

Abstract

Let $G$ be a finite group and $A$ be a subgroup of $G$. Then $A$ is called a $p$-$CAP$-subgroup of $G$, if $A$ covers or avoids every $pd$-chief factor of $G$. A subgroup $H$ of $G$ is said to be an $ICPC$-subgroup of $G$, if $H\cap [H,G]\le H_{pcG}$, where $H_{pcG}$ is a $p$-$CAP$-subgroup of $G$ contained in $H$. In this paper, we investigate the structure of $G$ under the assumption that certain subgroups are $ICPC$-subgroups of $G$, and characterization of $p$-nilpotency and other results are obtained.