On the $ICPC$-property of finite subgroups

TL;DR

This paper investigates the structure of finite groups by analyzing the properties of $ICPC$-subgroups, leading to new characterizations of $p$-nilpotency and related structural results.

Contribution

It introduces the concept of $ICPC$-subgroups and provides new structural characterizations of finite groups based on these subgroups.

Findings

Characterization of $p$-nilpotency using $ICPC$-subgroups

Structural results under $ICPC$-subgroup assumptions

Conditions for $G$ to be $p$-nilpotent

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 $H \cap [H,G] \leq H_{pcG}$, 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.