On finite groups with given $IC\Phi$-subgroups

TL;DR

This paper investigates the structure of finite groups with specific $IC ext{-} ext{Phi}$-subgroups, providing new characterizations and criteria for properties like abelianness, nilpotence, and solvable formations.

Contribution

It introduces new structural insights and criteria for finite groups based on $IC ext{-} ext{Phi}$-subgroups, including characterizations of abelian groups and conditions for nilpotence.

Findings

New characterization of finite abelian groups

Criteria for 2-nilpotence and nilpotence of finite groups

Criteria for groups to belong to certain solvably saturated formations

Abstract

A subgroup $H$ of a group $G$ is said to be an $IC\Phi$-subgroup of $G$ if $H \cap [H,G] \le \Phi(H)$. We analyze the structure of a finite group $G$ under the assumption that some given subgroups of $G$ are $IC\Phi$-subgroups of $G$. A new characterization of finite abelian groups and some new criteria for $2$-nilpotence and nilpotence of finite groups will be obtained. Moreover, we will obtain two criteria for a finite group to lie in a given solvably saturated formation containing the class of finite supersolvable groups.