On one generalization of finite $\frak U$-critical groups

TL;DR

This paper characterizes finite groups where every non-identity subgroup is either $P$-subnormal or $P$-abnormal, extending the understanding of subgroup structure and generalizations of critical groups.

Contribution

It provides a classification of finite groups with the property that all non-identity subgroups are either $P$-subnormal or $P$-abnormal, generalizing known results on $rak U$-critical groups.

Findings

Finite groups with all non-identity subgroups $P$-subnormal or $P$-abnormal are characterized.

The work extends the theory of $rak U$-critical groups to a broader class.

Structural properties of these groups are elucidated.

Abstract

A proper subgroup $H$ of a group $G$ is said to be: $\Bbb{P}$-subnormal in $G$ if there exists a chain of subgroups $H=H_0 < H_1< ... < H_{n}=G$ such that $|H_{i}:H_{i-1}|$ is a prime for $i=1,...,n$; $\Bbb{P}$-abnormal in $G$ if for every two subgroups $K\leq L$ of $G$, where $H\leq K$, $|L:K|$ is not a prime. In this paper we describe finite groups in which every non-identity subgroup is either $\Bbb{P}$-subnormal or $\Bbb{P}$-abnormal.