The continuation of finite $E_\mathfrak{F}$-groups theory

TL;DR

This paper investigates the structure of finite groups with specific subgroup properties within a certain class of groups, proving their solubility under these conditions.

Contribution

It characterizes finite groups with $rak{F}$-subnormal or self-normalizing primary cyclic subgroups for a broad class of formations, extending existing theory.

Findings

Groups with these properties are soluble.

Structural descriptions of such groups are provided.

The results apply to a wide class of subgroup-closed formations.

Abstract

We describe the structure of finite groups with $\mathfrak{F}$-subnormal or self-normalizing primary cyclic subgroups when $\mathfrak{F}$ is a subgroup-closed saturate superradical formation containing all nilpotent groups. We prove that groups with absolutely $\mathfrak{F}$-subnormal or self-normalizing primary cyclic subgroups are soluble when $\mathfrak{F}$ is a subgroup-closed saturate formation containing all nilpotent groups.