# Finite groups with $\mathfrak{F}$-subnormal normalizers of Sylow subgroups

## TL;DR

This paper investigates a class of finite groups where Sylow subgroup normalizers are -subnormal, establishing formation properties and identifying specific hereditary saturated formations where this class coincides with .

## Contribution

It introduces and analyzes the class *_ of groups with -subnormal Sylow normalizers, proving it forms a formation and characterizing when it equals certain hereditary saturated formations.

## Key findings

- *_ is a formation.
- Identifies hereditary saturated formations where *_ equals .
- Provides structural properties of groups with -subnormal Sylow normalizers.

## Abstract

Let $\pi$ be a set of primes and $\mathfrak{F}$ be a formation. In this article a properties of the class ${\rm w}^{*}_{\pi}\mathfrak{F}$ of all groups $G$, such that $\pi(G)\subseteq \pi(\mathfrak{F})$ and the normalizers of all Sylow $p$-subgroups of $G$ are $\mathfrak{F}$-subnormal in $G$ for every $p\in\pi\cap\pi(G)$ are investigated. It is established that ${\rm w}^{*}_{\pi}\mathfrak{F}$ is a formation. Some hereditary saturated formations $\mathfrak{F}$ for which ${\rm w}^{*}_{\pi}\mathfrak{F}=\mathfrak{F}$ are founded.