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

Abstract

Let $\pi$ be a set of primes and $\mathfrak{F}$ be a formation. In this article a properties of the class ${\mathrm{w}}_{\pi }^{\ast }\mathfrak{F}$ 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 ${\mathrm{w}}_{\pi }^{\ast }\mathfrak{F}$ is a formation. Some hereditary saturated formations $\mathfrak{F}$ for which ${\mathrm{w}}_{\pi }^{\ast }\mathfrak{F}=\mathfrak{F}$ are founded.