Finite Groups with Formation Subnormal Normalizers of Sylow Subgroups

Abstract

Let $$\mathfrak{F}$$ be a formation. Properties of the class $$\mathrm{w}^{*}\mathfrak{F}$$ of all groups $$G$$ for which $$\pi(G)\subseteq\pi(\mathfrak{F})$$ and the normalizers of all Sylow subgroups are $$\mathfrak{F}$$ -subnormal in $$G$$ are studied. In particular, it is established that this class is a formation closed with respect to taking Hall subgroups. Hereditary saturated formations $$\mathfrak{F}$$ coinciding with $$\mathrm{w}^{*}\mathfrak{F}$$ are found.