Finite groups with $H_{\mathcal{L}}$-embedded subgroups

Abstract

Let $G$ be a finite soluble group and let $\mathfrak{F}$ be a class of groups. A chief factor $H/K$ of $G$ is said to be $\mathfrak{F}$-central (in $G$) if $(H/K)⋊(G/C_G(H/K))\in \mathfrak{F}$; we write ${\mathcal{L}}_{c\mathfrak{F}}(G)$ to denote the set of all subgroups $A$ of $G$ such that every chief factor $H/K$ of $G$ between $A_G$ and $A^G$ is $\mathfrak{F}$-central in $G$. Let $\mathcal{L}$ be a set of subgroups of $G$. We say that a subgroup $A$ of $G$ is $H_{\mathcal{L}}$-embedded in $G$ provided $A$ is a Hall subgroup of some subgroup $E\in \mathcal{L}$. In this paper, we study the structure of $G$ under the condition that every subgroup of $G$ is $H_{\mathcal{L}}$-embedded in $G$, where $\mathcal{L}={\mathcal{L}}_{c\mathfrak{F}}(G)$ for some hereditary saturated formation $\mathfrak{F}$. Some known results are generalized.