On $H_{\sigma }$-permutably embedded subgroups of finite groups

Abstract

Let $G$ be a finite group. Let $\sigma =\{{\sigma }_i|i\in I\}$ be a partition of the set of all primes $\mathbb{P}$ and $n$ an integer. We write $\sigma (n)=\{{\sigma }_i|{\sigma }_i\cap \pi (n)\ne \varnothing \}$, $\sigma (G)=\sigma (|G|)$. A set $\mathcal{H}$ of subgroups of $G$ is said to be a complete Hall $\sigma$-set of $G$ if every member of $\mathcal{H}\setminus \{1\}$ is a Hall ${\sigma }_i$-subgroup of $G$ for some ${\sigma }_i$ and $calH$ contains exact one Hall ${\sigma }_i$-subgroup of $G$ for every ${\sigma }_i\in \sigma (G)$. A subgroup $A$ of $G$ is called (i) a $\sigma$-Hall subgroup of $G$ if $\sigma (A)\cap \sigma (|G:A|)=\varnothing$; (ii) $\sigma$-permutable in $G$ if $G$ possesses a complete Hall $\sigma$-set $\mathcal{H}$ such that $A{H}^x=H^{x}A$ for all $H\in \mathcal{H}$ and all $x\in G$. We say that a subgroup $A$ of $G$ is $H_{\sigma }$-permutably embedded in $G$ if $A$ is a $\sigma$-Hall subgroup of some $\sigma$-permutable subgroup of $G$. We study finite groups $G$ having an $H_{\sigma }$-permutably embedded subgroup of order $|A|$ for each subgroup $A$ of $G$. Some known results are generalized.