# On $H_{σ}$-permutably embedded subgroups of finite groups

### Wenbin Guo

University of Science and Technology of China, Hefei, Anhui, China### Chi Zhang

University of Science and Technology of China, Hefei, Anhui, China### Alexander N. Skiba

Francisk Skorina Gomel State University, Gomel, Belarus### D. A. Sinitsa

Francisk Skorina Gomel State University, Gomel, Belarus

## Abstract

Let $G$ be a finite group. Let $σ={σ_{i}∣i∈I}$ be a partition of the set of all primes $P$ and $n$ an integer. We write $σ(n)={σ_{i}∣σ_{i}∩π(n)=∅}$, $σ(G)=σ(∣G∣)$. A set $H$ of subgroups of $G$ is said to be a *complete Hall $σ$-set* of $G$ if every member of $H∖{1}$ is a Hall $σ_{i}$-subgroup of $G$ for some $σ_{i}$ and $calH$ contains exact one Hall $σ_{i}$-subgroup of $G$ for every $σ_{i}∈σ(G)$. A subgroup $A$ of $G$ is called (i) a $σ$-*Hall subgroup* of $G$ if $σ(A)∩σ(∣G:A∣)=∅$; (ii) $σ$*-permutable* in $G$ if $G$ possesses a complete Hall $σ$-set $H$ such that $AH_{x}=H_{x}A$ for all $H∈H$ and all $x∈G$. We say that a subgroup $A$ of $G$ is $H_{σ}$-*permutably embedded* in $G$ if $A$ is a $σ$-Hall subgroup of some $σ$-permutable subgroup of $G$. We study finite groups $G$ having an $H_{σ}$-permutably embedded subgroup of order $∣A∣$ for each subgroup $A$ of $G$. Some known results are generalized.

## Cite this article

Wenbin Guo, Chi Zhang, Alexander N. Skiba, D. A. Sinitsa, On $H_{σ}$-permutably embedded subgroups of finite groups. Rend. Sem. Mat. Univ. Padova 139 (2018), pp. 143–158

DOI 10.4171/RSMUP/139-4