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

## TL;DR

This paper investigates the structure of finite groups with specific embedded subgroups called $H_{\sigma}$-permutably embedded subgroups, generalizing known results in subgroup embedding theory.

## Contribution

It introduces the concept of $H_{\sigma}$-permutably embedded subgroups and extends existing theorems to this broader context.

## Key findings

- Characterization of finite groups with $H_{\sigma}$-permutably embedded subgroups
- Generalization of known subgroup embedding results
- Conditions for the existence of such subgroups in 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 $\Bbb{P}$ and $n$ an integer. We write $\sigma (n) =\{\sigma_{i} |\sigma_{i}\cap \pi (n)\ne \emptyset \}$, $\sigma (G) =\sigma (|G|)$. A set $ {\cal H}$ of subgroups of $G$ is said to be a complete Hall $\sigma $-set of $G$ if every member of ${\cal H}\setminus \{1\}$ is a Hall $\sigma_{i}$-subgroup of $G$ for some $\sigma_{i}$ and ${\cal H}$ 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 $\sigma (|A|) \cap \sigma (|G:A|)=\emptyset$; (ii) ${\sigma}$-permutable in $G$ if $G$ possesses a complete Hall $\sigma$-set ${\cal H}$ such that $AH^{x}=H^{x}A$ for all $H\in {\cal H}$ 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.