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# Rendiconti del Seminario Matematico della Università di Padova

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Volume 139, 2018, pp. 143–158
DOI: 10.4171/RSMUP/139-4

Published online: 2018-06-06

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

Wenbin Guo, Chi Zhang, Alexander N. Skiba and D. A. Sinitsa

(1) University of Science and Technology of China, Hefei, Anhui, China
(2) University of Science and Technology of China, Hefei, Anhui, China
(3) Francisk Skorina Gomel State University, Gomel, Belarus
(4) Francisk Skorina Gomel State University, Gomel, Belarus

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 $\sigma (n) =\{\sigma_{i} |\sigma_{i}\cap \pi (n)\ne \emptyset \}$, $\sigma (G) =\sigma (|G|)$. 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 ${\mathcal cal H}$ 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 ${\mathcal H}$ such that $AH^x=H^xA$ 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.

Keywords: Finite group, $\sigma$-Hall subgroup, $\sigma$-subnormal subgroup, $\sigma$-nilpotent group, $H_{\sigma}$-permutably embedded subgroup

Guo Wenbin, Zhang Chi, Skiba Alexander, Sinitsa D.: On $H_{\sigma}$-permutably embedded subgroups of finite groups. Rend. Sem. Mat. Univ. Padova 139 (2018), 143-158. doi: 10.4171/RSMUP/139-4