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

Guo, Wenbin; Zhang, Chi; Skiba, Alexander; Sinitsa, D.

Rendiconti del Seminario Matematico della Università di Padova, Tome 139 (2018), pp. 143-158.

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

Classification : 20, 00
Mots-clés : Finite group, $\sigma$-Hall subgroup, $\sigma$-subnormal subgroup, $\sigma$-nilpotent group, $H_{\sigma}$-permutably embedded subgroup