Czechoslovak Mathematical Journal, Vol. 69, No. 1, pp. 11-24, 2019


On $\sigma$-permutably embedded subgroups of finite groups

Chenchen Cao, Li Zhang, Wenbin Guo

Received March 23, 2017.   Published online May 10, 2018.

Abstract:  Let $\sigma=\{\sigma_i\colon i\in I\}$ be some partition of the set of all primes $\mathbb{P}$, $G$ be a finite group and $\sigma(G)=\{\sigma_i\colon\sigma_i\cap\pi(G)\neq\emptyset\}$. A set $\mathcal{H}$ of subgroups of $G$ is said to be a complete Hall $\sigma$-set of $G$ if every non-identity member of $\mathcal{H}$ is a Hall $\sigma_i$-subgroup of $G$ and $\mathcal{H}$ contains exactly one Hall $\sigma_i$-subgroup of $G$ for every $\sigma_i\in\sigma(G)$. $G$ is said to be $\sigma$-full if $G$ possesses a complete Hall $\sigma$-set. A subgroup $H$ of $G$ is $\sigma$-permutable in $G$ if $G$ possesses a complete Hall $\sigma$-set $\mathcal{H}$ such that $HA^x$= $A^xH$ for all $A\in\mathcal{H}$ and all $x\in G$. A subgroup $H$ of $G$ is $\sigma$-permutably embedded in $G$ if $H$ is $\sigma$-full and for every $\sigma_i\in\sigma(H)$, every Hall $\sigma_i$-subgroup of $H$ is also a Hall $\sigma_i$-subgroup of some $\sigma$-permutable subgroup of $G$. By using the $\sigma$-permutably embedded subgroups, we establish some new criteria for a group $G$ to be soluble and supersoluble, and also give the conditions under which a normal subgroup of $G$ is hypercyclically embedded. Some known results are generalized.
Keywords:  finite group; $\sigma$-subnormal subgroup; $\sigma$-permutably embedded subgroup; $\sigma$-soluble group; supersoluble group
Classification MSC:  20D10, 20D20, 20D35
DOI:  10.21136/CMJ.2018.0148-17

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Affiliations:   Chenchen Cao (corresponding author), Wenbin Guo, Department of Mathematics, University of Science and Technology of China, No. 96 JinZhai Road, Baohe, Hefei, Anhui, 230026, P. R. China, e-mail: cccao@mail.ustc.edu.cn, wbguo@ustc.edu.cn; Li Zhang, School of Mathmatics and Physics, Anhui Jianzhu University, No. 856 Jin Tahi Road, Baohe, Hefei, Auhui, 230022, P. R. China, e-mail: zhang12@mail.ustc.edu.cn


 
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