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Article

Cao, Chenchen ; Zhang, Li ; Guo, Wenbin
On $\sigma $-permutably embedded subgroups of finite groups. (English). Czechoslovak Mathematical Journal, vol. 69 (2019), issue 1, pp. 11-24
MSC: 20D10, 20D20, 20D35 | MR 3923570 | Zbl 07088765 | DOI: 10.21136/CMJ.2018.0148-17
Keywords:
finite group; $\sigma $-subnormal subgroup; $\sigma $-permutably embedded subgroup; \hbox {$\sigma $-soluble} group; supersoluble group
Summary:
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 $G$. \endgraf 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.
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