# Article

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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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