Abstract:
Let $G$ be a finite group and $\pi(G)$ denote the set of all primes dividing the order of $G$. Let $\sigma=\{\sigma_i\mid i\in I\}$ be some partition of the set $\mathbb{P}$ of all primes and $$ \sigma(G) =\{\sigma_i\mid \sigma_i\cap\pi(G)\neq\emptyset, i\in I\}. $$ A set $\mathcal{H}$ of subgroups of $G$ is said to be a complete Hall $\sigma$-set of $G$ if every member $\neq 1$ of $\mathcal{H}$ is a Hall $\sigma_i$-subgroup of $G$ for some $\sigma_i\in\sigma$ and $\mathcal{H}$ contains exactly one Hall $\sigma_i$-subgroup of $G$ for every $\sigma_i\in \sigma(G)$. A group $G$ is said to be a $\sigma$-full group if $G$ possesses a complete Hall $\sigma$-set. A subgroup $H$ of $G$ is called $\sigma$-permutable in $G$ if $G$ possesses a complete Hall $\sigma$-set $\mathcal{H}$ such that $$ HA^x=A^xH \quad \text{for all} \ \ A\in\mathcal{H}\quad \text{and all}\ \ x\in G. $$ 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 $G$. A subgroup $H$ of $G$ is said to be $s\sigma$-quasinormal in $G$ if there exists a $\sigma$-full subgroup $T$ of $G$ such that $G=HT$ and for all $\sigma_i\in\sigma(T)$, $H$ permutes with every Hall $\sigma_i$-subgroup of $T$. In this paper, we investigate the structure of finite groups by $\sigma$-permutably embedded and $s\sigma$-quasinormal subgroups. In particular, some new criterias of $\sigma$-solvability, $p$-nilpotency, supersolubility of a group are obtained.
