On $$\sigma$$-Permutably Embedded and $$s\sigma$$-Quasinormal Subgroups of Finite Groups

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.