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.
