Finite groups with some σ-primary subgroups sσ-quasinormal

LetGbe a finite group,sigma={sigma(i)vertical bar i is an element of I} a partition of the set of all primes P and sigma(G) = {sigma(i)vertical bar sigma(i) boolean AND pi(vertical bar G vertical bar) not equal null set}.A set H of subgroups of G is said to be a complete Hall sigma-set of G if every nonidentity member of H is a H all sigma(i)-subgroup of G for some i is an element of I and H contains exactly one H all sigma(i)-subgroup of G for every sigma(i) is an element of sigma(G). G is said to be sigma-full if G possesses a complete H all sigma-set. We say a subgroup H of G is s sigma-quasinormal (supplement-sigma-quasinormal) in G if there exists a sigma-full subgroup T of G such that G = HT and H permutes with every H all sigma(i)-subgroup of T for all sigma(i) is an element of sigma(T).In this article, we obtain some results about the s sigma-quasinormal subgroups and use them to determine the structure of finite groups. In particular, some new criteria of p-nilpotency, solubility, supersolubility of a group are obtained.