Finite groups with sσ-quasinormal subgroups

Let G be a finite group and sigma = {sigma(i) | i is an element of I} be some partition of the set P of all primes. A set H of subgroups of G is said to be a complete Hall sigma-set of G if every member not equal 1 of H is a Hall sigma(i)-subgroup of G for some sigma(i) is an element of sigma and H contains exactly one Hall sigma(i)-subgroup of G for every sigma(i) is an element of 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 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) is an element of sigma(T), H permutes with every Hall sigma(i)-subgroup of T. In this paper, we main investigate finite groups all of whose cyclic subgroups of order 2 or 4 are s sigma-quasinormal subgroups. Some new criteria for p-nilpotency and supersolubilities are established.