On σ-Normal Subgroups of Finite σ-Soluble Groups

In this paper, we establish some results that confirm that σ σ -normal and σ σ -permutable subgroups play an important role in the structural analysis of finite groups associated with σ σ -partitions of the set of all primes. We show that the difference between these two subgroup embedding properties and their corresponding associated classes in the σ σ -soluble universe is just the Hall σ σ -structure. Furthermore, we prove that the projectors of every σ σ -soluble group associated with the saturated formation of all σ σ -nilpotent groups are conjugate and coincide with the covering subgroups. Some relations between these projectors and the σ σ -nilpotent and σ σ -self-normalising subgroups of a σ σ -soluble group are also exhibited.