On weakly $sigma$-permutable subgroups of finite groups

Abstract

Let G be a finite group and σ = {σ i |i ∈ I} be a partition of the set of all primesH for all A ∈ H and all x ∈ G. We say that a subgroup H of G is weakly σ-permutable in G if there exists a σ-subnormal subgroup T of G such that G = HT and H ∩ T ≤ H σG , where H σG is the subgroup of H generated by all those subgroups of H which are σpermutable in G. By using this new notion, we establish some new criterias for a group G to be a σ-soluble and supersoluble, and also we give the conditions under which a normal subgroup of G is hypercyclically embedded.