A generalization of an Agrawal theorem on soluble PST -groups

Let σ={σi⏧i∈I} be some partition of the set of all prime numbers and G a finite group. A subgroup A of G is said to be σ-permutable in G if A permutes with all Hall σi-subgroups H, that is, AH = HA for all i∈I. The group G is said to be a PσT-group if σ-permutability is a transitive relation in G, that is, if K is a σ-permutable subgroup of H and H is a σ-permutable subgroup of G, then K is a σ-permutable subgroup of G. In the case when σ=σ1={{2},{3},{5},…}, a PσT-group is called a PST-group. We study the structure of finite soluble groups G in which every subnormal subgroup is σ-permutable and GNσ∩H≤Z∞(H) for every Hall σi-subgroup H of G and all i. In particular, we obtain a new characterization of soluble PσT-groups which is a generalization of an Agrawal theorem on soluble PST-groups.