A generalization of an Agrawal theorem on soluble PST-groups

Abstract

Let σ = { σ i ⏧ 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 P σ T -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 = σ 1 = { { 2 } , { 3 } , { 5 } , … } , a P σ T -group is called a PST-group. We study the structure of finite soluble groups G in which every subnormal subgroup is σ-permutable and G N σ ∩ H ≤ Z ∞ ( H ) for every Hall σi -subgroup H of G and all i. In particular, we obtain a new characterization of soluble P σ T -groups which is a generalization of an Agrawal theorem on soluble PST-groups.