On partially T-quasinormal subgroups of finite groups

Let H be a subgroup of a group G. We say that: (1) H is T-quasinormal in G if H permutes with every Sylow subgroup Q of G such that (vertical bar H vertical bar, vertical bar Q vertical bar) = 1 and (vertical bar H vertical bar,vertical bar Q(G)vertical bar) not equal 1; (2) H is partially T-quasinormal in G if G has a normal subgroup T such that HT is S-quasinormal in G and H boolean AND T <= H-TG, where H-TG is the subgroup generated by all those subgroups of H which are T-quasinormal in G. In this paper, we find a condition under which every chief factor of G below a normal subgroup E of G is cyclic by using the partial T-quasinormality of some subgroups.