On partially τ-quasinormal subgroups of finite groups

Let H be a subgroup of a group G. We say that: insert ignore into journalissuearticles values(1); H is τ -quasinormal in G if H permutes with every Sylow subgroup Q of G such that insert ignore into journalissuearticles values(|H|, |Q|); = 1 and insert ignore into journalissuearticles values(|H|, |Q G|); 6= 1; insert ignore into journalissuearticles values(2); H is partially τ -quasinormal in G if G has a normal subgroup T such that HT is S-quasinormal in G and H ∩ T ≤ HτG, where HτG is the subgroup generated by all those subgroups of H which are τ -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 τ -quasinormality of some subgroups.