On partially τ-quasinormal subgroups of finite groups

Abstract

Let H be a subgroup of a group G. We say that: (1) H is τ -quasinormal in G if H permutes with every Sylow subgroup Q of G such that (|H|, |Q|) = 1 and (|H|, |Q G|) 6= 1; (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.