c * -Supplemented subgroups and p -nilpotency of finite groups - Wang, Youyu; Wei, H. (Ukrains’kyi Matematychnyi Zhurnal, 2007)
A subgroup H of a finite group G is said to be c^*-supplemented in G if there exists a subgroup K such that G = HK and H ⋂ K is permutable in G. It is proved that a finite group G that is S_4-free is p-nilpotent if N_G (P) is p-nilpotent and, for all x ∈ G N_G (P), every minimal subgroup of P ∩ P^x ∩ G^N_p is c^*-supplemented in P and (if p = 2) one of the following conditions is satisfied: (a) every cyclic subgroup of P ∩ P^x ∩ G^N_p of order 4 is c^*-supplemented in P, (b) [Ω2(P ∩ P^x ∩ G^N_p),P] ⩽ Z(P ∩ G^N_p), (c) P is quaternion-free, where P a Sylow p-subgroup of G and G^N_p is the p-nilpotent residual of G. This extends and improves some known results.