ON NEARLY S-PERMUTABLE AND NEARLY HALL S-EMBEDDED SUBGROUPS: IMPLICATIONS FOR SUPERSOLVABILITY AND -NILPOTENCY IN GROUPS

A subgroup T of G is said to be nearly S-permutable if , for some prime . Also, for every containing T the normalizer contain some Sylow p-subgroups of H. Furthermore, T is called nearly Hall S-embedded if for some normal subgroup s.t TN ( ) is a Hall and , where is the largest nearly S-embedded of G contained in T and T is nearly S-permutable provided supplement such that T permutes with every Sylow of B. Now we will show supersolvability and p-nil potency of some groups.