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

Abstract

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.