The Effects Of Weakly NS~*-Permutable Subgroups The Structure Of Finite Groups

Let G be a finite group.A subgroup H of G is called nearly S-permutable in G if for every prime p with(p,|H|)=1 and for every subgroup K of G containing H the normalizer NK(H)contains some Sylow p-subgroup of K.A subgroup H of G is said to be NS*-permutable subgroup of G if there exists a normal subgroup T of G such that HT(?)G,?(G/T)(?)?(H)and H?T is nearly S-permutable subgroup of G.A subgroup H of G is said to be weakly NS*-permutable(weakly NS*-supplemented)subgroup of G if there exists a subnormal subgroup T(a subgroup T)of G such that G=HT,and H?T?HnG,where HnG is the subgroup of H generated by all those subgroups of H which are nearly S-permutable in G.we can check that NS*-permutable subgroup is wider than nearly S-permutable subgroup,and a special case of NS*-permutable subgroup is that exists a normal subgroup T of G such that HT=G,H?T is nearly S-permutable subgroup of G.Clearly,weakly NS*-per mutable(weakly NS*-supplemented)subgroup is an extension of this case.In this paper,we investigate the structure and the property of a finite group according to its subgroups.Some recent results of S-permutable subgroups,c-normal subgroups,weakly S-permutable(weakly S-supplemented)subgroups and nearly S-permutable subgroups are generalized.