On the Normal Index of Maximal Subgroups in Finite Groups

The connection between the structure of finite group and character of subgroup in finite group is important in research.A series of results of solubility,super solubility,nilpotent of groups is obtained by researching the maximal subgroup,normal subgroup,subnormal subgroup,and normal index of maximal subgroup in finite group.With the method of the normal index of maximal subgroup of a finite group and minimal inverse example,one sufficient condition of finite group as π-soluble group,and two necessary and sufficient conditions of finite group as π-soluble group are obtained.