On the Intersection of a Class of Maximal Subgroups of a Finite Group

Of late there has been considerable interest in the study of analogs of the Frattini subgroup of a finite group and the investigation of their properties, particularly their influence on the structure of the group, see [2-11], [14-16] and [18]. Gaschütz [11] and more recently Bechtell [2] and Rose [18] have considered extensively the intersection of the family of all non-normal, maximal subgroups of a finite group. Deskins [8] has discussed the intersection of the family of all maximal subgroups of a finite group whose indices are not divisible by a given prime. Bhatia [7] considered the intersection of the class of all maximal subgroups of a given group whose indices are composites. In this paper we investigate the intersection of another class of maximal subgroups and its relationship with the structure of the group. The subgroup we consider here contains the Frattini subgroup and also the two subgroups introduced in [8] and [7].