On the Intersection of Families of Maximal Subgroups of a Finite Group*

All groups considered are finite. Given a group G, the Frattini subgroup of G, Q(G) is defined to be the intersection of all maximal subgroups of G. There has been much interest in generalizing the Frattini subgroup in various ways, and in investigating their influence on the structure of the group (see [3,.5,8,12,13]). These generalizations were done taking into account the following question: if 2(G) is a family of maximal subgroups A4 of a group G determined by some ‘external’ relationship between A4 and G, what is the nature of @(2(G)), the intersection of all the A4 of 2(G)? Now, suppose that P is a group theoretic property which can be characterized in the following way: A group G verifies P if and only if the family 2(G) is empty. Can we assure that the subgroup @(2(G)) verifies the property P? This last question has been answered satisfactorily in many cases: Let 2(G) be the family of all non-normal maximal subgroups of a group G and let P be the property of nilpotence. It is well known that a group G verifies P if and only if every maximal subgroup of G is normal in G, i.e., the family 2(G) is empty. Gaschutz [8], proved that for every group G, the subgroup @(S(G)) is a nilpotent group. Let U(G) be the family of all maximal subgroups of a group whose indices are