Frattini subgroups ofp-closed andp-supersolvable groups

Abstract

A nonabelianp-group with cyclic center cannot occur as a normal subgroup contained in the Frattini subgroup of ap-closed group. If a nonabelian normal subgroup of orderpn and nilpotence classk is contained in the Frattini subgroup of ap-closed group, then its exponent is a divisor ofpn−k. This fact is used to derive a relation among the order, number of generators, exponent, and class of the Frattini subgroup, forp-groups. Finally, it is conjectured that a nonabelianp-group having center of orderp cannot occur as a normal subgroup contained in the Frattini subgroup of any finite group. A proof is given forp-supersolvable groups.