Normal Subgroups of Groups of Prime-Power Order

Abstract

In [5] Gaschütz noted that the dihedral and quaternion groups of order 8 cannot be Frattini subgroups of 2-groups. This, together with results in [6], raised the question of what limitations there may be on the structure of those normal subgroups which are contained in the Frattini subgroup of a p-group. Partial answers have been obtained in [l, 2, 3, 6, 7]. It has been shown that the subgroups considered cannot have cyclic centre, and that many such subgroups, if they have two generators, must be metacyclic. It is of interest to remove the restriction that the embedded normal subgroups have two generators. Under some weaker restrictions it may be shown that often such a subgroup N has the property N′ ≤ N ^P, or N′ ≤ N ⁴ if p = 2. The parallel results obtained when the embedded normal subgroup has two generators strengthen the results mentioned above.