On prime-power groups with two generators

Abstract

Let G denote a group of order a power of the prime p, and let G′ be the derived group of G. The lower central series of G will be writtenFor any subgroup H of G we denote by P(H) the subgroup of H generated by all elements xp as x runs through H, and by Φ(H) the Frattini subgroup of H. We write (H:Φ(H)) = pd(H); thus d(H) is the minimal number of generators of H.