ON FINITE METACYCLIC p-GROUPS

Abstract

If a metacyclic group G of exponent pe contains a normal cyclic subgroup A of order pe and, in the case p = 2, has no nonabelian sections of order 8, then there exists in G a cyclic subgroup B such that G = AB and A ∩ B = {1}. Some consequences of the stated result are also proved. For example, if a metacyclic group of exponent pe is noncyclic and has no nonabelian sections of order 8 and A < G is cyclic of order pe, then there is a cyclic B < G such that G = AB and A ∩ B = {1}. Next, it is proved that if G > {1} is a metacyclic p-group, then exp (Φ(G)) < exp (G).