Bound on the diameter of metacyclic groups

摘要

Let G(m,n,k) = Z(m) proportional to(k) Z(n) be the split metacyclic group, where k is a unit modulo n. We derive an upper bound for the diameter of G(m,n,k) using an arithmetic parameter called the weight, which depends on n, k, and the order of k. As an application, we show how this would determine a bound on the diameter of an arbitrary metacyclic group.